Introduction
Quantum error correction (QEC) is a concept of paramount importance in modern quantum physics. Besides its manifest practical importance for realizing large-scale fault-tolerant quantum computation [1 , 2 , 3 ] , QEC has also emerged as an invaluable conceptual framework for understanding a diverse array of problems across many-body physics. Indeed, the robust protection of quantum information in a degenerate ground-state manifold has long been considered a fundamental definition of topological quantum order [4 , 5 , 6 ] . More recently, classic works on QEC in topological codes [7 , 8 ] have been reinterpreted through the lens of mixed-state topological order [9 , 10 ] , prompting vigorous investigation into new mixed-state phases of matter [11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 ] . In addition to its central role in topologically ordered phases, QEC has provided new perspectives on diverse problems such as thermalization in unitary and monitored quantum dynamics [22 , 23 , 24 , 25 , 26 , 27 ] , classical and quantum spin glasses [28 , 29 , 30 ] , and the AdS-CFT correspondence [31 , 32 , 33 ] .
Although the protection of quantum information is widely believed to be an intrinsic property of topological phases [34 ] , QEC is typically understudied in detail only for especially simple toy models [7 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 ] . The most well-studied example is the toric code [4 ] , where a theory of optimal decoding has been thoroughly developed in terms of the statistical physics of the Nishimori
Figure 1. Schematic cartoon of bit-flip decoherence on the transverse-field toric code (TFTC). Each ground state ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ m m m τ < 0 \tau< 0 τ < 0 X e X_e X e τ = 0 \tau= 0 τ = 0 m m m
random-bond Ising model (RBIM) [7 , 8 , 9 , 35 , 43 ] . In recent works [44 , 45 , 46 , 47 , 48 , 49 ] , the intrinsic error protection properties of certain analytically tractable nonstabilizer codes have been similarly investigated using statistical physics techniques. Unfortunately, although the existence of a decoder which recovers the protected information is guaranteed in these codes when the decoherence strength is below a critical threshold, it is unclear how to devise a conceptually simple or algorithmically efficient decoding algorithm which recovers the information. We are thus led to ask if it is ever possible to efficiently decode, i.e., carry out QEC, in these physically motivated nonstabilizer codes.
In this work, we develop decoding protocols for a class of topological quantum memories deformed away from their stabilizer limit. Our central focus is on the toric code perturbed by a transverse field [50 , 51 , 52 , 53 , 54 , 55 , 56 ] , which provides quantum fluctuations to the m m m h c h_{\mathrm{c}} h c 1 h c h_{\mathrm{c}} h c m m m
Since the TFTC ground states themselves exhibit coherent quantum fluctuations of m m m [61 , 62 ] ; in particular, its implementation requires the precise ground-state wavefunctions of the non-solvable TFTC Hamiltonian. Remarkably, we will nevertheless demonstrate that ground states of the TFTC under sufficiently weak bit-flip and phase-flip decoherence can be reliably decoded using projective measurements of the “bare” stabilizers of the unperturbed toric code, even when the transverse field is tuned close to criticality. As we will elaborate, the success of this simple-minded decoding strategy relies crucially on a special property of the TFTC Hamiltonian: namely, the Hamiltonian is stoquastic [63 ] . Conceptually, stoquasticity allows for a physical representation of the TFTC ground states using the classical statistical physics of lattice gauge theories [50 , 51 , 64 ] . More concretely, stoquasticity leads to a basis of approximate ground states which are strictly “sign-free”, leading to several technical simplifications in the construction of optimal decoders for the postmeasurement states.
As a first step, we investigate the preservation of quantum information in pure TFTC ground states under projective stabilizer measurements. One motivation for this problem arises from previous investigations of the stabilizer toric code subjected to “coherent errors” [65 , 66 , 67 , 68 , 69 , 70 , 71 ] and in particular the work of Refs. [72 , 73 ] . Specifically, Refs. [65 , 66 , 67 , 68 , 69 , 70 , 71 ] considered projective stabilizer measurements of toric code states deformed by a low-depth unitary circuit, and found that certain low-depth circuits (for example, uniform single-qubit rotations) followed by stabilizer measurements were sufficient to irreversibly destroy the encoded information. In contrast, Ref. [72 ] studied quantum information encoded in the ground-state manifold of the Castelnovo-Chamon model [57 , 59 ] , an analytically tractable (albeit non-generic) deformation of the toric code Hamiltonian. There it was shown that the encoded information was protected from stabilizer measurements throughout the entire topological phase of the model. Ref. [73 ] extended these results in several directions, and drew connections between QEC and emergent higher-form symmetries. Since these Hamiltonian deformations correspond to unitary circuits whose depth diverges as the phase boundary is approached [74 ] , it is natural to ask why this class of coherent errors can be effectively corrected via stabilizer measurements, while generic shallow-depth coherent errors cannot.
Here we extend the results of Refs. [72 , 73 ] to the TFTC, a more generic Hamiltonian deformation of the toric code, showing once more that the encoded information is protected from stabilizer measurements throughout its entire topological phase. Furthermore, we explain the preservation of information following syndrome measurements in all of these models as arising from a common mechanism, namely, the stoquasticity of the Hamiltonian deformations in question. We first prove (under very mild physical assumptions) that quantum information is preserved under syndrome measurements of the TFTC throughout its topological phase. Then, we explicitly construct and numerically investigate the optimal decoder for the postmeasurement states. This decoder is described by a three-dimensional lattice gauge theory with a constraint along a single imaginary time slice arising from syndrome measurements. We can efficiently simulate this model, and thereby sample syndromes and implement the optimal postmeasurement decoder, using the worm algorithm (WA) [75 , 76 , 77 , 78 , 79 ] . From our simulations, we explicitly verify our theoretical prediction that the fidelity of recovered information is exponentially close to unity throughout the topological phase.
Our second motivation for studying stabilizer measurements of TFTC ground states is the construction of an efficient decoder for the TFTC under Pauli noise. In a companion work [49 ] , we argued that a broad class of Hamiltonian-deformed toric codes exhibit a finite intrinsic error threshold to Pauli noise as a critical point is approached. This intrinsic threshold is defined by the behavior of the coherent information [61 , 62 , 80 ] , which guarantees the existence of a decoder below threshold. Inspired by the preceding result in the undecohere[d] code states, we investigate the efficacy of stabilizer measurements for performing QEC on the decohered code states. Specifically, we construct and numerically implement the optimal postmeasurement decoder for the TFTC under independent bit-flip and phase-flip decoherence and subsequent stabilizer measurements. The optimal postmeasurement decoder in the presence of decoherence is described by a somewhat more complex statistical physics model, involving a three-dimensional gauge theory coupled to a two-dimensional independent-spin model by a constraint, which can again be simulated efficiently via the WA. We demonstrate2 Figure 2 (a)], thereby providing direct numerical support for the results of Ref. [49 ] .
Figure 2. Phase diagrams of optimal postmeasurement decoding in the TFTC under independent bit-flip and phase-flip decoherence, with strengths p X p_X p X p Z p_Z p Z h h h h c h_{\mathrm{c}} h c p X p_X p X h h h h c h_{\mathrm{c}} h c Figure 10 (c) for quantitative error thresholds in a closely related spacetime-isotropic model. (b) In contrast, the error threshold for phase-flip decoherence remains exactly constant and equal to its stabilizer value throughout the topological phase.
While the decoders we construct are optimal given that stabilizer measurements have been performed, a natural question is whether stabilizer measurements themselves are optimal—that is, whether the decoding threshold using stabilizer measurements agrees with the ‘intrinsic’ threshold set by the ‘true’ optimal recovery channel. Although we are unable to construct a fully rigorous proof for the case of bit-flip decoherence (where the transverse field and decoherence “work together” to create m m m p c N i s h i m o r i ≈ 0.109 p_{\mathrm{c}}^{\mathrm{Nishimori}}\approx0.109 p c Nishimori ≈ 0.109 Figure 2 (b)]. Roughly speaking, this behavior arises because the TFTC ground states contain no e e e e e e
To further illustrate the power of our approach, we also investigate the d d d [55 , 56 ] in its ferromagnetic phase as a Hamiltonian deformation of the repetition code [72 ] . Such a code can be viewed as a “symmetry-protected” quantum code, in the sense that it exhibits excellent protection against decoherence which respects the Ising symmetry, but no protection whatsoever against decoherence which breaks the symmetry. As we will show explicitly, the TFIM code exhibits a maximal threshold to bit-flip decoherence throughout the ferromagnetic phase, and QEC can be performed efficiently by measuring the repetition code stabilizers and utilizing a majority-vote decoder. In short, this decoder succeeds with high fidelity because each symmetry-broken code state exhibits a nonzero magnetization density with vanishingly small fluctuations, for all non-maximal decoherence strengths; consequently, the domain walls observed via stabilizer measurement can be unambiguously annihilated to recover a repetition code state.
This work is organized as follows. In Sec. II, we review the necessary background on the TFTC, and particularly its connection to Z 2 \mathbb{Z}_2 Z 2
Background
We begin by fixing our notation and conventions on the stabilizer toric code, and by reviewing the transverse-field toric code (TFTC) and its relation to Z 2 \mathbb{Z}_2 Z 2 [50 , 51 , 64 ] . We additionally review the quantum-classical mapping for this model [51 , 55 , 56 , 81 ] , which will provide significant physical intuition for many results throughout this work.
Stabilizer Toric Code
We consider an L × L L \times L L × L ℓ \ell ℓ X ℓ X_\ell X ℓ Z ℓ Z_\ell Z ℓ [4 ]
H 0 = − ∑ v A v − ∑ p B p , (1) H_0 = -\sum_v A_v - \sum_p B_p, \tag*{(1)} H 0 = − v ∑ A v − p ∑ B p , (1) where the vertex stabilizers A v A_v A v B p B_p B p
A v ≡ ∏ ℓ ∋ v X ℓ , B p ≡ ∏ ℓ ∈ p Z ℓ . (2) A_v \equiv\prod_{\ell\ni v} X_\ell,\qquad B_p \equiv\prod_{\ell\in p} Z_\ell. \tag*{(2)} A v ≡ ℓ ∋ v ∏ X ℓ , B p ≡ ℓ ∈ p ∏ Z ℓ . (2) The former product is taken over the four links ℓ \ell ℓ v v v ℓ \ell ℓ p p p C 0 \mathcal{C}_0 C 0 H 0 H_0 H 0 + 1 +1 + 1 A v A_v A v B p B_p B p A v = − 1 A_v=-1 A v = − 1 B p = − 1 B_p=-1 B p = − 1 e e e m m m
The logical operators of the stabilizer toric code generate two microscopic Z 2 \mathbb{Z}_2 Z 2 [82 , 83 ] . They are given by the product of Pauli-Z Z Z X X X
Z ‾ μ ≡ ∏ ℓ ∈ C μ Z ℓ , X ‾ μ ≡ ∏ ℓ ∈ C ~ μ X ℓ , (3) \overline{Z}_\mu\equiv\prod_{\ell\in C_\mu} Z_\ell,\qquad\overline{X}_\mu\equiv\prod_{\ell\in\widetilde{C}_\mu} X_\ell, \tag*{(3)} Z μ ≡ ℓ ∈ C μ ∏ Z ℓ , X μ ≡ ℓ ∈ C μ ∏ X ℓ , (3) where C μ C_\mu C μ C ~ μ \widetilde{C}_\mu C μ μ \mu μ Figure 3 (a)], with μ = x , y \mu=x,y μ = x , y Z ‾ μ \overline{Z}_\mu Z μ X ‾ μ \overline{X}_\mu X μ A v A_v A v B p B_p B p { X ‾ x , Z ‾ y } = { X ‾ y , Z ‾ x } = 0 \{\overline{X}_x,\overline{Z}_y\}=\{\overline{X}_y,\overline{Z}_x\}=0 { X x , Z y } = { X y , Z x } = 0 [ X ‾ x , Z ‾ x ] = [ X ‾ y , Z ‾ y ] = 0 [\overline{X}_x,\overline{Z}_x]=[\overline{X}_y,\overline{Z}_y]=0 [ X x , Z x ] = [ X y , Z y ] = 0 Z ‾ μ \overline{Z}_\mu Z μ X ‾ μ \overline{X}_\mu X μ C 0 \mathcal{C}_0 C 0 3 C μ C_\mu C μ C ~ μ \widetilde{C}_\mu C μ 4 C 0 \mathcal{C}_0 C 0 ∣ ψ ⟩ , ∣ ϕ ⟩ ∈ C 0 |\psi\rangle,|\phi\rangle\in\mathcal{C}_0 ∣ ψ ⟩ , ∣ ϕ ⟩ ∈ C 0 O O O
Figure 3. (a) Stabilizers A v , B p A_v,B_p A v , B p Z ‾ x , X ‾ y \overline{Z}_x,\overline{X}_y Z x , X y Z Z Z X X X Z 2 \mathbb{Z}_2 Z 2 (10) ], obtained from the TFTC via the quantum-classical mapping. Classical Ising spins U i j U_{ij} U ij i , j , k , l i,j,k,l i , j , k , l U i j = − 1 U_{ij}=-1 U ij = − 1 2 ε 2\varepsilon 2 ε 2 J 2J 2 J h h h h < h c h<h_c h < h c C h \mathcal{C}_h C h h > h c h>h_c h > h c m m m
⟨ ψ ∣ O ∣ ϕ ⟩ = c O ⟨ ψ ∣ ϕ ⟩ , (4) \langle\psi|O|\phi\rangle=c_O\langle\psi|\phi\rangle, \tag*{(4)} ⟨ ψ ∣ O ∣ ϕ ⟩ = c O ⟨ ψ ∣ ϕ ⟩ , (4) where c O c_O c O O O O [80 , 84 , 85 ] , underlies the stability of the toric code’s topological order to arbitrary local perturbations [6 , 86 , 87 , 88 ] .
Throughout this work, we let { ∣ ψ 0 σ ⟩ } \{|\psi_0^\sigma\rangle\} { ∣ ψ 0 σ ⟩} Z Z Z C 0 \mathcal{C}_0 C 0 σ = ( σ x , σ y ) \sigma=(\sigma_x,\sigma_y) σ = ( σ x , σ y ) σ μ = 0 , 1 \sigma_\mu=0,1 σ μ = 0 , 1 μ = x , y \mu=x,y μ = x , y Z Z Z
A v ∣ ψ 0 σ ⟩ = B p ∣ ψ 0 σ ⟩ = + ∣ ψ 0 σ ⟩ , (5) A_v|\psi_0^\sigma\rangle=B_p|\psi_0^\sigma\rangle=+|\psi_0^\sigma\rangle, \tag*{(5)} A v ∣ ψ 0 σ ⟩ = B p ∣ ψ 0 σ ⟩ = + ∣ ψ 0 σ ⟩ , (5) Z ‾ μ ∣ ψ 0 σ ⟩ = ( − 1 ) σ μ ∣ ψ 0 σ ⟩ . (6) \overline{Z}_\mu|\psi_0^\sigma\rangle=(-1)^{\sigma_\mu}|\psi_0^\sigma\rangle. \tag*{(6)} Z μ ∣ ψ 0 σ ⟩ = ( − 1 ) σ μ ∣ ψ 0 σ ⟩ . (6) Each state ∣ ψ 0 σ ⟩ |\psi_0^\sigma\rangle ∣ ψ 0 σ ⟩ Z Z Z Z ℓ = − 1 Z_\ell=-1 Z ℓ = − 1 Z Z Z Z ‾ μ \overline{Z}_\mu Z μ C μ C_\mu C μ ∣ ψ 0 ( 0 , 0 ) ⟩ |\psi_0^{(0,0)}\rangle ∣ ψ 0 ( 0 , 0 ) ⟩ ∣ ψ 0 ( 1 , 0 ) ⟩ |\psi_0^{(1,0)}\rangle ∣ ψ 0 ( 1 , 0 ) ⟩ Z ‾ x = − 1 \overline{Z}_x=-1 Z x = − 1 y y y
Finally, it will occasionally be convenient to utilize an alternate notation X ‾ σ \overline{X}^\sigma X σ X X X
X ‾ σ ∣ ψ 0 σ ′ ⟩ = ∣ ψ 0 σ ⊕ σ ′ ⟩ , (7) \overline{X}^\sigma|\psi_0^{\sigma'}\rangle=|\psi_0^{\sigma\oplus\sigma'}\rangle, \tag*{(7)} X σ ∣ ψ 0 σ ′ ⟩ = ∣ ψ 0 σ ⊕ σ ′ ⟩ , (7) where σ ⊕ σ ′ = ( σ x ⊕ σ x ′ , σ y ⊕ σ y ′ ) \sigma\oplus\sigma'=(\sigma_x\oplus\sigma'_x,\sigma_y\oplus\sigma'_y) σ ⊕ σ ′ = ( σ x ⊕ σ x ′ , σ y ⊕ σ y ′ ) ⊕ \oplus ⊕ X ‾ ( 0 , 0 ) = 1 \overline{X}^{(0,0)}=1 X ( 0 , 0 ) = 1 X ‾ ( 1 , 0 ) = X ‾ y \overline{X}^{(1,0)}=\overline{X}_y X ( 1 , 0 ) = X y
Transverse-Field Toric Code
In this work, we will study QEC protocols for quantum information stored in the ground-state manifold of the transverse-field toric code (TFTC) [50 , 51 , 52 , 53 , 54 , 55 , 56 ] , a physically natural and well-studied deformation of the toric code. The Hamiltonian is
H h = − ∑ v A v − ∑ p B p − h ∑ ℓ X ℓ , (8) H_h=-\sum_v A_v-\sum_p B_p-h\sum_\ell X_\ell, \tag*{(8)} H h = − v ∑ A v − p ∑ B p − h ℓ ∑ X ℓ , (8) where we assume h ≥ 0 h\ge0 h ≥ 0 h h h H h H_h H h C h \mathcal{C}_h C h H h H_h H h L L L L L L [6 , 86 , 87 ] . Furthermore, although the logical-Z Z Z Z Z Z [86 ] , and the ground states in C h \mathcal{C}_h C h (4) up to corrections exponentially small in L L L C h \mathcal{C}_h C h H h H_h H h
Physically, the effect of the transverse field is to provide dynamics and quantum fluctuations to the m m m h c ≈ 0.328 h_c\approx0.328 h c ≈ 0.328 [89 ] , the gap to creating m m m h > h c h>h_c h > h c 5 h c h_c h c 3 d 3d 3 d 6 ( 2 + 1 ) d (2+1)d ( 2 + 1 ) d [50 , 51 , 55 , 56 ] .
Note that the vertex stabilizer A v A_v A v H h H_h H h h h h C h \mathcal{C}_h C h A v = + 1 A_v=+1 A v = + 1 H h H_h H h ( 2 + 1 ) d (2+1)d ( 2 + 1 ) d Z 2 \mathbb{Z}_2 Z 2 [50 , 51 ] , defined by the Hamiltonian
H L G T = − ∑ p B p − h ∑ ℓ X ℓ (9) H_{\mathrm{LGT}}=-\sum_p B_p-h\sum_\ell X_\ell \tag*{(9)} H LGT = − p ∑ B p − h ℓ ∑ X ℓ (9) together with the Hilbert space constraint A v ! = + 1 A_v\mathrel{!}=+1 A v ! = + 1 Z ℓ Z_\ell Z ℓ X ℓ X_\ell X ℓ Z 2 \mathbb{Z}_2 Z 2 m m m B p = − 1 B_p=-1 B p = − 1 A v = 1 A_v=1 A v = 1 X ℓ = − 1 X_\ell=-1 X ℓ = − 1 e e e m m m
Quantum-Classical Mapping
Significant physical intuition for the QEC protocols we investigate will come from the quantum-classical mapping [51 , 55 , 56 , 81 ] . As reviewed in Appendix A, matrix elements of the form e − β H h e^{-\beta H_h} e − β H h Z 2 \mathbb{Z}_2 Z 2 τ \tau τ 7
S L G T [ U ] = − ε ∑ [ i j k l ] ⊥ U i j U j k U k l U l i − J ∑ [ i j k l ] ∥ U i j U j k U k l U l i S_{\mathrm{LGT}}[U]=-\varepsilon\sum_{[ijkl]_\perp}U_{ij}U_{jk}U_{kl}U_{li}-J\sum_{[ijkl]_\parallel}U_{ij}U_{jk}U_{kl}U_{li} S LGT [ U ] = − ε [ ij k l ] ⊥ ∑ U ij U j k U k l U l i − J [ ij k l ] ∥ ∑ U ij U j k U k l U l i S L G T [ U ] = − ∑ [ i j k l ] J i j k l U i j U j k U k l U l i , (10) \phantom{S_{\mathrm{LGT}}[U]}=-\sum_{[ijkl]}J_{ijkl}U_{ij}U_{jk}U_{kl}U_{li}, \tag*{(10)} S LGT [ U ] = − [ ij k l ] ∑ J ij k l U ij U j k U k l U l i , (10) where U i j = ± 1 U_{ij}=\pm1 U ij = ± 1 ⟨ i j ⟩ \langle ij\rangle ⟨ ij ⟩ i ≡ ( v , τ ) i\equiv(v,\tau) i ≡ ( v , τ ) τ = 0 , ± ε , ± 2 ε , … \tau=0,\pm\varepsilon,\pm2\varepsilon,\ldots τ = 0 , ± ε , ± 2 ε , … L τ L_\tau L τ ε ≡ β L τ \varepsilon\equiv\frac{\beta}{L_\tau} ε ≡ L τ β [ i j k l ] [ijkl] [ ij k l ] [ i j k l ] ⊥ [ijkl]_\perp [ ij k l ] ⊥ ε \varepsilon ε [ i j k l ] ∥ [ijkl]_\parallel [ ij k l ] ∥ J ≡ − 1 2 log tanh ( ε h ) J\equiv-\frac{1}{2}\log\tanh(\varepsilon h) J ≡ − 2 1 log tanh ( ε h ) J i j k l ≡ ε J_{ijkl}\equiv\varepsilon J ij k l ≡ ε J i j k l = J J_{ijkl}=J J ij k l = J [ i j k l ] [ijkl] [ ij k l ]
The statistical physics model (10) can be regarded as a model of one-dimensional Ising magnetic flux loops L \mathcal{L} L U i j U j k U k l U l i = − 1 U_{ij}U_{jk}U_{kl}U_{li}=-1 U ij U j k U k l U l i = − 1 2 ε 2\varepsilon 2 ε 2 J 2J 2 J
Z L G T ≡ ∑ { U } e − S L G T [ U ] ∝ ∑ L e − E [ L ] , (11) Z_{\mathrm{LGT}}\equiv\sum_{\{U\}}e^{-S_{\mathrm{LGT}}[U]}\propto\sum_{\mathcal{L}}e^{-E[\mathcal{L}]}, \tag*{(11)} Z LGT ≡ { U } ∑ e − S LGT [ U ] ∝ L ∑ e − E [ L ] , (11) where ∑ L \sum_{\mathcal{L}} ∑ L L \mathcal{L} L E [ L ] ≡ 2 ε ∣ L ∣ ∥ + 2 J ∣ L ∣ ⊥ E[\mathcal{L}]\equiv2\varepsilon|\mathcal{L}|_\parallel+2J|\mathcal{L}|_\perp E [ L ] ≡ 2 ε ∣ L ∣ ∥ + 2 J ∣ L ∣ ⊥ L \mathcal{L} L ∣ L ∣ ∥ |\mathcal{L}|_\parallel ∣ L ∣ ∥ ∣ L ∣ ⊥ |\mathcal{L}|_\perp ∣ L ∣ ⊥ L \mathcal{L} L h < h c h<h_c h < h c h > h c h>h_c h > h c ( 2 + 1 ) d (2+1)d ( 2 + 1 ) d 3 d 3d 3 d [50 ] , and the phase transition is therefore once again in the 3 d 3d 3 d
The classical action in Eq. (10) can be regarded as a path-integral description of the TFTC, where the flux loops precisely correspond to the imaginary-time worldlines of m m m X X X
Z h σ ≡ ⟨ ψ 0 σ ⊕ σ ′ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ (12) Z_{h}^{\boldsymbol{\sigma}} \equiv\langle\psi_{0}^{\sigma\oplus\sigma'}|e^{-\beta H_h}|\psi_{0}^{\sigma'}\rangle \tag*{(12)} Z h σ ≡ ⟨ ψ 0 σ ⊕ σ ′ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ (12) can be expressed as a LGT partition function over the imaginary-time interval τ ∈ [ − β 2 , β 2 ] \tau\in\left[-\frac{\beta}{2},\frac{\beta}{2}\right] τ ∈ [ − 2 β , 2 β ] τ = ± β 2 \tau= \pm\frac{\beta}{2} τ = ± 2 β X X X X σ X^{\boldsymbol{\sigma}} X σ Z h ≡ Z h ( 0 , 0 ) Z_h \equiv Z_h^{(0,0)} Z h ≡ Z h ( 0 , 0 ) Figure 4 (a)], whereas Z h σ Z_h^{\boldsymbol{\sigma}} Z h σ σ ≠ ( 0 , 0 ) \boldsymbol{\sigma} \ne(0,0) σ = ( 0 , 0 ) Figure 4 (b)]. Note that Eq. (12) is independent of the choice of σ ′ \boldsymbol{\sigma}' σ ′ H h H_h H h X X X
Figure 4. (a) Schematic depiction of an anyon worldline configuration contributing to the partition function Z h ≡ ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ⟩ Z_h \equiv\langle\psi_0^\sigma|e^{-\beta H_h}|\psi_0^\sigma\rangle Z h ≡ ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ⟩ contractible , so as to not implement a nontrivial logical-X X X Z h σ Z_h^{\boldsymbol{\sigma}} Z h σ (12) for σ ≠ ( 0 , 0 ) \boldsymbol{\sigma} \ne(0,0) σ = ( 0 , 0 ) σ = ( 0 , 1 ) \boldsymbol{\sigma}=(0,1) σ = ( 0 , 1 ) x x x Z y Z_y Z y O ( L ) \mathcal{O}(L) O ( L ) h < h c h<h_c h < h c Z h σ Z_h^{\boldsymbol{\sigma}} Z h σ σ ≠ ( 0 , 0 ) \boldsymbol{\sigma}\ne(0,0) σ = ( 0 , 0 ) Z h ≡ Z h ( 0 , 0 ) Z_h\equiv Z_h^{(0,0)} Z h ≡ Z h ( 0 , 0 )
Throughout the topological phase h < h c h < h_c h < h c L L L
Z h σ Z h = { 1 , σ = ( 0 , 0 ) O ( e − L / ξ ) , σ ≠ ( 0 , 0 ) ( h < h c ) . \frac{Z_h^{\boldsymbol{\sigma}}}{Z_h} = \begin{cases} 1, & \boldsymbol{\sigma}=(0,0)\\ \mathcal{O}(e^{-L/\xi}), & \boldsymbol{\sigma}\ne(0,0) \end{cases} \quad(h<h_c). Z h Z h σ = { 1 , O ( e − L / ξ ) , σ = ( 0 , 0 ) σ = ( 0 , 0 ) ( h < h c ) .
Although we are unaware of a rigorous proof of Eq. (12) for all h < h c h<h_c h < h c isotropic 3 d 3d 3 d [92 ] . Furthermore, essentially identical observables have been numerically studied extensively in the literature [93 , 94 ] . In Appendix § 1 1, we also numerically verify this statement in an isotropic 3 d 3d 3 d h < h c h<h_c h < h c
Setup
The primary focus of this work is to construct and study QEC protocols for the TFTC under Pauli decoherence. In Sec. III A, we broadly define the QEC problem we will investigate. In Sec. III B we introduce imaginary-time evolution (ITE) as a technique for constructing a useful approximate representation of TFTC code states, and in Sec. III C we describe a reformulation of our QEC protocol in terms of the code states constructed via ITE.
Definition of QEC Protocol
Suppose two qubits of quantum information are encoded in the TFTC ground state ∣ ψ h ⟩ ∈ C h |\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h
E ℓ ( ρ ) = ( 1 − p ) ρ + p X X ℓ ρ X ℓ + p Y Y ℓ ρ Y ℓ + p Z Z ℓ ρ Z ℓ (13) \mathcal{E}_{\ell}(\rho)=(1-p)\rho+p_X X_{\ell}\rho X_{\ell}+p_Y Y_{\ell}\rho Y_{\ell}+p_Z Z_{\ell}\rho Z_{\ell} \tag*{(13)} E ℓ ( ρ ) = ( 1 − p ) ρ + p X X ℓ ρ X ℓ + p Y Y ℓ ρ Y ℓ + p Z Z ℓ ρ Z ℓ (13) on each link ℓ \ell ℓ 8 p ≡ p X + p Y + p Z p\equiv p_X+p_Y+p_Z p ≡ p X + p Y + p Z ρ h = E ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) \rho_h=\mathcal{E}(|\psi_h\rangle\langle\psi_h|) ρ h = E ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) E ≡ ∏ ℓ E ℓ \mathcal{E}\equiv\prod_{\ell}\mathcal{E}_{\ell} E ≡ ∏ ℓ E ℓ ( p X , p Y , p Z ) (p_X,p_Y,p_Z) ( p X , p Y , p Z ) p X p_X p X p Z p_Z p Z E ℓ \mathcal{E}_{\ell} E ℓ p Y = p X p Z / ( 1 − p ) p_Y=p_Xp_Z/(1-p) p Y = p X p Z / ( 1 − p ) E ℓ = E ℓ X ∘ E ℓ Z \mathcal{E}_{\ell}=\mathcal{E}_{\ell}^{X}\circ\mathcal{E}_{\ell}^{Z} E ℓ = E ℓ X ∘ E ℓ Z
As described in Sec. III B, we expect the code space C h \mathcal{C}_h C h h < h c h<h_c h < h c [49 ] , we showed that a finite strength of Pauli decoherence is required to irreversibly destroy information encoded in C h \mathcal{C}_h C h h h h h c h_c h c
Nevertheless, even if quantum information is in principle protected below some intrinsic decoherence threshold, a central question is whether this encoded information can be practically recovered—that is, whether there exists an efficient and conceptually simple QEC protocol which recovers the encoded information. Motivated by the results of Refs. [72 , 73 ] , we will investigate the efficacy of stabilizer measurements for performing QEC on the decohered TFTC states. Starting from the decohered state ρ h \rho_h ρ h A v A_v A v B p B_p B p
ρ h ( s ) ≡ Π ^ s ρ h Π ^ s P s (14) \rho_h(s) \equiv\frac{\hat{\Pi}_s\rho_h\hat{\Pi}_s}{P_s} \tag*{(14)} ρ h ( s ) ≡ P s Π ^ s ρ h Π ^ s (14) with probability P s ≡ tr [ Π ^ s ρ h ] P_s \equiv\operatorname{tr}[\hat{\Pi}_s\rho_h] P s ≡ tr [ Π ^ s ρ h ]
Π ^ s ≡ ∏ v ( 1 + ( − 1 ) s v A v 2 ) ∏ p ( 1 + ( − 1 ) s p B p 2 ) . (15) \hat{\Pi}_s \equiv\prod_v \left(\frac{1+(-1)^{s_v}A_v}{2}\right)\prod_p \left(\frac{1+(-1)^{s_p}B_p}{2}\right). \tag*{(15)} Π ^ s ≡ v ∏ ( 2 1 + ( − 1 ) s v A v ) p ∏ ( 2 1 + ( − 1 ) s p B p ) . (15) is the measurement projector. The syndrome s ≡ { s v , s p } s \equiv\{s_v,s_p\} s ≡ { s v , s p } s v , s p = 0 , 1 s_v,s_p=0,1 s v , s p = 0 , 1 e e e m m m s v = 1 s_v=1 s v = 1 s p = 1 s_p=1 s p = 1 s s s V s V_s V s ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ ρ h ( s ) \rho_h(s) ρ h ( s ) F s ( ∣ ψ h ⟩ ) F_s(|\psi_h\rangle) F s ( ∣ ψ h ⟩)
F s ( ∣ ψ h ⟩ ) ≡ ⟨ ψ h ∣ V s ρ h ( s ) V s † ∣ ψ h ⟩ . (16) F_s(|\psi_h\rangle) \equiv\langle\psi_h|V_s\rho_h(s)V_s^\dagger|\psi_h\rangle. \tag*{(16)} F s ( ∣ ψ h ⟩) ≡ ⟨ ψ h ∣ V s ρ h ( s ) V s † ∣ ψ h ⟩ . (16) This question can be addressed both for individual syndromes s s s ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ ρ h ( s ) \rho_h(s) ρ h ( s )
F ‾ ( ∣ ψ h ⟩ ) ≡ ∑ s P s F s ( ∣ ψ h ⟩ ) = ∑ s ⟨ ψ h ∣ V s Π ^ s ρ h Π ^ s V s † ∣ ψ h ⟩ . (17) \begin{aligned}
\overline{F}(|\psi_h\rangle) \equiv\sum_s P_sF_s(|\psi_h\rangle) \\
&= \sum_s \langle\psi_h|V_s\hat{\Pi}_s\rho_h\hat{\Pi}_sV_s^\dagger|\psi_h\rangle. \tag*{(17)}
\end{aligned} F ( ∣ ψ h ⟩) ≡ s ∑ P s F s ( ∣ ψ h ⟩) = s ∑ ⟨ ψ h ∣ V s Π ^ s ρ h Π ^ s V s † ∣ ψ h ⟩ . (17) Note that F ‾ \overline{F} F F s F_s F s [95 ] yields
P [ F s ≤ 1 − ϵ ] ≤ 1 − F ‾ ϵ (18) \mathbb{P}[F_s\le1-\epsilon]\le\frac{1-\overline{F}}{\epsilon} \tag*{(18)} P [ F s ≤ 1 − ϵ ] ≤ ϵ 1 − F (18) for any ϵ > 0 \epsilon>0 ϵ > 0 P [ ⋅ ] \mathbb{P}[\cdot] P [ ⋅ ] ( ⋅ ) (\cdot) ( ⋅ )
In QEC algorithms for the stabilizer toric code under Pauli decoherence [7 , 35 ] , one seeks to pairwise annihilate the observed anyons to recover a stabilizer state ∣ ψ 0 ⟩ ∈ C 0 |\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0 C 0 \mathcal{C}_0 C 0
then we adiabatically re-encode into the deformed code space C h \mathcal{C}_h C h V s V_s V s 9
V s = U h X s Z s , V_s=U_hX_sZ_s, V s = U h X s Z s , U h ≡ T exp ( − i ∫ 0 T d t H h ( t ) ) . (19) U_h\equiv\mathcal{T}\exp\left(-i\int_0^T\mathrm{d}t\,H_h(t)\right). \tag*{(19)} U h ≡ T exp ( − i ∫ 0 T d t H h ( t ) ) . (19) Here10 U h U_h U h C 0 \mathcal{C}_0 C 0 C h \mathcal{C}_h C h T T T H h H_h H h X s X_s X s Z s Z_s Z s X X X Z Z Z m m m e e e s s s X s X_s X s Z s Z_s Z s
X s B p = ( − 1 ) s p B p X s , Z s A v = ( − 1 ) s v A v Z s . (20) X_sB_p=(-1)^{s_p}B_pX_s,\qquad Z_sA_v=(-1)^{s_v}A_vZ_s. \tag*{(20)} X s B p = ( − 1 ) s p B p X s , Z s A v = ( − 1 ) s v A v Z s . (20) Such an error correction protocol will succeed with high fidelity if X s Z s ρ h ( s ) Z s X s X_sZ_s\rho_h(s)Z_sX_s X s Z s ρ h ( s ) Z s X s ∣ ψ 0 ⟩ = U h † ∣ ψ h ⟩ |\psi_0\rangle=U_h^\dagger|\psi_h\rangle ∣ ψ 0 ⟩ = U h † ∣ ψ h ⟩
Figure 21 (PDF p. 8).
Throughout this work, we will assume that no additional errors arise during the course of stabilizer measurement, anyon pairing, or adiabatic re-encoding: we assume that each of these operations can be performed perfectly, and focus our attention on determining how to best pair the anyons (i.e., how to choose X s X_s X s Z s Z_s Z s X s X_s X s Z s Z_s Z s homologous choices differ by stabilizers and act identically on the postmeasurement states, while non-homologous choices differ by logical operators and lead to distinct recovery unitaries. Our central task is therefore to determine the optimal homology classes for the anyon-pairing unitaries X s X_s X s Z s Z_s Z s s s s s s s decoder . We will refer to an algorithm which maximizes the decoding fidelity (16) for each s s s optimal postmeasurement decoder , to distinguish from the true optimal recovery channel which may not involve measurements of the “bare” stabilizers A v A_v A v B p B_p B p [61 , 62 ] , due to the nontrivial quantum fluctuations within the code space C h \mathcal{C}_h C h
It is useful to briefly mention an alternative interpretation of the QEC problem depicted in the diagram (21). Rather than viewing C h \mathcal{C}_h C h U h U_h U h C 0 \mathcal{C}_0 C 0 U h U_h U h coherent error acting on C 0 \mathcal{C}_0 C 0 E \mathcal{E} E [65 , 66 , 67 , 68 , 69 , 70 ] have focused on finite-depth (typically depth-one) unitary circuits; in contrast, U h U_h U h h → h c h \to h_c h → h c p = 0 p = 0 p = 0
The central result of this work is to demonstrate that stabilizer measurements can in fact be utilized as part of an effective decoding protocol, even in close vicinity to the quantum critical point at h c h_c h c p = 0 p = 0 p = 0 h < h c h < h_c h < h c
Imaginary Time Evolution
To analyze the quantum code C h \mathcal{C}_h C h imaginary time evolution (ITE) to construct an excellent approximate basis for C h \mathcal{C}_h C h
Let ∣ ψ h ⟩ ∈ C h |\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ H h H_h H h ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ ∣ ψ 0 ⟩ ∈ C 0 |\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0
∣ ψ h ⟩ ≈ ∣ φ h ⟩ ≡ e − β H h / 2 ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ . (21) |\psi_h\rangle\approx|\varphi_h\rangle\equiv\frac{e^{-\beta H_h/2}|\psi_0\rangle}{\sqrt{\langle\psi_0|e^{-\beta H_h}|\psi_0\rangle}}. \tag*{(21)} ∣ ψ h ⟩ ≈ ∣ φ h ⟩ ≡ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ e − β H h /2 ∣ ψ 0 ⟩ . (21) Here β \beta β β ∼ poly ( L ) \beta\sim\operatorname{poly}(L) β ∼ poly ( L ) e − β H h / 2 e^{-\beta H_h/2} e − β H h /2 C h \mathcal{C}_h C h β ≪ exp ( L ) \beta\ll\exp(L) β ≪ exp ( L ) C h \mathcal{C}_h C h
a straightforward exercise to show11 ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ C h \mathcal{C}_h C h L L L h < h c h < h_c h < h c H h H_h H h β \beta β β \beta β L L L
Throughout the topological phase12 h < h c h < h_c h < h c (21) acts approximately unitarily on C 0 \mathcal{C}_0 C 0
∣ φ h σ ⟩ = 1 Z h e − β H h / 2 ∣ ψ 0 σ ⟩ (22) |\varphi_h^{\boldsymbol{\sigma}}\rangle= \frac{1}{\sqrt{\mathcal{Z}_h}}e^{-\beta H_h/2}|\psi_0^{\boldsymbol{\sigma}}\rangle \tag*{(22)} ∣ φ h σ ⟩ = Z h 1 e − β H h /2 ∣ ψ 0 σ ⟩ (22) denote the logical-Z Z Z Z h ≡ Z h ( 0 , 0 ) \mathcal{Z}_h \equiv\mathcal{Z}_h^{(0,0)} Z h ≡ Z h ( 0 , 0 ) (12) . Using Eq. (12), these four states are approximately orthonormal for all h < h c h < h_c h < h c
⟨ φ h σ ∣ φ h σ ′ ⟩ = Z h σ ⊕ σ ′ Z h = { 1 , σ = σ ′ O ( e − L / ξ ) , σ ≠ σ ′ . (23) \langle\varphi_h^{\boldsymbol{\sigma}}|\varphi_h^{\boldsymbol{\sigma}'}\rangle= \frac{\mathcal{Z}_h^{\boldsymbol{\sigma}\oplus\boldsymbol{\sigma}'}}{\mathcal{Z}_h} =
\begin{cases}
1, & \boldsymbol{\sigma}=\boldsymbol{\sigma}' \\
\mathcal{O}(e^{-L/\xi}), & \boldsymbol{\sigma}\ne\boldsymbol{\sigma}'.
\end{cases} \tag*{(23)} ⟨ φ h σ ∣ φ h σ ′ ⟩ = Z h Z h σ ⊕ σ ′ = { 1 , O ( e − L / ξ ) , σ = σ ′ σ = σ ′ . (23) In other words, ITE preserves the inner products between the logical-Z Z Z ∣ φ h σ ⟩ |\varphi_h^{\boldsymbol{\sigma}}\rangle ∣ φ h σ ⟩ m m m Figure 4 (b)], and these processes are exponentially suppressed throughout the topological phase.
Eq. (23) , combined with the logical-X X X H h H_h H h C 0 \mathcal{C}_0 C 0 Π ^ 0 \hat{\Pi}_0 Π ^ 0 C 0 \mathcal{C}_0 C 0 (15) with trivial syndrome s v , s p = 0 s_v,s_p=0 s v , s p = 0 e − β H h / 2 Π ^ 0 e^{-\beta H_h/2}\hat{\Pi}_0 e − β H h /2 Π ^ 0
1 Z h e − β H h / 2 Π ^ 0 = V h I h 1 / 2 , (24) \frac{1}{\sqrt{\mathcal{Z}_h}}e^{-\beta H_h/2}\hat{\Pi}_0 = \mathcal{V}_h\mathcal{I}_h^{1/2}, \tag*{(24)} Z h 1 e − β H h /2 Π ^ 0 = V h I h 1/2 , (24) where the Hermitian operator I h \mathcal{I}_h I h
I h ≡ Π ^ 0 e − β H h Π ^ 0 Z h , (25) \mathcal{I}_h \equiv\frac{\hat{\Pi}_0 e^{-\beta H_h}\hat{\Pi}_0}{\mathcal{Z}_h}, \tag*{(25)} I h ≡ Z h Π ^ 0 e − β H h Π ^ 0 , (25) and the isometry V h \mathcal{V}_h V h 13
V h ≡ 1 Z h e − β H h / 2 Π ^ 0 I h − 1 / 2 Π ^ 0 . (26) \mathcal{V}_h \equiv\frac{1}{\sqrt{\mathcal{Z}_h}}e^{-\beta H_h/2}\hat{\Pi}_0\mathcal{I}_h^{-1/2}\hat{\Pi}_0. \tag*{(26)} V h ≡ Z h 1 e − β H h /2 Π ^ 0 I h − 1/2 Π ^ 0 . (26) which satisfies V h † V h = Π ^ 0 \mathcal{V}_h^\dagger\mathcal{V}_h=\hat{\Pi}_0 V h † V h = Π ^ 0 V h \mathcal{V}_h V h (21) . Notably, V h \mathcal{V}_h V h I h \mathcal{I}_h I h C 0 \mathcal{C}_0 C 0 I h \mathcal{I}_h I h C 0 \mathcal{C}_0 C 0 X X X
I h = Π ^ 0 [ 1 + ∑ σ ≠ ( 0 , 0 ) Z h σ Z h X ˉ σ ] Π ^ 0 , (27) \mathcal{I}_h=\hat{\Pi}_0\left[1+\sum_{\sigma\ne(0,0)}\frac{Z_h^\sigma}{Z_h}\bar{X}^\sigma\right]\hat{\Pi}_0, \tag*{(27)} I h = Π ^ 0 1 + σ = ( 0 , 0 ) ∑ Z h Z h σ X ˉ σ Π ^ 0 , (27) where the coefficients can be verified by taking matrix elements of Eq. (25) using the logical-Z Z Z ∣ ψ 0 σ ⟩ |\psi_0^\sigma\rangle ∣ ψ 0 σ ⟩ Z h σ / Z h Z_h^\sigma/Z_h Z h σ / Z h L L L h < h c h<h_c h < h c I h \mathcal{I}_h I h Π ^ 0 \hat{\Pi}_0 Π ^ 0 V h ∣ ψ 0 ⟩ \mathcal{V}_h|\psi_0\rangle V h ∣ ψ 0 ⟩ ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ (21) , and therefore ITE acts approximately unitarily on C 0 \mathcal{C}_0 C 0
Throughout the remainder of this work, we will utilize the ITE states ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ C h \mathcal{C}_h C h C h \mathcal{C}_h C h ∣ ψ h ⟩ ∈ C h |\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ ⟨ φ h ∣ ψ h ⟩ ≥ 1 − O ( e − L / ξ ) \langle\varphi_h|\psi_h\rangle\ge1-O(e^{-L/\xi}) ⟨ φ h ∣ ψ h ⟩ ≥ 1 − O ( e − L / ξ ) C h \mathcal{C}_h C h
∣ F ‾ ( ∣ ψ h ⟩ ) − F ‾ ( ∣ φ h ⟩ ) ∣ ≤ 4 1 − ∣ ⟨ φ h ∣ ψ h ⟩ ∣ 2 . (28) \left|\overline{F}(|\psi_h\rangle)-\overline{F}(|\varphi_h\rangle)\right|\le4\sqrt{1-|\langle\varphi_h|\psi_h\rangle|^2}. \tag*{(28)} F ( ∣ ψ h ⟩) − F ( ∣ φ h ⟩) ≤ 4 1 − ∣ ⟨ φ h ∣ ψ h ⟩ ∣ 2 . (28) Furthermore, in Appendix D we argue that the ITE state in Eq. (21) for each ∣ ψ 0 ⟩ ∈ C 0 |\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0 U h ∣ ψ 0 ⟩ U_h|\psi_0\rangle U h ∣ ψ 0 ⟩
We briefly mention an interesting alternative method of promoting ITE to a bona-fide unitary encoding, which we call “pruning” and describe in Appendix E. In short, since the nearly orthogonal states ∣ φ h σ ⟩ |\varphi_h^\sigma\rangle ∣ φ h σ ⟩ § 3 A), they must also exhibit nearly disjoint support in the computational basis. We can therefore “prune” these four code states to construct a basis of exactly orthonormal and strictly non-negative code states, which once again approximately span C h \mathcal{C}_h C h h < h c h<h_c h < h c
Modified QEC Protocol
The discussion of the preceding subsection motivates the following slight modification of the QEC problem described in Sec. III A. First, instead of starting from the TFTC state ∣ ψ h ⟩ = U h ∣ ψ 0 ⟩ ∈ C h |\psi_h\rangle=U_h|\psi_0\rangle\in\mathcal{C}_h ∣ ψ h ⟩ = U h ∣ ψ 0 ⟩ ∈ C h ∣ ψ 0 ⟩ ∈ C 0 |\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0 ∣ φ h ⟩ ∝ e − β H h / 2 ∣ ψ 0 ⟩ |\varphi_h\rangle\propto e^{-\beta H_h/2}|\psi_0\rangle ∣ φ h ⟩ ∝ e − β H h /2 ∣ ψ 0 ⟩ ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ ρ h = E ( ∣ φ h ⟩ ⟨ φ h ∣ ) \rho_h=\mathcal{E}(|\varphi_h\rangle\langle\varphi_h|) ρ h = E ( ∣ φ h ⟩ ⟨ φ h ∣ ) A v , B p A_v,B_p A v , B p ρ h ( s ) \rho_h(s) ρ h ( s ) (14) . Finally, we attempt to recover ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ m m m e e e X s X_s X s Z s Z_s Z s
∣ φ h ⟩ → E ρ h ↑ e − β H h / 2 ↓ Π ^ s ∣ ψ 0 ⟩ ← X s Z s ρ h ( s ) (29) \begin{array}{ccccc}
|\varphi_h\rangle& \xrightarrow{\mathcal{E}} & \rho_h \\
\uparrow^{e^{-\beta H_h/2}} & & \downarrow^{\hat{\Pi}_s} \\
|\psi_0\rangle& \xleftarrow{X_s Z_s} & \rho_h(s) \tag*{(29)}
\end{array} ∣ φ h ⟩ ↑ e − β H h /2 ∣ ψ 0 ⟩ E X s Z s ρ h ↓ Π ^ s ρ h ( s ) (29) So long as h < h c h<h_c h < h c (28) guarantees that the average recovery fidelity of this protocol,
F ‾ I T E ( ∣ ψ 0 ⟩ ) ≡ ∑ s p s F s I T E ( ∣ ψ 0 ⟩ ) , (30) \overline{\mathcal{F}}^{\mathrm{ITE}}(|\psi_0\rangle)\equiv\sum_s p_s\mathcal{F}^{\mathrm{ITE}}_s(|\psi_0\rangle), \tag*{(30)} F ITE ( ∣ ψ 0 ⟩) ≡ s ∑ p s F s ITE ( ∣ ψ 0 ⟩) , (30) F s I T E ( ∣ ψ 0 ⟩ ) ≡ ⟨ ψ 0 ∣ X s Z s ρ h ( s ) Z s X s ∣ ψ 0 ⟩ , (31) \mathcal{F}^{\mathrm{ITE}}_s(|\psi_0\rangle)\equiv\langle\psi_0|X_sZ_s\rho_h(s)Z_sX_s|\psi_0\rangle, \tag*{(31)} F s ITE ( ∣ ψ 0 ⟩) ≡ ⟨ ψ 0 ∣ X s Z s ρ h ( s ) Z s X s ∣ ψ 0 ⟩ , (31) is equal to the average recovery fidelity of the original protocol up to exponentially small corrections in L L L F ‾ I T E \overline{\mathcal{F}}^{\mathrm{ITE}} F ITE F s I T E \mathcal{F}^{\mathrm{ITE}}_s F s ITE ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ ∣ ψ h ⟩ = U h ∣ ψ 0 ⟩ |\psi_h\rangle=U_h|\psi_0\rangle ∣ ψ h ⟩ = U h ∣ ψ 0 ⟩ (16) . In turn, the optimal post-measurement decoders we will construct for the ITE code states are guaranteed to be asymptotically optimal for the original protocol of Sec. III A, in the sense that no other decoder can achieve a larger fidelity on the true code states by more than an O ( e − L / ξ ) O(e^{-L/\xi}) O ( e − L / ξ ) (31) .
As we will see in § 3.2 D, it will be sufficient to construct optimal postmeasurement decoders for bit-flip and phase-flip decoherence separately. We will therefore set p Z = 0 p_Z=0 p Z = 0 Z s = 1 Z_s=1 Z s = 1 arbitrary code state ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ specific logical-Z Z Z ∣ ψ 0 ( 0 , 0 ) ⟩ |\psi_0^{(0,0)}\rangle ∣ ψ 0 ( 0 , 0 ) ⟩ (29) exhibits a strong logical-X X X E \mathcal{E} E X X X (30) for ∣ ψ 0 ⟩ = ∣ ψ 0 ( 0 , 0 ) ⟩ |\psi_0\rangle=|\psi_0^{(0,0)}\rangle ∣ ψ 0 ⟩ = ∣ ψ 0 ( 0 , 0 ) ⟩ any code state. In other words, our QEC protocol will succeed with high fidelity on an arbitrary code state if and only if it succeeds with high fidelity on ∣ ψ 0 ( 0 , 0 ) ⟩ |\psi_0^{(0,0)}\rangle ∣ ψ 0 ( 0 , 0 ) ⟩
Stabilizer Measurements of Transverse-Field Toric Code States
In this section, we will demonstrate that projective measurements of the “bare” toric code stabilizers preserve information encoded within the TFTC code space with high fidelity. This section can be regarded as the p = 0 p=0 p = 0 § 3.2 ), and serves as a useful first benchmark for the efficacy of stabilizer measurements in performing QEC on the TFTC states.
Starting from an arbitrary ITE code state ∣ φ h ⟩ ∝ e − β H h / 2 ∣ ψ 0 ⟩ |\varphi_h\rangle\propto e^{-\beta H_h/2}|\psi_0\rangle ∣ φ h ⟩ ∝ e − β H h /2 ∣ ψ 0 ⟩ (21) ], we perform a projective measurement of the stabilizers B p B_p B p
∣ φ h ( s ) ⟩ ≡ 1 P s Π ^ s ∣ φ h ⟩ (32) |\varphi_h(s)\rangle\equiv\frac{1}{\sqrt{P_s}}\hat{\Pi}_s|\varphi_h\rangle \tag*{(32)} ∣ φ h ( s )⟩ ≡ P s 1 Π ^ s ∣ φ h ⟩ (32) with probability
P s ≡ ⟨ φ h ∣ Π ^ s ∣ φ h ⟩ , (33) P_s \equiv\langle\varphi_h|\hat{\Pi}_s|\varphi_h\rangle, \tag*{(33)} P s ≡ ⟨ φ h ∣ Π ^ s ∣ φ h ⟩ , (33) where the projector Π ^ s \hat{\Pi}_s Π ^ s (15) . Given the syndrome s s s m m m X s X_s X s ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ ∣ φ h ( s ) ⟩ |\varphi_h(s)\rangle ∣ φ h ( s )⟩ A v = 1 A_v=1 A v = 1 e e e Z s Z_s Z s
In Appendix G, we prove that quantum information is perfectly preserved by stabilizer measurement with syndrome s s s V s V_s V s
The measurement probabilities P s σ = ⟨ φ h σ ∣ Π ^ s ∣ φ h σ ⟩ P_s^\sigma=\langle\varphi_h^\sigma|\hat{\Pi}_s|\varphi_h^\sigma\rangle P s σ = ⟨ φ h σ ∣ Π ^ s ∣ φ h σ ⟩ P s σ P_s^\sigma P s σ σ \sigma σ
The basis ITE states remain orthonormal after measurement, i.e., ⟨ φ h σ ′ ( s ) ∣ φ h σ ( s ) ⟩ = δ σ σ ′ \langle\varphi_h^{\sigma'}(s)|\varphi_h^\sigma(s)\rangle=\delta_{\sigma\sigma'} ⟨ φ h σ ′ ( s ) ∣ φ h σ ( s )⟩ = δ σ σ ′
Similarly, a high-fidelity approximate recovery unitary exists if these two criteria are approximately satisfied. Intuitively, the first condition guarantees that the measurement does not reveal any information about the underlying code state, while the second condition guarantees that superpositions of code states are not collapsed by syndrome measurement.14 Π ^ s \hat{\Pi}_s Π ^ s unitarily on the code space.
For the logical-Z Z Z ∣ φ h σ ⟩ |\varphi_h^\sigma\rangle ∣ φ h σ ⟩ (22) , the first criterion above holds exactly for all h h h X X X § 3 A, we will explicitly demonstrate that the second criterion is satisfied throughout the topological phase up to exponentially small corrections, with high probability in the syndrome measurement, and subsequently in § 3 B we determine the optimal anyon pairing X s X_s X s s s s § 3 C, we numerically implement this optimal postmeasurement decoder and demonstrate its efficacy throughout the topological phase.
Decodability via Stoquasticity
Although H h H_h H h (8) represents a rather natural perturbation to the stabilizer toric code, it also possesses a special property called stoquasticity . A Hamiltonian is called stoquastic in a given basis if all off-diagonal matrix elements in this basis are real and non-positive [6 ] . By this definition, H h H_h H h Z Z Z X X X Z Z Z ∣ z ⟩ |z\rangle ∣ z ⟩ z ≡ { z ℓ } z\equiv\{z_\ell\} z ≡ { z ℓ } z ℓ = 0 , 1 z_\ell=0,1 z ℓ = 0 , 1 Z ℓ ∣ z ⟩ = ( − 1 ) z ℓ ∣ z ⟩ Z_\ell|z\rangle=(-1)^{z_\ell}|z\rangle Z ℓ ∣ z ⟩ = ( − 1 ) z ℓ ∣ z ⟩ e − β H h e^{-\beta H_h} e − β H h 15
⟨ z ∣ e − β H h ∣ z ′ ⟩ ≥ 0 ∀ z , z ′ . (34) \langle z|e^{-\beta H_h}|z'\rangle\ge0\quad\forall z,z'. \tag*{(34)} ⟨ z ∣ e − β H h ∣ z ′ ⟩ ≥ 0 ∀ z , z ′ . (34) Note that this property was already implicitly used in Sec. II C to construct the quantum-classical mapping for H h H_h H h
We now show that stoquasticity directly implies that stabilizer measurements preserve the approximate orthogonality of the logical-Z Z Z ∣ φ h σ ⟩ |\varphi_h^\sigma\rangle ∣ φ h σ ⟩
∣ φ h σ ( s ) ⟩ ≡ 1 P s Π ^ s ∣ φ h σ ⟩ = 1 Z h ( s ) Π ^ s e − β H h / 2 ∣ ψ 0 σ ⟩ (35) |\varphi_h^\sigma(s)\rangle\equiv\frac{1}{\sqrt{P_s}}\hat{\Pi}_s|\varphi_h^\sigma\rangle
=\frac{1}{\sqrt{Z_h(s)}}\hat{\Pi}_s e^{-\beta H_h/2}|\psi_0^\sigma\rangle \tag*{(35)} ∣ φ h σ ( s )⟩ ≡ P s 1 Π ^ s ∣ φ h σ ⟩ = Z h ( s ) 1 Π ^ s e − β H h /2 ∣ ψ 0 σ ⟩ (35) with probability
P s = ⟨ φ h σ ∣ Π ^ s ∣ φ h σ ⟩ = Z h ( s ) Z h , (36) P_s=\langle\varphi_h^\sigma|\hat{\Pi}_s|\varphi_h^\sigma\rangle=\frac{Z_h(s)}{Z_h}, \tag*{(36)} P s = ⟨ φ h σ ∣ Π ^ s ∣ φ h σ ⟩ = Z h Z h ( s ) , (36) where we have defined the postmeasurement partition function
Z h ( s ) ≡ ⟨ ψ h 0 ∣ e − β H h / 2 Π ^ s e − β H h / 2 ∣ ψ h 0 ⟩ . (37) Z_h(s) \equiv\langle\psi_h^0|e^{-\beta H_h/2}\hat{\Pi}_s e^{-\beta H_h/2}|\psi_h^0\rangle. \tag*{(37)} Z h ( s ) ≡ ⟨ ψ h 0 ∣ e − β H h /2 Π ^ s e − β H h /2 ∣ ψ h 0 ⟩ . (37) Note that, since both H h H_h H h Π ^ s \hat{\Pi}_s Π ^ s X X X X μ X_\mu X μ Z h ( s ) Z_h(s) Z h ( s ) P s P_s P s σ \sigma σ ∣ φ h σ ⟩ |\varphi_h^\sigma\rangle ∣ φ h σ ⟩ ∣ φ h σ ( s ) ⟩ |\varphi_h^\sigma(s)\rangle ∣ φ h σ ( s )⟩ ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s ) ⟩ \langle\varphi_h^\sigma(s)|\varphi_h^{\sigma'}(s)\rangle ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s )⟩
∑ s P s ∣ ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s ) ⟩ ∣ = ∑ s P s [ ⟨ φ h σ ∣ Π ^ s ∣ φ h σ ′ ⟩ P s ] = ⟨ φ h σ ∣ φ h σ ′ ⟩ , (38) \sum_s P_s\left|\langle\varphi_h^\sigma(s)|\varphi_h^{\sigma'}(s)\rangle\right|
= \sum_s P_s\left[\frac{\langle\varphi_h^\sigma|\hat{\Pi}_s|\varphi_h^{\sigma'}\rangle}{P_s}\right]
= \langle\varphi_h^\sigma|\varphi_h^{\sigma'}\rangle, \tag*{(38)} s ∑ P s ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s )⟩ = s ∑ P s [ P s ⟨ φ h σ ∣ Π ^ s ∣ φ h σ ′ ⟩ ] = ⟨ φ h σ ∣ φ h σ ′ ⟩ , (38) where we have employed the resolution of identity ∑ s Π ^ s = 1 \sum_s\hat{\Pi}_s=1 ∑ s Π ^ s = 1 [95 ] , we obtain the result
P [ ∣ ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s ) ⟩ ∣ > ϵ ] ≤ ⟨ φ h σ ∣ φ h σ ′ ⟩ ϵ . (39) \mathbb{P}\left[\left|\langle\varphi_h^\sigma(s)|\varphi_h^{\sigma'}(s)\rangle\right|>\epsilon\right]\leq\frac{\langle\varphi_h^\sigma|\varphi_h^{\sigma'}\rangle}{\epsilon}. \tag*{(39)} P [ ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s )⟩ > ϵ ] ≤ ϵ ⟨ φ h σ ∣ φ h σ ′ ⟩ . (39) Since the right-hand side is O ( e − L / ξ ) O(e^{-L/\xi}) O ( e − L / ξ ) σ ≠ σ ′ \sigma\ne\sigma' σ = σ ′ (23) ], by choosing ϵ = O ( e − L / ξ ′ ) \epsilon=O(e^{-L/\xi'}) ϵ = O ( e − L / ξ ′ ) ξ ′ > ξ \xi'>\xi ξ ′ > ξ L L L
The preceding result has a simple physical interpretation within the quantum-classical mapping. Recall from Sec. III B that the overlaps ⟨ φ h σ ∣ φ h σ ′ ⟩ \langle\varphi_h^\sigma|\varphi_h^{\sigma'}\rangle ⟨ φ h σ ∣ φ h σ ′ ⟩ Z h σ ⊕ σ ′ Z_h^{\sigma\oplus\sigma'} Z h σ ⊕ σ ′ (12) ], which demand that m m m σ ≠ ( 0 , 0 ) \sigma\ne(0,0) σ = ( 0 , 0 ) h < h c h<h_c h < h c ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s ) ⟩ \langle\varphi_h^\sigma(s)|\varphi_h^{\sigma'}(s)\rangle ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s )⟩
Z h σ ⊕ σ ′ ( s ) = ⟨ ψ 0 σ ⊕ σ ′ ∣ e − β H h / 2 Π ^ s e − β H h / 2 ∣ ψ 0 σ ′ ⟩ , (40) Z_h^{\sigma\oplus\sigma'}(s)=\langle\psi_0^{\sigma\oplus\sigma'}|e^{-\beta H_h/2}\hat{\Pi}_s e^{-\beta H_h/2}|\psi_0^{\sigma'\!}\rangle, \tag*{(40)} Z h σ ⊕ σ ′ ( s ) = ⟨ ψ 0 σ ⊕ σ ′ ∣ e − β H h /2 Π ^ s e − β H h /2 ∣ ψ 0 σ ′ ⟩ , (40) which are independent of σ ′ \sigma' σ ′ X X X m m m τ = 0 \tau=0 τ = 0 τ = 0 \tau=0 τ = 0 p p p m m m s p = 1 s_p=1 s p = 1 Figure 5 . Eq. (39) therefore expresses the simple statement that, so long as system-spanning worldlines are exponentially rare in the unconditional ensemble, they must also be exponentially rare after conditioning on s s s s s s
Figure 5. (a) Schematic depiction of an anyon worldline configuration contributing to the postmeasurement partition function Z h ( s ) Z_h(s) Z h ( s ) (37) . Worldlines are required to pierce the τ = 0 \tau=0 τ = 0 m m m Z h σ ( s ) Z_h^{\sigma}(s) Z h σ ( s ) (40) , for σ ≠ ( 0 , 0 ) \sigma\ne(0,0) σ = ( 0 , 0 ) X X X s s s m m m Z h σ ( s ) Z_h^{\sigma}(s) Z h σ ( s ) Z h ( s ) Z_h(s) Z h ( s )
⟨ φ h σ ( s ) ∣ φ h σ ′ ( s ) ⟩ = Z h σ ⊕ σ ′ ( s ) Z h ( s ) = { 1 , σ = σ ′ , O ( e − L / ξ ′ ) , σ ≠ σ ′ , (41) \langle\varphi_h^\sigma(s)|\varphi_h^{\sigma'}(s)\rangle
=\frac{Z_h^{\sigma\oplus\sigma'}(s)}{Z_h(s)}
=\begin{cases}
1, & \sigma=\sigma',\\
O(e^{-L/\xi'}), & \sigma\ne\sigma',
\end{cases} \tag*{(41)} ⟨ φ h σ ( s ) ∣ φ h σ ′ ( s )⟩ = Z h ( s ) Z h σ ⊕ σ ′ ( s ) = { 1 , O ( e − L / ξ ′ ) , σ = σ ′ , σ = σ ′ , (41) where the latter expression holds with high probability.
In summary, we have demonstrated that both criteria 1 and 2 from the preceding subsection are satisfied with high probability, and therefore stabilizer measurements act approximately unitarily on an arbitrary ITE code state ∣ φ h ⟩ = ∑ σ b σ ∣ φ h σ ⟩ |\varphi_h\rangle=\sum_\sigma b_\sigma|\varphi_h^\sigma\rangle ∣ φ h ⟩ = ∑ σ b σ ∣ φ h σ ⟩ P s P_s P s s s s
⟨ φ h ∣ Π ^ s ∣ φ h ⟩ = Z h ( s ) Z h [ 1 + O ( e − L / ξ ′ ) ] , (42) \langle\varphi_h|\hat{\Pi}_s|\varphi_h\rangle
=\frac{Z_h(s)}{Z_h}\left[1+O(e^{-L/\xi'})\right], \tag*{(42)} ⟨ φ h ∣ Π ^ s ∣ φ h ⟩ = Z h Z h ( s ) [ 1 + O ( e − L / ξ ′ ) ] , (42) and the corresponding postmeasurement state is
∣ φ h ( s ) ⟩ = ∑ σ b σ ∣ φ h σ ( s ) ⟩ + O ( e − L / ξ ′ ) . (43) |\varphi_h(s)\rangle=\sum_\sigma b_\sigma|\varphi_h^\sigma(s)\rangle+O(e^{-L/\xi'}). \tag*{(43)} ∣ φ h ( s )⟩ = σ ∑ b σ ∣ φ h σ ( s )⟩ + O ( e − L / ξ ′ ) . (43) Our remaining goal is to determine an anyon pairing X s X_s X s ∣ φ h σ ( s ) ⟩ |\varphi_h^\sigma(s)\rangle ∣ φ h σ ( s )⟩ ∣ φ h σ ⟩ |\varphi_h^\sigma\rangle ∣ φ h σ ⟩
Optimal Postmeasurement Decoder
Having shown that stabilizer measurements preserve the encoded information with high fidelity for h < h c h<h_c h < h c X s X_s X s § 3.11 ) that, due to the logical-X X X ∣ φ h ( 0 , 0 ) ( s ) ⟩ |\varphi_h^{(0,0)}(s)\rangle ∣ φ h ( 0 , 0 ) ( s )⟩ A v A_v A v B p B_p B p + 1 +1 + 1 ( − 1 ) s p (-1)^{s_p} ( − 1 ) s p
∣ φ h ( 0 , 0 ) ( s ) ⟩ = X s ∑ σ c σ ( s ) ∣ ψ 0 σ ⟩ . (44) |\varphi_h^{(0,0)}(s)\rangle= X_s \sum_{\sigma} c_\sigma(s)|\psi_0^\sigma\rangle. \tag*{(44)} ∣ φ h ( 0 , 0 ) ( s )⟩ = X s σ ∑ c σ ( s ) ∣ ψ 0 σ ⟩ . (44) Here X s X_s X s X X X s s s ∣ ψ 0 σ ⟩ |\psi_0^\sigma\rangle ∣ ψ 0 σ ⟩ X s B p = ( − 1 ) s p B p X s X_sB_p=(-1)^{s_p}B_pX_s X s B p = ( − 1 ) s p B p X s X s X_s X s X s X_s X s A v A_v A v ∣ ψ 0 σ ⟩ |\psi_0^\sigma\rangle ∣ ψ 0 σ ⟩ X X X c σ ( s ) c_\sigma(s) c σ ( s ) X s X_s X s c ( 0 , 0 ) ( s ) c_{(0,0)}(s) c ( 0 , 0 ) ( s ) X s = X s X_s=X_s X s = X s
F s I T E = ∣ ⟨ ψ 0 ( 0 , 0 ) ∣ X s ∣ φ h ( 0 , 0 ) ( s ) ⟩ ∣ 2 = [ c ( 0 , 0 ) ( s ) ] 2 = max σ [ c σ ( s ) ] 2 . (45) \begin{aligned}
\mathcal{F}^{\mathrm{ITE}}_s=\left|\langle\psi_0^{(0,0)}|X_s|\varphi_h^{(0,0)}(s)\rangle\right|^2=[c_{(0,0)}(s)]^2 \\
&=\max_{\sigma}[c_\sigma(s)]^2. \tag*{(45)}
\end{aligned} F s ITE = ⟨ ψ 0 ( 0 , 0 ) ∣ X s ∣ φ h ( 0 , 0 ) ( s )⟩ 2 = [ c ( 0 , 0 ) ( s ) ] 2 = σ max [ c σ ( s ) ] 2 . (45) We now demonstrate that this optimal postmeasurement decoder achieves a high decoding fidelity with high probability. To this end, we write the four coefficients c σ ( s ) c_\sigma(s) c σ ( s )
c σ ( s ) = ⟨ ψ 0 σ ∣ X s ∣ φ h ( 0 , 0 ) ( s ) ⟩ = 1 Z h ( s ) ⟨ ψ 0 σ ∣ X s e − β H n / 2 ∣ ψ 0 ( 0 , 0 ) ⟩ . (46) c_\sigma(s)=\langle\psi_0^\sigma|X_s|\varphi_h^{(0,0)}(s)\rangle
=\frac{1}{\sqrt{Z_h(s)}}\langle\psi_0^\sigma|X_se^{-\beta H_n/2}|\psi_0^{(0,0)}\rangle. \tag*{(46)} c σ ( s ) = ⟨ ψ 0 σ ∣ X s ∣ φ h ( 0 , 0 ) ( s )⟩ = Z h ( s ) 1 ⟨ ψ 0 σ ∣ X s e − β H n /2 ∣ ψ 0 ( 0 , 0 ) ⟩ . (46) Notably, these coefficients are strictly non-negative due to stoquasticity. Furthermore, using logical-X X X
⟨ φ h σ ( s ) ∣ φ h ( 0 , 0 ) ( s ) ⟩ = ⟨ φ h ( 0 , 0 ) ( s ) ∣ X ‾ σ ∣ φ h ( 0 , 0 ) ( s ) ⟩ = ∑ σ ′ c σ ′ ( s ) c σ ⊕ σ ′ ( s ) . (47) \langle\varphi_h^\sigma(s)|\varphi_h^{(0,0)}(s)\rangle=\langle\varphi_h^{(0,0)}(s)|\overline{X}^\sigma|\varphi_h^{(0,0)}(s)\rangle
=\sum_{\sigma'}c_{\sigma'}(s)c_{\sigma\oplus\sigma'}(s). \tag*{(47)} ⟨ φ h σ ( s ) ∣ φ h ( 0 , 0 ) ( s )⟩ = ⟨ φ h ( 0 , 0 ) ( s ) ∣ X σ ∣ φ h ( 0 , 0 ) ( s )⟩ = σ ′ ∑ c σ ′ ( s ) c σ ⊕ σ ′ ( s ) . (47) Finally, the results of the preceding section imply that the left-hand side of Eq. (47) is exponentially small for σ ≠ ( 0 , 0 ) \sigma\ne(0,0) σ = ( 0 , 0 )
∑ σ ′ c σ ′ ( s ) c σ ⊕ σ ′ ( s ) = { 1 , σ = ( 0 , 0 ) O ( e − L / ξ ′ ) , σ ≠ ( 0 , 0 ) . (48) \sum_{\sigma'}c_{\sigma'}(s)c_{\sigma\oplus\sigma'}(s)=
\begin{cases}
1, & \sigma=(0,0) \\
\mathcal{O}(e^{-L/\xi'}), & \sigma\ne(0,0).
\end{cases} \tag*{(48)} σ ′ ∑ c σ ′ ( s ) c σ ⊕ σ ′ ( s ) = { 1 , O ( e − L / ξ ′ ) , σ = ( 0 , 0 ) σ = ( 0 , 0 ) . (48) Since each c σ ( s ) ≥ 0 c_\sigma(s)\ge0 c σ ( s ) ≥ 0 c ( 0 , 0 ) ( s ) c_{(0,0)}(s) c ( 0 , 0 ) ( s ) X s X_s X s
F s I T E = [ c ( 0 , 0 ) ( s ) ] 2 = 1 − O ( e − L / ξ ′ ) (49) \mathcal{F}^{\mathrm{ITE}}_s=[c_{(0,0)}(s)]^2=1-\mathcal{O}(e^{-L/\xi'}) \tag*{(49)} F s ITE = [ c ( 0 , 0 ) ( s ) ] 2 = 1 − O ( e − L / ξ ′ ) (49) with high probability.
We can once again construct a simple physical interpretation of this result through the quantum-classical mapping. By rewriting the ITE in Eq. (46) , we can express each amplitude c σ ( s ) c_\sigma(s) c σ ( s ) τ ∈ [ − β 2 , 0 ] \tau\in[-\frac{\beta}{2},0] τ ∈ [ − 2 β , 0 ] τ = 0 \tau=0 τ = 0 p p p s p = 1 s_p = 1 s p = 1 Figure 6 . The collection of all such spacetime worldline configurations L \mathcal{L} L L , L ′ \mathcal{L},\mathcal{L}^{\prime} L , L ′ X X X c σ ( s ) c_{\sigma}(s) c σ ( s ) h < h c h < h_c h < h c s s s X s X_s X s X s X_s X s τ = 0 \tau= 0 τ = 0 L \mathcal{L} L
Figure 6. Schematic depiction of the anyon worldline configurations contributing to the postmeasurement ITE states ∣ φ h ( s ) ⟩ |\varphi_h(s)\rangle ∣ φ h ( s )⟩ s s s m m m τ ∈ [ − β 2 , 0 ] \tau\in[-\frac{\beta}{2},0] τ ∈ [ − 2 β , 0 ] ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ X s X_s X s homologous to that of (a). Both worldlines induce the same value of the logical-Z Z Z ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ non-homologous to the preceding two. The net effect of this spacetime process, and the pairing X s X_s X s X X X ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ ∣ φ h ( s ) ⟩ |\varphi_h(s)\rangle ∣ φ h ( s )⟩ X s ∣ φ h ( s ) ⟩ X_s|\varphi_h(s)\rangle X s ∣ φ h ( s )⟩ ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩
In practice, to implement the optimal postmeasurement decoder, we must first numerically compute the amplitudes c σ ( s ) c_{\sigma}(s) c σ ( s ) X s X_s X s 16 c σ ( s ) c_{\sigma}(s) c σ ( s ) (11) that the partition function Z h Z_h Z h m m m L \mathcal{L} L § 3 A, the conditional partition function Z h ( s ) Z_h(s) Z h ( s ) (37) can be expressed as a statistical sum of anyon worldlines which are constrained by the observed syndrome s s s
Z h ( s ) ∝ ∑ L ∣ s e − E [ L ] . (50) Z_h(s) \propto\sum_{\mathcal{L}\mid s} e^{-E[\mathcal{L}]}. \tag*{(50)} Z h ( s ) ∝ L ∣ s ∑ e − E [ L ] . (50) Here the sum is performed over all anyon worldlines L \mathcal{L} L s s s τ = 0 \tau= 0 τ = 0 p p p s p = 1 s_p = 1 s p = 1 L \mathcal{L} L τ = ± β 2 \tau= \pm\frac{\beta}{2} τ = ± 2 β X X X τ ≤ 0 \tau\le0 τ ≤ 0 [ c σ ( s ) ] 2 [c_{\sigma}(s)]^2 [ c σ ( s ) ] 2 L \mathcal{L} L Z h ( s ) Z_h(s) Z h ( s )
Numerical Implementation of the Optimal Postmeasurement Decoder
We now demonstrate that the optimal postmeasurement decoder described in the preceding section can be efficiently implemented using Monte Carlo numerical simulations. Rather than sampling from the extremely anisotropic partition functions Z ~ h \widetilde{Z}_h Z h Z ~ h ( s ) \widetilde{Z}_h(s) Z h ( s ) isotropic Z 2 \mathbb{Z}_2 Z 2
S ~ L G T [ U ] = − 1 λ ∑ [ i j k l ] U i j U j k U k l U l i , (51) \widetilde{S}_{\mathrm{LGT}}[U] = -\frac{1}{\lambda}\sum_{[ijkl]} U_{ij}U_{jk}U_{kl}U_{li}, \tag*{(51)} S LGT [ U ] = − λ 1 [ ij k l ] ∑ U ij U j k U k l U l i , (51) and the (unconditional) partition function
Z ~ λ ≡ ∑ { U } e − S ~ L G T [ U ] \widetilde{Z}_{\lambda} \equiv\sum_{\{U\}} e^{-\widetilde{S}_{\mathrm{LGT}}[U]} Z λ ≡ { U } ∑ e − S LGT [ U ] ∝ ∑ L e − 2 ∣ L ∣ / λ , (52) \propto\sum_{\mathcal{L}} e^{-2|\mathcal{L}|/\lambda}, \tag*{(52)} ∝ L ∑ e − 2∣ L ∣/ λ , (52) where in the second line, each L \mathcal{L} L Z 2 \mathbb{Z}_2 Z 2 Z ~ h \widetilde{Z}_h Z h Z ~ h ( s ) \widetilde{Z}_h(s) Z h ( s ) x x x y y y τ \tau τ U i j U j k U k l U l i = + 1 U_{ij}U_{jk}U_{kl}U_{li} = +1 U ij U j k U k l U l i = + 1 τ = ± L τ / 2 \tau= \pm L_\tau/2 τ = ± L τ /2 L τ L_\tau L τ τ \tau τ τ = ± L τ / 2 \tau= \pm L_\tau/2 τ = ± L τ /2 ∏ ⟨ i j ⟩ ∈ γ U i j = 1 \prod_{\langle ij\rangle\in\gamma} U_{ij} = 1 ∏ ⟨ ij ⟩ ∈ γ U ij = 1 γ \gamma γ τ = ± β 2 \tau= \pm\frac{\beta}{2} τ = ± 2 β
The isotropic Z 2 \mathbb{Z}_2 Z 2 λ \lambda λ λ c ≈ 1.313 \lambda_c \approx1.313 λ c ≈ 1.313 3 d 3d 3 d β c I ≈ 0.221 \beta_c^{\mathrm{I}} \approx0.221 β c I ≈ 0.221 [98 ] via the Wegner duality relation tanh β c I = e − 2 / λ c \tanh\beta_c^{\mathrm{I}} = e^{-2/\lambda_c} tanh β c I = e − 2/ λ c [50 ] . While an extremely anisotropic limit (i.e., a continuous-time limit) is necessary for an exact description of the ITE states ∣ φ h ⟩ \lvert\varphi_h\rangle ∣ φ h ⟩ (21) , we expect that all of the universal properties near the critical point, as well as the qualitative features within the two phases, should be completely identical to the isotropic case17
To simulate the LGT, we utilize the worm algorithm (WA) [75 , 76 , 77 , 78 ] , which we describe in detail in Appendix H. The WA is a highly efficient nonlocal Monte Carlo update for statistical physics systems whose configuration space consists of closed loops, and has recently been utilized in other QEC contexts [42 , 79 , 99 , 100 ] . A crucial fact about the WA is that it naturally and efficiently tunnels between all four homology classes of worldlines, and can therefore be used to estimate their relative probabilities [79 ] from their observed relative frequencies.
Our numerical protocol first simulates the unconstrained 3 d 3d 3 d (51) in an L × L × L τ L \times L \times L_\tau L × L × L τ L \mathcal{L} L e − 2 ∣ L ∣ / λ e^{-2|\mathcal{L}|/\lambda} e − 2∣ L ∣/ λ L \mathcal{L} L s s s τ = 0 \tau= 0 τ = 0
Z ~ λ ( s ) ∝ ∑ L ∣ s e − 2 ∣ L ∣ / λ , (53) \widetilde{Z}_{\lambda}(s) \propto\sum_{\mathcal{L}|s} e^{-2|\mathcal{L}|/\lambda}, \tag*{(53)} Z λ ( s ) ∝ L ∣ s ∑ e − 2∣ L ∣/ λ , (53) where L ∣ s \mathcal{L}|s L ∣ s L \mathcal{L} L s s s τ < 0 \tau< 0 τ < 0 τ > 0 \tau> 0 τ > 0 τ = 0 \tau= 0 τ = 0 τ = 0 \tau= 0 τ = 0
As described in the preceding section, the flux configurations in the conditional ensemble for each s s s τ ≤ 0 \tau\leq0 τ ≤ 0 Figure 6 ). As we sample further flux loops from the conditional ensemble, we track their homology classes and obtain statistics for the relative frequency of each class. The per-syndrome decoding fidelity F s I s o \mathcal{F}^{\mathrm{Iso}}_s F s Iso
F s I s o ≡ max σ n σ ( s ) ∑ σ n σ ( s ) , (54) \mathcal{F}^{\mathrm{Iso}}_s \equiv\frac{\max_{\sigma} n_\sigma(s)}{\sum_{\sigma} n_\sigma(s)}, \tag*{(54)} F s Iso ≡ ∑ σ n σ ( s ) max σ n σ ( s ) , (54) where n σ ( s ) n_\sigma(s) n σ ( s ) σ \sigma σ n σ ( s ) / ∑ σ ′ n σ ′ ( s ) n_\sigma(s)/\sum_{\sigma'} n_{\sigma'}(s) n σ ( s ) / ∑ σ ′ n σ ′ ( s ) [ c σ ( s ) ] 2 [c_\sigma(s)]^2 [ c σ ( s ) ] 2 F ‾ I s o \overline{\mathcal{F}}^{\mathrm{Iso}} F Iso
Figure 7 depicts the average decoding fidelity F ‾ I s o \overline{\mathcal{F}}^{\mathrm{Iso}} F Iso λ \lambda λ s s s λ < λ c \lambda< \lambda_c λ < λ c F ‾ I s o \overline{\mathcal{F}}^{\mathrm{Iso}} F Iso 0.25 0.25 0.25 λ > λ c \lambda> \lambda_c λ > λ c ∣ φ η σ ⟩ \lvert\varphi^\sigma_\eta\rangle ∣ φ η σ ⟩ F ‾ I s o \overline{\mathcal{F}}^{\mathrm{Iso}} F Iso λ \lambda λ F ‾ I s o \overline{\mathcal{F}}^{\mathrm{Iso}} F Iso 0.25 0.25 0.25 probabilistic likelihood decoder [71 , 72 ] in place of our “maximum likelihood” decoder.
Figure 7. Average optimal postmeasurement decoding fidelity F ‾ I s o \overline{\mathcal{F}}^{\mathrm{Iso}} F Iso Z 2 \mathbb{Z}_2 Z 2 (51) ], for several linear system sizes L L L λ < λ c \lambda< \lambda_c λ < λ c F ‾ I s o \overline{\mathcal{F}}^{\mathrm{Iso}} F Iso 3 d 3d 3 d ν ≈ 0.630 \nu\approx0.630 ν ≈ 0.630 [98 ] .
Near the critical point, we find that the onset of decoding failures is well-described by the finite-size scaling ansatz
F ‾ I s o = Φ ( ( λ − λ c ) L 1 / ν ) , (55) \overline{\mathcal{F}}^{\mathrm{Iso}} = \Phi\bigl((\lambda-\lambda_c)L^{1/\nu}\bigr), \tag*{(55)} F Iso = Φ ( ( λ − λ c ) L 1/ ν ) , (55) where ν ≈ 0.630 \nu\approx0.630 ν ≈ 0.630 3 d 3d 3 d 3 d 3d 3 d [98 ] , and Φ \Phi Φ Figure 7 inset.
In practice, any implementation of the optimal postmeasurement decoder crucially relies on knowledge of the transverse field strength h h h λ \lambda λ [7 , 35 ] , can successfully recover the encoded information at least for sufficiently small h h h h < h c h < h_c h < h c Figure 17 (a)]. We are not aware of any theoretical reason that MWPM should perform as well as the optimal postmeasurement decoder in this p = 0 p = 0 p = 0 § 3.2 C, where we introduce bit-flip decoherence on top of the TFTC ground states, we will see numerically that MWPM exhibits a slightly smaller bit-flip decoding threshold [ p c M W P M ( λ ) < p c ⋆ ( λ ) ] [p_c^{\mathrm{MWPM}}(\lambda) < p_c^\star(\lambda)] [ p c MWPM ( λ ) < p c ⋆ ( λ )]
Figure 17. MWPM decoding fidelity F ‾ M W P M \overline{F}^{\mathrm{MWPM}} F MWPM p p p λ \lambda λ L L L p = 0 p = 0 p = 0 λ < λ c \lambda< \lambda_c λ < λ c p = 0.08 p = 0.08 p = 0.08 F ‾ M W P M \overline{F}^{\mathrm{MWPM}} F MWPM λ < λ c \lambda< \lambda_c λ < λ c p = 0.1 p = 0.1 p = 0.1 λ c ( p = 0.1 ) \lambda_c(p = 0.1) λ c ( p = 0.1 ) 0.98 λ c 0.98\lambda_c 0.98 λ c
In summary, we have shown that quantum information in the ground-state manifold of the TFTC is preserved under stabilizer measurements with high fidelity, with high probability in the syndrome measurement. Motivated by this result, we now investigate the utility of stabilizer measurements in decoding TFTC states subjected to Pauli decoherence.
QEC of Pauli Decoherence via Stabilizer Measurements
Let us again suppose that two qubits of quantum information are encoded in the ITE state ∣ ϕ h ⟩ ∝ e − β H h / 2 ∣ ψ 0 ⟩ \lvert\phi_h\rangle\propto e^{-\beta H_h/2}\lvert\psi_0\rangle ∣ ϕ h ⟩ ∝ e − β H h /2 ∣ ψ 0 ⟩ ℓ \ell ℓ (13) , resulting in the decohered state ρ h ≡ E ( ∣ ϕ h ⟩ ⟨ ϕ h ∣ ) \rho_h \equiv\mathcal{E}(\lvert\phi_h\rangle\langle\phi_h\rvert) ρ h ≡ E (∣ ϕ h ⟩ ⟨ ϕ h ∣) ∣ ϕ h ⟩ \lvert\phi_h\rangle ∣ ϕ h ⟩ A v , B p A_v,B_p A v , B p
ρ h ( s ) ≡ Π ^ s ρ h Π ^ s P s (56) \rho_h(s) \equiv\frac{\widehat{\Pi}_s\rho_h\widehat{\Pi}_s}{P_s} \tag*{(56)} ρ h ( s ) ≡ P s Π s ρ h Π s (56) with probability
P s ≡ tr [ Π ^ s ρ h ] . (57) P_s \equiv\operatorname{tr}[\widehat{\Pi}_s\rho_h]. \tag*{(57)} P s ≡ tr [ Π s ρ h ] . (57) Similar to § 3 , we seek to determine a Pauli correction X s Z s X_sZ_s X s Z s s s s ∣ ψ 0 ⟩ \lvert\psi_0\rangle ∣ ψ 0 ⟩ ρ h ( s ) \rho_h(s) ρ h ( s )
As a first hint that stabilizer measurements can serve as an effective QEC strategy, we will show in § 3.2 A that the probability P s P_s P s s s s ∣ ψ 0 ⟩ \lvert\psi_0\rangle ∣ ψ 0 ⟩ § 3.2 B, we derive the optimal postmeasurement decoder for the bit-flip decohered ITE states. We then numerically implement (a spacetime isotropic version of) this decoder in § 3.2 C, demonstrating explicitly that this decoder exhibits a nonzero threshold even as the TFTC’s critical point is approached. In § 3.2 D we reinstate phase-flip decoherence, where we prove that the phase-flip threshold of the TFTC is identical to that of the stabilizer toric code. Finally, in § 3.2 E we will argue (without a fully rigorous proof) that stabilizer measurements are asymptotically optimal for decoding bit-flip decoherence—that is, the decoding threshold for the optimal postmeasurement decoder coincides with the “intrinsic” error threshold for bit-flip decoherence.
To demonstrate that the syndrome probabilities (57) are approximately independent of ∣ ϕ h ⟩ \lvert\phi_h\rangle ∣ ϕ h ⟩ ∣ ϕ h ⟩ = ∣ ϕ h σ ⟩ \lvert\phi_h\rangle=\lvert\phi_h^\sigma\rangle ∣ ϕ h ⟩ = ∣ ϕ h σ ⟩ Z Z Z s s s
P s σ ≡ tr [ Π ^ s E ( ∣ ϕ h σ ⟩ ⟨ ϕ h σ ∣ ) ] = Z h , p σ ( s ) Z h . (58) P_s^\sigma\equiv\operatorname{tr}[\widehat{\Pi}_s\mathcal{E}(\lvert\phi_h^\sigma\rangle\langle\phi_h^\sigma\rvert)] = \frac{Z_{h,p}^\sigma(s)}{Z_h}. \tag*{(58)} P s σ ≡ tr [ Π s E (∣ ϕ h σ ⟩ ⟨ ϕ h σ ∣)] = Z h Z h , p σ ( s ) . (58) Similar to § 3 A, we have defined the postmeasurement partition function
Z h , p σ ( s ) ≡ ⟨ ψ 0 σ ∣ e − β H h / 2 E ( Π ^ s ) e − β H h / 2 ∣ ψ 0 σ ⟩ , (59) Z_{h,p}^\sigma(s) \equiv\langle\psi_0^\sigma\lvert e^{-\beta H_h/2}\mathcal{E}(\widehat{\Pi}_s)e^{-\beta H_h/2}\rvert\psi_0^\sigma\rangle, \tag*{(59)} Z h , p σ ( s ) ≡ ⟨ ψ 0 σ ∣ e − β H h /2 E ( Π s ) e − β H h /2 ∣ ψ 0 σ ⟩ , (59) where we have used the fact that the channel E \mathcal{E} E P s σ P_s^\sigma P s σ σ \sigma σ X X X
For a more general state ∣ ϕ h ⟩ = ∑ σ b σ ∣ ϕ h σ ⟩ \lvert\phi_h\rangle=\sum_\sigma b_\sigma\lvert\phi_h^\sigma\rangle ∣ ϕ h ⟩ = ∑ σ b σ ∣ ϕ h σ ⟩ P s P_s P s
tr [ Π ^ s E ( ∣ ϕ h σ ⟩ ⟨ ϕ h σ ′ ∣ ) ] = ⟨ ψ 0 σ ′ ∣ e − β H h / 2 E ( Π ^ s ) e − β H h / 2 ∣ ψ 0 σ ⟩ Z h = Z h , p σ ⊕ σ ′ ( s ) Z h , (60) \operatorname{tr}[\widehat{\Pi}_s\mathcal{E}(\lvert\phi_h^\sigma\rangle\langle\phi_h^{\sigma'}\rvert)] = \frac{\langle\psi_0^{\sigma'}\lvert e^{-\beta H_h/2}\mathcal{E}(\widehat{\Pi}_s)e^{-\beta H_h/2}\rvert\psi_0^\sigma\rangle}{Z_h}
= \frac{Z_{h,p}^{\sigma\oplus\sigma'}(s)}{Z_h}, \tag*{(60)} tr [ Π s E (∣ ϕ h σ ⟩ ⟨ ϕ h σ ′ ∣)] = Z h ⟨ ψ 0 σ ′ ∣ e − β H h /2 E ( Π s ) e − β H h /2 ∣ ψ 0 σ ⟩ = Z h Z h , p σ ⊕ σ ′ ( s ) , (60) where we have defined the twisted partition functions
Z h , p σ ⊕ σ ′ ( s ) ≡ ⟨ ψ 0 σ ′ ∣ e − β H h / 2 E ( Π ^ s ) e − β H h / 2 ∣ ψ 0 σ ⟩ (61) Z_{h,p}^{\sigma\oplus\sigma'}(s) \equiv\langle\psi_0^{\sigma'}\lvert e^{-\beta H_h/2}\mathcal{E}(\widehat{\Pi}_s)e^{-\beta H_h/2}\rvert\psi_0^\sigma\rangle \tag*{(61)} Z h , p σ ⊕ σ ′ ( s ) ≡ ⟨ ψ 0 σ ′ ∣ e − β H h /2 E ( Π s ) e − β H h /2 ∣ ψ 0 σ ⟩ (61) which are independent of σ ′ \sigma' σ ′ Z h , p σ ( s ) Z_{h,p}^{\sigma}(s) Z h , p σ ( s )
∑ s Z h , p σ ( s ) = ⟨ ψ 0 σ ⊕ σ ′ ∣ e − β H h / 2 E ( 1 ) e − β H h / 2 ∣ ψ 0 σ ′ ⟩ = Z h σ , (62) \sum_s Z_{h,p}^{\sigma}(s) = \langle\psi_0^{\sigma\oplus\sigma'}\lvert e^{-\beta H_h/2}\mathcal{E}(1)e^{-\beta H_h/2}\rvert\psi_0^{\sigma'}\rangle
= Z_h^\sigma, \tag*{(62)} s ∑ Z h , p σ ( s ) = ⟨ ψ 0 σ ⊕ σ ′ ∣ e − β H h /2 E ( 1 ) e − β H h /2 ∣ ψ 0 σ ′ ⟩ = Z h σ , (62) where Z h σ Z_h^\sigma Z h σ (12) . We therefore find
∑ s P s σ ∣ Z h , p σ ′ ( s ) Z h , p σ ( s ) ∣ = ∑ s Z h , p σ ( s ) Z h Z h , p σ ′ ( s ) Z h , p σ ( s ) = Z h σ ′ Z h . (63) \sum_s P_s^\sigma\left|\frac{Z_{h,p}^{\sigma'}(s)}{Z_{h,p}^{\sigma}(s)}\right| = \sum_s \frac{Z_{h,p}^\sigma(s)}{Z_h}\frac{Z_{h,p}^{\sigma'}(s)}{Z_{h,p}^\sigma(s)}
= \frac{Z_h^{\sigma'}}{Z_h}. \tag*{(63)} s ∑ P s σ Z h , p σ ( s ) Z h , p σ ′ ( s ) = s ∑ Z h Z h , p σ ( s ) Z h , p σ ( s ) Z h , p σ ′ ( s ) = Z h Z h σ ′ . (63) Recall from Eq. (12) that the final expression is exponentially small for σ ≠ ( 0 , 0 ) \boldsymbol{\sigma}\ne(0,0) σ = ( 0 , 0 )
Z h , p σ ( s ) Z h , p ( s ) = { 1 , σ = ( 0 , 0 ) O ( e − L / ξ ′ ) , σ ≠ ( 0 , 0 ) (64) \frac{Z_{h,p}^{\sigma}(s)}{Z_{h,p}(s)} =
\begin{cases}
1, & \boldsymbol{\sigma}=(0,0)\\
O(e^{-L/\xi'}), & \boldsymbol{\sigma}\ne(0,0)
\end{cases} \tag*{(64)} Z h , p ( s ) Z h , p σ ( s ) = { 1 , O ( e − L / ξ ′ ) , σ = ( 0 , 0 ) σ = ( 0 , 0 ) (64) with probability exponentially close to unity. Therefore, for an arbitrary ITE state ∣ ϕ h ⟩ \lvert\phi_h\rangle ∣ ϕ h ⟩ (60) to the syndrome probability P s P_s P s
P s = Z h , p ( s ) Z h [ 1 + O ( e − L / ξ ′ ) ] (65) P_s = \frac{Z_{h,p}(s)}{Z_h}\left[1+O(e^{-L/\xi'})\right] \tag*{(65)} P s = Z h Z h , p ( s ) [ 1 + O ( e − L / ξ ′ ) ] (65) with high probability. In other words, stabilizer measurements reveal vanishingly little information encoded in the decohered states ρ h \rho_h ρ h
Physically, we can once again regard each Z h , p σ ( s ) Z_{h,p}^\sigma(s) Z h , p σ ( s ) X ‾ σ \overline{X}^{\sigma} X σ Z h , p σ ( s ) Z_{h,p}^{\sigma}(s) Z h , p σ ( s ) (40) , where the worldlines were constrained to rigidly agree with the observed syndrome s s s τ = 0 \tau=0 τ = 0 E \mathcal{E} E Figure 8 for the case with p Z = 0 p_Z=0 p Z = 0 s s s (64) .
Figure 8. (a) Schematic depiction of an anyon worldline and bit-flip error configuration contributing to the postmeasurement partition function Z h , p ( s ) Z_{h,p}(s) Z h , p ( s ) (59) . Contractible anyon worldlines (blue) piercing the τ = 0 \tau=0 τ = 0 s s s Z h , p σ ( s ) Z_{h,p}^{\sigma}(s) Z h , p σ ( s ) (61) , for σ ≠ ( 0 , 0 ) \sigma\ne(0,0) σ = ( 0 , 0 ) X X X s s s Z h , p σ ( s ) Z_{h,p}^{\sigma}(s) Z h , p σ ( s ) Z h , p ( s ) Z_{h,p}(s) Z h , p ( s )
Bit-Flip Decoherence: Optimal Decoding of Stabilizer Measurements
We now move onto the construction of the optimal postmeasurement decoder for the ITE states. We begin with the case of bit-flip decoherence, where the transverse field of H h H_h H h m m m § 3.2 D that it is sufficient to decode bit-flips and phase-flips independently, and so here we set p Z = 0 p_Z=0 p Z = 0
To construct the optimal postmeasurement decoder in this setting, it will be convenient to unravel ρ h ≡ E ( ∣ φ h ⟩ ⟨ φ h ∣ ) \rho_h\equiv\mathcal{E}(|\varphi_h\rangle\langle\varphi_h|) ρ h ≡ E ( ∣ φ h ⟩ ⟨ φ h ∣ ) e e e
ρ h = E ( ∣ φ h ⟩ ⟨ φ h ∣ ) = ∑ e p e X e ∣ φ h ⟩ ⟨ φ h ∣ X e . (66) \rho_h=\mathcal{E}(|\varphi_h\rangle\langle\varphi_h|)=\sum_e p_e X_e|\varphi_h\rangle\langle\varphi_h|X_e. \tag*{(66)} ρ h = E ( ∣ φ h ⟩ ⟨ φ h ∣ ) = e ∑ p e X e ∣ φ h ⟩ ⟨ φ h ∣ X e . (66) Here e ≡ { e ℓ } e\equiv\{e_\ell\} e ≡ { e ℓ } e ℓ = 0 , 1 e_\ell=0,1 e ℓ = 0 , 1 ℓ \ell ℓ e ℓ = 1 e_\ell=1 e ℓ = 1 X e ≡ ∏ ℓ X ℓ e ℓ X_e\equiv\prod_\ell X_\ell^{e_\ell} X e ≡ ∏ ℓ X ℓ e ℓ p e ≡ ( 1 − p ) 2 L 2 − ∣ e ∣ p ∣ e ∣ p_e\equiv(1-p)^{2L^2-|e|}p^{|e|} p e ≡ ( 1 − p ) 2 L 2 − ∣ e ∣ p ∣ e ∣
Notably, the bit-flip channel E \mathcal{E} E strong Z 2 \mathbb{Z}_2 Z 2 X X X X ‾ μ \overline{X}_\mu X μ X e X_e X e X ‾ μ \overline{X}_\mu X μ Π ^ s \hat{\Pi}_s Π ^ s m m m X s X_s X s X ‾ μ \overline{X}_\mu X μ § 3.11 , this observation implies that it is sufficient to focus on decoding the specific stabilizer code state ∣ ψ 0 ( 0 , 0 ) ⟩ |\psi_0^{(0,0)}\rangle ∣ ψ 0 ( 0 , 0 ) ⟩ ∣ ψ 0 ( 0 , 0 ) ⟩ |\psi_0^{(0,0)}\rangle ∣ ψ 0 ( 0 , 0 ) ⟩ ( 0 , 0 ) (0,0) ( 0 , 0 ) ∣ φ h ⟩ ≡ ∣ φ h ( 0 , 0 ) ⟩ |\varphi_h\rangle\equiv|\varphi_h^{(0,0)}\rangle ∣ φ h ⟩ ≡ ∣ φ h ( 0 , 0 ) ⟩ ∣ ψ 0 ⟩ ≡ ∣ ψ 0 ( 0 , 0 ) ⟩ |\psi_0\rangle\equiv|\psi_0^{(0,0)}\rangle ∣ ψ 0 ⟩ ≡ ∣ ψ 0 ( 0 , 0 ) ⟩
Using Eq. (22) , the decohered state ρ h = E ( ∣ φ h ⟩ ⟨ φ h ∣ ) \rho_h=\mathcal{E}(|\varphi_h\rangle\langle\varphi_h|) ρ h = E ( ∣ φ h ⟩ ⟨ φ h ∣ )
ρ h = 1 Z h E ( e − β H h / 2 ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ e − β H h / 2 ) = 1 Z h ∑ e p e X e e − β H h / 2 ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ e − β H h / 2 X e . (67) \begin{aligned}
\rho_h=\frac{1}{Z_h}\mathcal{E}\left(e^{-\beta H_h/2}|\psi_0\rangle\langle\psi_0|e^{-\beta H_h/2}\right) \\
&=\frac{1}{Z_h}\sum_e p_eX_e e^{-\beta H_h/2}|\psi_0\rangle\langle\psi_0|e^{-\beta H_h/2}X_e. \tag*{(67)}
\end{aligned} ρ h = Z h 1 E ( e − β H h /2 ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ e − β H h /2 ) = Z h 1 e ∑ p e X e e − β H h /2 ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ e − β H h /2 X e . (67) We now measure the stabilizers B p B_p B p s s s § 3 , Λ v = + 1 \Lambda_v=+1 Λ v = + 1 e e e ρ h ( s ) \rho_h(s) ρ h ( s ) A v A_v A v B p B_p B p + 1 +1 + 1 ( − 1 ) s p (-1)^{s_p} ( − 1 ) s p 4 × 4 4\times4 4 × 4 M σ σ ′ ( s ) M_{\sigma\sigma'}(s) M σ σ ′ ( s )
ρ h ( s ) = Π ^ s ρ h Π ^ s P s = X s ( ∑ σ σ ′ M σ σ ′ ( s ) ∣ ψ 0 σ ⟩ ⟨ ψ 0 σ ′ ∣ ) X s . (68) \begin{aligned}
\rho_h(s)=\frac{\hat{\Pi}_s\rho_h\hat{\Pi}_s}{P_s} \\
&=X_s\left(\sum_{\sigma\sigma'}M_{\sigma\sigma'}(s)|\psi_0^\sigma\rangle\langle\psi_0^{\sigma'}|\right)X_s. \tag*{(68)}
\end{aligned} ρ h ( s ) = P s Π ^ s ρ h Π ^ s = X s ( σ σ ′ ∑ M σ σ ′ ( s ) ∣ ψ 0 σ ⟩ ⟨ ψ 0 σ ′ ∣ ) X s . (68) As in § 3 B, X s X_s X s X X X s s s ∣ ψ 0 σ ⟩ |\psi_0^\sigma\rangle ∣ ψ 0 σ ⟩ M σ σ ′ ( s ) M_{\sigma\sigma'}(s) M σ σ ′ ( s ) ρ h ( s ) \rho_h(s) ρ h ( s )
M σ σ ′ ( s ) ≡ ⟨ ψ 0 σ ∣ X s ρ h ( s ) X s ∣ ψ 0 σ ′ ⟩ = ⟨ ψ 0 ∣ e − β H h / 2 E ( X s ∣ ψ 0 σ ′ ⟩ ⟨ ψ 0 σ ∣ X s ) e − β H h / 2 ∣ ψ 0 ⟩ Z h , p ( s ) . (69) \begin{aligned}
M_{\sigma\sigma'}(s)\equiv\langle\psi_0^{\sigma}|X_s\rho_h(s)X_s|\psi_0^{\sigma'}\rangle\\
&=\frac{\langle\psi_0|e^{-\beta H_h/2}\mathcal{E}(X_s|\psi_0^{\sigma'}\rangle\langle\psi_0^{\sigma}|X_s)e^{-\beta H_h/2}|\psi_0\rangle}{Z_{h,p}(s)}. \tag*{(69)}
\end{aligned} M σ σ ′ ( s ) ≡ ⟨ ψ 0 σ ∣ X s ρ h ( s ) X s ∣ ψ 0 σ ′ ⟩ = Z h , p ( s ) ⟨ ψ 0 ∣ e − β H h /2 E ( X s ∣ ψ 0 σ ′ ⟩ ⟨ ψ 0 σ ∣ X s ) e − β H h /2 ∣ ψ 0 ⟩ . (69) Using ∑ σ X s ∣ ψ 0 σ ⟩ ⟨ ψ 0 σ ∣ X s = Π ^ s \sum_\sigma X_s|\psi_0^\sigma\rangle\langle\psi_0^\sigma|X_s=\hat{\Pi}_s ∑ σ X s ∣ ψ 0 σ ⟩ ⟨ ψ 0 σ ∣ X s = Π ^ s
∑ σ ′ M σ ′ , σ ⊕ σ ′ ( s ) = Z h , p σ ( s ) Z h , p ( s ) . (70) \sum_{\sigma'}M_{\sigma'\!,\sigma\oplus\sigma'}(s)=\frac{Z_{h,p}^{\sigma}(s)}{Z_{h,p}(s)}. \tag*{(70)} σ ′ ∑ M σ ′ , σ ⊕ σ ′ ( s ) = Z h , p ( s ) Z h , p σ ( s ) . (70) where we have defined both the postmeasurement partition function Z h , p ( s ) Z_{h,p}(s) Z h , p ( s ) Z h , p σ ( s ) Z_{h,p}^{\sigma}(s) Z h , p σ ( s ) (59) and (61) respectively. Recalling from Eq. (64) that the right-hand side is exponentially small for σ ≠ ( 0 , 0 ) \sigma\ne(0,0) σ = ( 0 , 0 )
∑ σ M σ σ ( s ) = 1 , (71) \sum_{\sigma} M_{\sigma\sigma}(s)=1, \tag*{(71)} σ ∑ M σ σ ( s ) = 1 , (71) M σ σ ′ ( s ) = O ( e − L / ξ ′ ) ( σ ≠ σ ′ ) . (72) M_{\sigma\sigma'}(s)=O(e^{-L/\xi'})\quad(\sigma\ne\sigma'). \tag*{(72)} M σ σ ′ ( s ) = O ( e − L / ξ ′ ) ( σ = σ ′ ) . (72) The latter equation, which holds with high probability, follows from noting once more that M σ σ ′ ( s ) ≥ 0 M_{\sigma\sigma'}(s)\ge0 M σ σ ′ ( s ) ≥ 0 (70) for σ ≠ ( 0 , 0 ) \sigma\ne(0,0) σ = ( 0 , 0 )
An immediate useful observation from the preceding discussion is that, whereas ρ h \rho_h ρ h ρ h ( s ) \rho_h(s) ρ h ( s ) ρ h ( s ) \rho_h(s) ρ h ( s ) ∣ ψ 0 ⟩ \lvert\psi_0\rangle ∣ ψ 0 ⟩
M s ( ρ ) = ∑ σ M σ σ ( s ) X s X σ ρ X ˉ σ X s , (73) \mathcal{M}_s(\rho)=\sum_{\sigma}M_{\sigma\sigma}(s)X_sX^\sigma\rho\bar{X}^\sigma X_s, \tag*{(73)} M s ( ρ ) = σ ∑ M σ σ ( s ) X s X σ ρ X ˉ σ X s , (73) where M σ σ ( s ) M_{\sigma\sigma}(s) M σ σ ( s ) X s X σ X_sX^\sigma X s X σ M σ σ ′ ( s ) M_{\sigma\sigma'}(s) M σ σ ′ ( s ) X s X_s X s M σ σ ( s ) M_{\sigma\sigma}(s) M σ σ ( s ) σ = ( 0 , 0 ) \sigma=(0,0) σ = ( 0 , 0 ) X s = X s X_s=\mathcal{X}_s X s = X s
F s I T E ( ∣ ψ 0 ⟩ ) = ⟨ ψ 0 ∣ X s ρ h ( s ) X s ∣ ψ 0 ⟩ = M ( 0 , 0 ) , ( 0 , 0 ) ( s ) . (74) \mathcal{F}^{\mathrm{ITE}}_s(\lvert\psi_0\rangle)=\langle\psi_0\vert X_s\rho_h(s)X_s\lvert\psi_0\rangle
=M_{(0,0),(0,0)}(s). \tag*{(74)} F s ITE (∣ ψ 0 ⟩) = ⟨ ψ 0 ∣ X s ρ h ( s ) X s ∣ ψ 0 ⟩ = M ( 0 , 0 ) , ( 0 , 0 ) ( s ) . (74) On the other hand, unlike in the p = 0 p=0 p = 0 § 3 , stoquasticity has not guaranteed that ∣ ψ 0 ⟩ \lvert\psi_0\rangle ∣ ψ 0 ⟩ ρ h ( s ) \rho_h(s) ρ h ( s ) p p p M ( 0 , 0 ) , ( 0 , 0 ) ( s ) M_{(0,0),(0,0)}(s) M ( 0 , 0 ) , ( 0 , 0 ) ( s )
We can once again construct a physical interpretation for the diagonal coefficients M σ σ ( s ) M_{\sigma\sigma}(s) M σ σ ( s ) Z h , p ( s ) \mathcal{Z}_{h,p}(s) Z h , p ( s ) Π ^ s \hat{\Pi}_s Π ^ s ∑ s Z h , p ( s ) = Z h \sum_s\mathcal{Z}_{h,p}(s)=\mathcal{Z}_h ∑ s Z h , p ( s ) = Z h m m m L \mathcal{L} L e e e e − E [ L ] e^{-E[\mathcal{L}]} e − E [ L ] (11) ], while the latter are weighted by the probabilities p e ∝ [ p / ( 1 − p ) ] ∣ e ∣ p_e\propto[p/(1-p)]^{|e|} p e ∝ [ p / ( 1 − p ) ] ∣ e ∣ Π ^ s \hat{\Pi}_s Π ^ s Z h , p ( s ) \mathcal{Z}_{h,p}(s) Z h , p ( s ) ( L , e ) (\mathcal{L},e) ( L , e ) Z h , p ( s ) \mathcal{Z}_{h,p}(s) Z h , p ( s ) s L s_{\mathcal{L}} s L τ = 0 \tau=0 τ = 0 s e ≡ ∂ e s_e\equiv\partial e s e ≡ ∂ e s s s
Z h , p ( s ) ∝ ∑ L , e p e e − E [ L ] δ ( s L ⊕ s e = s ) = ∑ L , e ∣ s p e e − E [ L ] . (75) \begin{aligned}
\mathcal{Z}_{h,p}(s)\propto\sum_{\mathcal{L},e}p_e e^{-E[\mathcal{L}]}\delta(s_{\mathcal{L}}\oplus s_e=s) \\
&=\sum_{\mathcal{L},e\mid s}p_e e^{-E[\mathcal{L}]}. \tag*{(75)}
\end{aligned} Z h , p ( s ) ∝ L , e ∑ p e e − E [ L ] δ ( s L ⊕ s e = s ) = L , e ∣ s ∑ p e e − E [ L ] . (75) where in the second line, the sum is performed over only the pairs ( L , e ) (\mathcal{L},e) ( L , e ) s s s τ = 0 \tau=0 τ = 0 τ = 0 \tau=0 τ = 0 Figure 8 ).
As in § 3 B, the collection of all anyon worldlines L \mathcal{L} L s L s_{\mathcal{L}} s L τ ∈ [ − β / 2 , 0 ] \tau\in[-\beta/2,0] τ ∈ [ − β /2 , 0 ] e e e e , e ′ e,e' e , e ′ X e X_e X e X e ′ X_{e'} X e ′ A v A_v A v ( L , e ) (\mathcal{L},e) ( L , e ) joint homology classes: two pairs ( L , e ) (\mathcal{L},e) ( L , e ) ( L ′ , e ′ ) (\mathcal{L}',e') ( L ′ , e ′ ) τ = 0 \tau=0 τ = 0 Figure 9 . By inspecting the second line of Eq. (69) , we see that the diagonal matrix elements M σ σ ( s ) M_{\sigma\sigma}(s) M σ σ ( s )
Figure 9. Schematic depiction of worldline and error configurations contributing to the postmeasurement decohered ITE states ρ h ( s ) \rho_h(s) ρ h ( s ) τ ≤ 0 \tau\le0 τ ≤ 0 L \mathcal{L} L τ < 0 \tau< 0 τ < 0 e e e s s s m m m ( L , e ) (\mathcal{L},e) ( L , e ) ∣ ψ 0 ⟩ \lvert\psi_0\rangle ∣ ψ 0 ⟩ X s X_s X s L \mathcal{L} L e e e ( L , e ) (\mathcal{L},e) ( L , e ) ( L , e ) (\mathcal{L},e) ( L , e ) L \mathcal{L} L e e e X s X_s X s X X X ∣ ψ 0 ⟩ \lvert\psi_0\rangle ∣ ψ 0 ⟩ ( L , e ) (\mathcal{L},e) ( L , e )
To implement the optimal postmeasurement decoder in practice, we must numerically compute each diagonal coefficient M σ σ ( s ) M_{\sigma\sigma}(s) M σ σ ( s ) § 3 C, these coefficients can be computed efficiently using Monte Carlo simulations of the constrained statistical physics model described by the partition function Z h , p ( s ) Z_{h,p}(s) Z h , p ( s )
Numerical Implementation of Optimal Postmeasurement Decoder for Bit-Flip Decoherence
We now numerically investigate the optimal postmeasurement decoder derived in the preceding section. We will once again utilize an isotropic Z 2 \mathbb{Z}_2 Z 2 ∣ φ h ⟩ \lvert\varphi_h\rangle ∣ φ h ⟩
To sample a syndrome s s s L \mathcal{L} L e − 2 ∣ L ∣ / λ e^{-2\lvert\mathcal{L}\rvert/\lambda} e − 2 ∣ L ∣ / λ e e e p e = ( 1 − p ) 2 L 2 − ∣ e ∣ p ∣ e ∣ p_e=(1-p)^{2L^2-\lvert e\rvert}p^{\lvert e\rvert} p e = ( 1 − p ) 2 L 2 − ∣ e ∣ p ∣ e ∣ s L s_{\mathcal{L}} s L s e s_e s e s = s L ⊕ s e s=s_{\mathcal{L}}\oplus s_e s = s L ⊕ s e
P ~ s ≡ Z ~ λ , p ( s ) Z ~ λ , Z ~ λ , p ( s ) ∝ ∑ L , e ∣ s p e e − 2 ∣ L ∣ / λ . (76) \widetilde{P}_s \equiv\frac{\widetilde{Z}_{\lambda,p}(s)}{\widetilde{Z}_{\lambda}},\qquad\widetilde{Z}_{\lambda,p}(s)\propto\sum_{\mathcal{L},e\mid s}p_e e^{-2\lvert\mathcal{L}\rvert/\lambda}. \tag*{(76)} P s ≡ Z λ Z λ , p ( s ) , Z λ , p ( s ) ∝ L , e ∣ s ∑ p e e − 2 ∣ L ∣ / λ . (76) To sample further pairs ( L , e ) (\mathcal{L},e) ( L , e ) Z ~ λ , p ( s ) \widetilde{Z}_{\lambda,p}(s) Z λ , p ( s )
First, we update L \mathcal{L} L § 3 C; i.e., we perform the WA restricted to the half-spaces τ > 0 \tau>0 τ > 0 τ < 0 \tau<0 τ < 0 s L s_{\mathcal{L}} s L s s s
Second, we update e e e e e e s e s_e s e s s s
The preceding two updates are not ergodic by themselves, since they never modify s L s_{\mathcal{L}} s L s e s_e s e s L s_{\mathcal{L}} s L s e s_e s e s = s L ⊕ s e s=s_{\mathcal{L}}\oplus s_e s = s L ⊕ s e ℓ \ell ℓ τ = 0 \tau=0 τ = 0 ℓ \ell ℓ ℓ \ell ℓ
P a c c e p t = min { 1 , p e ′ e − 2 ∣ L ′ ∣ / λ p e e − 2 ∣ L ∣ / λ } . (77) P_{\mathrm{accept}}=\min\left\{1,\frac{p_{e'}e^{-2\lvert\mathcal{L}'\rvert/\lambda}}{p_e e^{-2\lvert\mathcal{L}\rvert/\lambda}}\right\}. \tag*{(77)} P accept = min { 1 , p e e − 2 ∣ L ∣ / λ p e ′ e − 2 ∣ L ′ ∣ / λ } . (77) where ( L , e ) (\mathcal{L},e) ( L , e ) ( L ′ , e ′ ) (\mathcal{L}',e') ( L ′ , e ′ )
Finally, similar to § 3 C, we obtain an estimate for the decoding fidelity F s I s o \mathcal{F}^{\mathrm{Iso}}_s F s Iso n σ ( s ) n_\sigma(s) n σ ( s ) ( L , e ) (\mathcal{L},e) ( L , e ) σ \sigma σ
F s I s o = max σ n σ ( s ) ∑ σ n σ ( s ) . (78) \mathcal{F}^{\mathrm{Iso}}_s=\frac{\max_\sigma n_\sigma(s)}{\sum_\sigma n_\sigma(s)}. \tag*{(78)} F s Iso = ∑ σ n σ ( s ) max σ n σ ( s ) . (78) We then average over many observed syndromes s s s F I s o ‾ \overline{\mathcal{F}^{\mathrm{Iso}}} F Iso
Figure 10 presents our main numerical results, which are obtained using the optimal postmeasurement decoder described above [except for the blue diamonds in Figure 10 (c)]. Additional numerical plots for the optimal postmeasurement decoder can be found in Appendix § 1 2. Numerical plots for the minimal-weight perfect matching (MWPM) decoder can be found in Appendix § 1 3.
Figure 10. (a) Average optimal postmeasurement decoding fidelity F ‾ I s o \overline{F}^{\mathrm{Iso}} F Iso p = 0.09 p=0.09 p = 0.09 λ \lambda λ (51) ], for several linear system sizes L L L 3 d 3d 3 d ν ≈ 0.630 \nu\approx0.630 ν ≈ 0.630 [98 ] . (b) Decoding fidelity F ‾ I s o \overline{F}^{\mathrm{Iso}} F Iso λ = 0.998 λ c \lambda=0.998\lambda_c λ = 0.998 λ c p p p p 0 ≡ p c o p t ( λ = 0.998 λ c ) ≈ 0.0961 p_0 \equiv p_c^{\mathrm{opt}}(\lambda=0.998\lambda_c)\approx0.0961 p 0 ≡ p c opt ( λ = 0.998 λ c ) ≈ 0.0961 ν N i s h i ≈ 1.534 \nu_{\mathrm{Nishi}}\approx1.534 ν Nishi ≈ 1.534 [101 ] . (c) Optimal and MWPM postmeasurement decoding thresholds, p c o p t ( λ ) p_c^{\mathrm{opt}}(\lambda) p c opt ( λ ) p c M W P M ( λ ) p_c^{\mathrm{MWPM}}(\lambda) p c MWPM ( λ ) λ \lambda λ λ \lambda λ λ c \lambda_c λ c λ = λ c \lambda=\lambda_c λ = λ c
Figure 10 (a) depicts the average decoding fidelity using the optimal postmeasurement decoder, at a fixed bit-flip decoherence strength p = 0.09 p=0.09 p = 0.09 λ \lambda λ λ < λ c \lambda<\lambda_c λ < λ c F I s o ‾ \overline{\mathcal{F}^{\mathrm{Iso}}} F Iso F I s o \mathcal{F}^{\mathrm{Iso}} F Iso L L L p p p λ c ≈ 1.313 \lambda_c\approx1.313 λ c ≈ 1.313 § 3 C) and the 3 d 3d 3 d ν ≈ 0.630 \nu\approx0.630 ν ≈ 0.630 [98 ] , using a scaling form identical to Eq. (55) ; similarly good scaling collapses are obtained at smaller values of p p p § 1 2). Note that the curves for different system sizes do not perfectly cross, and in fact appear to cross at λ \lambda λ λ c \lambda_c λ c pairwise crossings between adjacent system sizes appears to drift towards smaller values of λ c \lambda_c λ c λ c \lambda_c λ c
Figure 10 (b) depicts the average decoding fidelity at the fixed LGT temperature λ = 0.998 λ c \lambda=0.998\lambda_c λ = 0.998 λ c p 0 ≈ 0.0961 p_0\approx0.0961 p 0 ≈ 0.0961 p c N i s h i ≈ 0.109 p_c^{\mathrm{Nishi}}\approx0.109 p c Nishi ≈ 0.109 [7 , 101 , 102 ] . We also achieve an excellent scaling collapse using the correlation length exponent ν N i s h i ≈ 1.534 \nu_{\mathrm{Nishi}}\approx1.534 ν Nishi ≈ 1.534 [101 ] . Similar scaling collapses are achieved for other values of λ \lambda λ λ c \lambda_c λ c § 1 2). This numerical evidence is consistent with the expectation that the entire phase boundary p c o p t ( λ ) p_c^{\mathrm{opt}}(\lambda) p c opt ( λ ) λ < λ c \lambda<\lambda_c λ < λ c Figure 2 (a)], exhibits a renormalization group flow towards the optimal decoding threshold in the λ = 0 \lambda=0 λ = 0 [7 ] ).
Finally, Figure 10 (c) depicts the λ \lambda λ p c o p t ( λ ) p_c^{\mathrm{opt}}(\lambda) p c opt ( λ ) p c M W P M ( λ ) p_c^{\mathrm{MWPM}}(\lambda) p c MWPM ( λ ) Figure 10 (b). Up to statistical fluctuations in the optimal thresholds, we find that the thresholds of both decoders monotonically decrease with increasing λ \lambda λ λ \lambda λ λ c \lambda_c λ c λ < λ c \lambda<\lambda_c λ < λ c λ c \lambda_c λ c p c o p t p_c^{\mathrm{opt}} p c opt p c M W P M p_c^{\mathrm{MWPM}} p c MWPM § 3.2 E), the latter must also remain finite as λ \lambda λ λ c \lambda_c λ c
Ref. [49 ] ’s theoretical prediction of a finite intrinsic error threshold as the TFTC’s quantum critical point is approached.
Phase-Flip Decoherence
Thus far, we have focused on decoding bit-flip decoherence in the TFTC. In that setting, both the transverse field and bit-flip decoherence cooperate to spread m m m e e e e e e h h h [7 ] ).
For simplicity, let us first set p X = 0 p_X=0 p X = 0 e e e ∣ φ h ⟩ \lvert\varphi_h\rangle ∣ φ h ⟩ e e e ρ h ≡ E ( ∣ φ h ⟩ ⟨ φ h ∣ ) \rho_h\equiv\mathcal{E}(\lvert\varphi_h\rangle\langle\varphi_h\rvert) ρ h ≡ E (∣ φ h ⟩ ⟨ φ h ∣) h h h A v A_v A v m m m Z s Z_s Z s e e e ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ ∏ ℓ ∈ C Z ℓ \prod_{\ell\in\mathcal{C}} Z_\ell ∏ ℓ ∈ C Z ℓ ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ h > 0 h>0 h > 0
The solution is to first measure the m m m § 3 , measuring each B p B_p B p ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ B p B_p B p m m m X s X_s X s ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ B p B_p B p
Π ^ s B ρ h Π ^ s B tr [ Π ^ s B ρ h ] = E ( Π ^ s B ∣ φ h ⟩ ⟨ φ h ∣ Π ^ s B ) ⟨ φ h ∣ Π ^ s B ∣ φ h ⟩ = E ( X s ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X s ) + O ( e − L / ξ ′ ) = X s E ( ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ) X s + O ( e − L / ξ ′ ) , (79) \frac{\hat{\Pi}_s^B \rho_h \hat{\Pi}_s^B}{\operatorname{tr}[\hat{\Pi}_s^B \rho_h]} =
\frac{\mathcal{E}(\hat{\Pi}_s^B |\varphi_h\rangle\langle\varphi_h| \hat{\Pi}_s^B)}{\langle\varphi_h|\hat{\Pi}_s^B|\varphi_h\rangle}
= \mathcal{E}(X_s|\psi_0\rangle\langle\psi_0|X_s) + O(e^{-L/\xi'})
= X_s\mathcal{E}(|\psi_0\rangle\langle\psi_0|)X_s + O(e^{-L/\xi'}), \tag*{(79)} tr [ Π ^ s B ρ h ] Π ^ s B ρ h Π ^ s B = ⟨ φ h ∣ Π ^ s B ∣ φ h ⟩ E ( Π ^ s B ∣ φ h ⟩ ⟨ φ h ∣ Π ^ s B ) = E ( X s ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X s ) + O ( e − L / ξ ′ ) = X s E ( ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ) X s + O ( e − L / ξ ′ ) , (79) where Π ^ s B \hat{\Pi}_s^B Π ^ s B B p B_p B p m m m X s X_s X s A v A_v A v [7 ] , which recovers ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ E ( ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ) \mathcal{E}(|\psi_0\rangle\langle\psi_0|) E ( ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ) p Z < p c N i s h i ≈ 0.109 p_Z<p_c^{\mathrm{Nishi}}\approx0.109 p Z < p c Nishi ≈ 0.109 p c N i s h i p_c^{\mathrm{Nishi}} p c Nishi h h h h < h c h<h_c h < h c
More generally, suppose E \mathcal{E} E (13) , with p Y = p X p Z / ( 1 − p ) p_Y=p_Xp_Z/(1-p) p Y = p X p Z / ( 1 − p ) § 3.2 B that after B p B_p B p ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ M s \mathcal{M}_s M s s s s (73) ]. Since the original bit-flip channel is independent of the phase-flip channel, we have
Π ^ s B ρ h Π ^ s B tr [ Π ^ s B ρ h ] = E Z ∘ M s ( ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ) + O ( e − L / ξ ′ ) (80) \frac{\hat{\Pi}_s^B \rho_h \hat{\Pi}_s^B}{\operatorname{tr}[\hat{\Pi}_s^B \rho_h]} = \mathcal{E}^Z \circ\mathcal{M}_s(|\psi_0\rangle\langle\psi_0|) + O(e^{-L/\xi'}) \tag*{(80)} tr [ Π ^ s B ρ h ] Π ^ s B ρ h Π ^ s B = E Z ∘ M s ( ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ) + O ( e − L / ξ ′ ) (80) with high probability, where E Z \mathcal{E}^Z E Z p c N i s h i p_c^{\mathrm{Nishi}} p c Nishi
Finally, since the m m m X s X_s X s e e e A v A_v A v e e e m m m m m m X s X_s X s § 3.2 B, and pairwise annihilate the e e e Z s Z_s Z s h < h c h<h_c h < h c
From this result, it is tempting to conclude that stabilizer measurements themselves serve as an optimal decoder for phase-flip decoherence. Strictly speaking, this conclusion does not follow from the preceding result: in principle, it is logically possible that the TFTC with nonzero h h h
Optimality of Stabilizer Measurements for Decoding Bit-Flip Decoherence
We have seen in § 3 that stabilizer measurements effectively act unitarily on the TFTC code space for all h < h c h<h_c h < h c § 3.2 C that the decoding threshold p c o p t ( h ) p_c^{\mathrm{opt}}(h) p c opt ( h ) h → h c h\to h_c h → h c p c o p t ( h ) p_c^{\mathrm{opt}}(h) p c opt ( h )
For our purposes, the intrinsic threshold p c i n t ( h ) p_c^{\mathrm{int}}(h) p c int ( h ) [61 , 62 , 103 ] . Let us denote our principal TFTC system as Q Q Q R R R ∣ Ψ Q R ⟩ |\Psi_{QR}\rangle ∣ Ψ QR ⟩ R R R Q Q Q
∣ Ψ Q R ⟩ = 1 4 ∑ σ ∣ ψ h σ ⟩ Q ⊗ ∣ σ ⟩ R , (81) |\Psi_{QR}\rangle= \frac{1}{\sqrt{4}}\sum_{\sigma}|\psi_h^\sigma\rangle_Q\otimes|\sigma\rangle_R, \tag*{(81)} ∣ Ψ QR ⟩ = 4 1 σ ∑ ∣ ψ h σ ⟩ Q ⊗ ∣ σ ⟩ R , (81) where ∣ ψ h σ ⟩ Q ≡ U h ∣ ψ σ ⟩ Q |\psi_h^\sigma\rangle_Q\equiv U_h|\psi^\sigma\rangle_Q ∣ ψ h σ ⟩ Q ≡ U h ∣ ψ σ ⟩ Q C h \mathcal{C}_h C h ∣ σ ⟩ R |\sigma\rangle_R ∣ σ ⟩ R R R R E \mathcal{E} E [80 ] by introducing an explicit environment subsystem E E E Q Q Q Q R E QRE QR E
∣ Ψ Q R E ⟩ = 1 4 ∑ σ , e p e X e ∣ ψ h σ ⟩ Q ⊗ ∣ σ ⟩ R ⊗ ∣ e ⟩ E , (82) |\Psi_{QRE}\rangle= \frac{1}{\sqrt{4}}\sum_{\sigma,e}\sqrt{p_e}X_e|\psi_h^\sigma\rangle_Q\otimes|\sigma\rangle_R\otimes|e\rangle_E, \tag*{(82)} ∣ Ψ QR E ⟩ = 4 1 σ , e ∑ p e X e ∣ ψ h σ ⟩ Q ⊗ ∣ σ ⟩ R ⊗ ∣ e ⟩ E , (82) where ∣ e ⟩ E |e\rangle_E ∣ e ⟩ E E E E h h h p p p p c i n t ( h ) p_c^{\mathrm{int}}(h) p c int ( h ) R R R E E E L → ∞ L\to\infty L → ∞ 18 ,
p c i n t ( h ) ≡ sup { p : ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 → L → ∞ 0 } . (83) p_c^{\mathrm{int}}(h)\equiv\sup\left\{p:\left\|\rho_{RE}-\rho_R\otimes\rho_E\right\|_1\xrightarrow{L\to\infty}0\right\}. \tag*{(83)} p c int ( h ) ≡ sup { p : ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 L → ∞ 0 } . (83) where the supremum is taken over the range p ∈ [ 0 , 1 2 ] p \in[0,\frac{1}{2}] p ∈ [ 0 , 2 1 ]
ρ S ≡ tr S c ∣ Ψ Q R E ⟩ ⟨ Ψ Q R E ∣ \rho_S \equiv\operatorname{tr}_{S^c}|\Psi_{QRE}\rangle\langle\Psi_{QRE}| ρ S ≡ tr S c ∣ Ψ QR E ⟩ ⟨ Ψ QR E ∣ S S S ∥ O ∥ 1 ≡ tr [ O † O ] \lVert O\rVert_1 \equiv\operatorname{tr}[\sqrt{O^\dagger O}] ∥ O ∥ 1 ≡ tr [ O † O ] ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 ≤ ε \lVert\rho_{RE}-\rho_R\otimes\rho_E\rVert_1 \le\varepsilon ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 ≤ ε R \mathcal{R} R Q Q Q [62 ]
F e ≡ ⟨ Ψ Q R ∣ R ( ρ Q R ) ∣ Ψ Q R ⟩ ≥ 1 − ε . (84) F_{\mathrm{e}} \equiv\langle\Psi_{QR}|\mathcal{R}(\rho_{QR})|\Psi_{QR}\rangle\ge1-\varepsilon. \tag*{(84)} F e ≡ ⟨ Ψ QR ∣ R ( ρ QR ) ∣ Ψ QR ⟩ ≥ 1 − ε . (84) The quantity F e F_{\mathrm{e}} F e entanglement fidelity [62 , 80 ] , can in turn be used to lower-bound the recovery fidelity ⟨ ψ h ∣ R ( ρ h ) ∣ ψ h ⟩ \langle\psi_h|\mathcal{R}(\rho_h)|\psi_h\rangle ⟨ ψ h ∣ R ( ρ h ) ∣ ψ h ⟩ ∣ ψ h ⟩ ∈ C h |\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h p c i n t ( h ) p_{\mathrm{c}}^{\mathrm{int}}(h) p c int ( h ) intrinsic protection of quantum information, i.e., the capacity for information to be recovered by some (perhaps extraordinarily complicated) channel R \mathcal{R} R
Now, suppose that a stabilizer measurement is performed on subsystem Q Q Q
∣ Ψ Q R E ( s ) ⟩ ≡ ∑ σ , e p e 4 P s Π ^ s X e ∣ ψ h σ ⟩ Q ⊗ ∣ σ ⟩ R ⊗ ∣ e ⟩ E (85) |\Psi_{QRE}(s)\rangle\equiv\sum_{\sigma,e}\sqrt{\frac{p_e}{4P_s}}\widehat{\Pi}_sX_e|\psi_h^\sigma\rangle_Q\otimes|\sigma\rangle_R\otimes|e\rangle_E \tag*{(85)} ∣ Ψ QR E ( s )⟩ ≡ σ , e ∑ 4 P s p e Π s X e ∣ ψ h σ ⟩ Q ⊗ ∣ σ ⟩ R ⊗ ∣ e ⟩ E (85) with probability P s ≡ tr [ Π ^ s ρ Q ] = 1 4 ∑ σ ⟨ ψ h σ ∣ E ( Π ^ s ) ∣ ψ h σ ⟩ P_s \equiv\operatorname{tr}[\widehat{\Pi}_s\rho_Q]=\frac{1}{4}\sum_\sigma\langle\psi_h^\sigma|\mathcal{E}(\widehat{\Pi}_s)|\psi_h^\sigma\rangle P s ≡ tr [ Π s ρ Q ] = 4 1 ∑ σ ⟨ ψ h σ ∣ E ( Π s ) ∣ ψ h σ ⟩ S S S ρ S ( s ) \rho_S(s) ρ S ( s ) p < p c i n t ( h ) p<p_{\mathrm{c}}^{\mathrm{int}}(h) p < p c int ( h ) s s s V s , Q V_{s,Q} V s , Q Q Q Q V s , Q ρ Q R ( s ) V s , Q † V_{s,Q}\rho_{QR}(s)V_{s,Q}^{\dagger} V s , Q ρ QR ( s ) V s , Q † ∣ Ψ Q R ⟩ ⟨ Ψ Q R ∣ |\Psi_{QR}\rangle\langle\Psi_{QR}| ∣ Ψ QR ⟩ ⟨ Ψ QR ∣
Since subsystem Q Q Q V s , Q V_{s,Q} V s , Q Q Q Q R R R
Subsystem R R R Q E QE QE ρ R ( s ) ≈ 1 4 ∑ σ ∣ σ ⟩ ⟨ σ ∣ R = 1 4 1 R \rho_R(s)\approx\frac{1}{4}\sum_\sigma|\sigma\rangle\langle\sigma|_R=\frac{1}{4}\mathbb{1}_R ρ R ( s ) ≈ 4 1 ∑ σ ∣ σ ⟩ ⟨ σ ∣ R = 4 1 1 R
Subsystems R R R E E E ∥ ρ R E ( s ) − ρ R ( s ) ⊗ ρ E ( s ) ∥ 1 \lVert\rho_{RE}(s)-\rho_R(s)\otimes\rho_E(s)\rVert_1 ∥ ρ R E ( s ) − ρ R ( s ) ⊗ ρ E ( s ) ∥ 1
Physically, the first criterion states that the syndrome measurement does not collapse or reveal any encoded information, similar to § 3 . The second criterion states that the measurement does not teleport any information previously contained within Q Q Q E E E R E RE R E Q Q Q
∣ Ψ Q R E ( s ) ⟩ ≈ ( V s , Q † ⊗ 1 R E ) ∣ Ψ Q R ⟩ ⊗ ∣ ψ E ( s ) ⟩ , (86) |\Psi_{QRE}(s)\rangle\approx(V_{s,Q}^{\dagger}\otimes\mathbb{1}_{RE})|\Psi_{QR}\rangle\otimes|\psi_E(s)\rangle, \tag*{(86)} ∣ Ψ QR E ( s )⟩ ≈ ( V s , Q † ⊗ 1 R E ) ∣ Ψ QR ⟩ ⊗ ∣ ψ E ( s )⟩ , (86) i.e., the stabilizer measurement has purified subsystem E E E Q R QR QR
To analyze these two criteria in the TFTC, it is convenient as in preceding sections to work with ITE states ∣ φ h σ ⟩ ∝ e − β H h / 2 ∣ ψ 0 σ ⟩ |\varphi_h^\sigma\rangle\propto e^{-\beta H_h/2}|\psi_0^\sigma\rangle ∣ φ h σ ⟩ ∝ e − β H h /2 ∣ ψ 0 σ ⟩ ∣ ψ h σ ⟩ |\psi_h^\sigma\rangle ∣ ψ h σ ⟩ ∣ Ψ Q R ⟩ |\Psi_{QR}\rangle ∣ Ψ QR ⟩
∣ Φ Q R ⟩ ≡ 1 4 ∑ σ ∣ φ h σ ⟩ Q ⊗ ∣ σ ⟩ R = 1 4 Z h ∑ σ e − β H h / 2 ∣ ψ 0 σ ⟩ Q ⊗ ∣ σ ⟩ R , (87) \begin{aligned}
|\Phi_{QR}\rangle\equiv\frac{1}{\sqrt{4}}\sum_\sigma|\varphi_h^\sigma\rangle_Q\otimes|\sigma\rangle_R \\
&=\frac{1}{\sqrt{4\mathcal{Z}_h}}\sum_\sigma e^{-\beta H_h/2}|\psi_0^\sigma\rangle_Q\otimes|\sigma\rangle_R, \tag*{(87)}
\end{aligned} ∣ Φ QR ⟩ ≡ 4 1 σ ∑ ∣ φ h σ ⟩ Q ⊗ ∣ σ ⟩ R = 4 Z h 1 σ ∑ e − β H h /2 ∣ ψ 0 σ ⟩ Q ⊗ ∣ σ ⟩ R , (87) which exhibits fidelity with ∣ Ψ Q R ⟩ |\Psi_{QR}\rangle ∣ Ψ QR ⟩ Q Q Q R R R P s P_s P s s s s
P s = 1 4 ∑ σ ⟨ φ h σ ∣ E ( Π ^ s ) ∣ φ h σ ⟩ = Z h , p ( s ) Z h , (88) P_s=\frac{1}{4}\sum_\sigma\langle\varphi_h^\sigma|\mathcal{E}(\widehat{\Pi}_s)|\varphi_h^\sigma\rangle=\frac{\mathcal{Z}_{h,p}(s)}{\mathcal{Z}_h}, \tag*{(88)} P s = 4 1 σ ∑ ⟨ φ h σ ∣ E ( Π s ) ∣ φ h σ ⟩ = Z h Z h , p ( s ) , (88) where Z h , p ( s ) \mathcal{Z}_{h,p}(s) Z h , p ( s ) (59) . Furthermore, a short calculation shows that the postmeasurement reduced density matrix on R R R
ρ R ( s ) = 1 4 ∑ σ σ ′ Z h , p σ ⊕ σ ′ ( s ) Z h , p ( s ) ∣ σ ⟩ ⟨ σ ′ ∣ , (89) \rho_R(s)=\frac{1}{4}\sum_{\sigma\sigma'}\frac{\mathcal{Z}_{h,p}^{\sigma\oplus\sigma'}(s)}{\mathcal{Z}_{h,p}(s)}|\sigma\rangle\langle\sigma'|, \tag*{(89)} ρ R ( s ) = 4 1 σ σ ′ ∑ Z h , p ( s ) Z h , p σ ⊕ σ ′ ( s ) ∣ σ ⟩ ⟨ σ ′ ∣ , (89) with Z h , p σ ⊕ σ ′ ( s ) \mathcal{Z}_{h,p}^{\sigma\oplus\sigma'}(s) Z h , p σ ⊕ σ ′ ( s ) (61) . From Eq. (64) , we see that the σ ≠ σ ′ \sigma\ne\sigma' σ = σ ′ X X X
∥ ρ R ( s ) − 1 4 1 R ∥ 1 ≲ O ( e − L / ξ ′ ) (90) \left\lVert\rho_R(s)-\frac{1}{4}\mathbb{1}_R\right\rVert_1\lesssim O(e^{-L/\xi'}) \tag*{(90)} ρ R ( s ) − 4 1 1 R 1 ≲ O ( e − L / ξ ′ ) (90) with high probability, and therefore R R R Q E QE QE p p p p > p c i n t ( h ) p>p_{\mathrm{c}}^{\mathrm{int}}(h) p > p c int ( h ) h < h c h<h_{\mathrm{c}} h < h c p < p c i n t ( h ) p<p_{\mathrm{c}}^{\mathrm{int}}(h) p < p c int ( h )
Unfortunately, we are unable to prove criterion 2 without further assumptions [i.e., besides our central assumption in Eq. (12)]. Here we will discuss an additional physically motivated assumption, which we conjecture (but cannot prove) holds for all p < p c i n t ( h ) p<p_{\mathrm{c}}^{\mathrm{int}}(h) p < p c int ( h )
To state our physical assumption, consider the four classical probability distributions over bit-strings z \boldsymbol{z} z
Q z σ = ∑ e p e ∣ ⟨ z ∣ X e ∣ φ h σ ⟩ ∣ 2 , (91) Q_{\boldsymbol{z}}^\sigma=\sum_e p_e|\langle\boldsymbol{z}|X_e|\varphi_h^\sigma\rangle|^2, \tag*{(91)} Q z σ = e ∑ p e ∣ ⟨ z ∣ X e ∣ φ h σ ⟩ ∣ 2 , (91) which are simply the diagonal matrix elements of the decohered basis states ρ h σ = E ( ∣ φ h σ ⟩ ⟨ φ h σ ∣ ) \rho_h^\sigma=\mathcal{E}(|\varphi_h^\sigma\rangle\langle\varphi_h^\sigma|) ρ h σ = E ( ∣ φ h σ ⟩ ⟨ φ h σ ∣ )
∑ z Q z σ Q z σ ′ ≤ ε ( σ ≠ σ ′ ) . (92) \sum_z \sqrt{Q_z^\sigma Q_z^{\sigma'}} \le\varepsilon\quad(\sigma\ne\sigma'). \tag*{(92)} z ∑ Q z σ Q z σ ′ ≤ ε ( σ = σ ′ ) . (92) where ε → 0 \varepsilon\to0 ε → 0 L → ∞ L \to\infty L → ∞ p = 0 p = 0 p = 0 h = 0 h = 0 h = 0 p < p c i n t ( h = 0 ) p < p_c^{\mathrm{int}}(h = 0) p < p c int ( h = 0 ) h h h p p p p < p c i n t ( h ) p < p_c^{\mathrm{int}}(h) p < p c int ( h )
To see first that Eq. (92) implies that p < p c i n t ( h ) p < p_c^{\mathrm{int}}(h) p < p c int ( h ) ρ R \rho_R ρ R
∥ ρ R E − ρ R ⊗ ρ E ∥ 1 ≤ ∥ ρ R E − 1 4 1 R ⊗ ρ E ∥ 1 + ∥ ρ R − 1 4 1 R ∥ 1 . (93) \left\lVert\rho_{RE} - \rho_R \otimes\rho_E \right\rVert_1
\le\left\lVert\rho_{RE} - \frac{1}{4}\mathbb{1}_R \otimes\rho_E \right\rVert_1 + \left\lVert\rho_R - \frac{1}{4}\mathbb{1}_R \right\rVert_1. \tag*{(93)} ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 ≤ ρ R E − 4 1 1 R ⊗ ρ E 1 + ρ R − 4 1 1 R 1 . (93) Noting that ρ R = 1 4 ∑ σ σ ′ ⟨ φ h σ ′ ∣ φ h σ ⟩ ∣ σ ⟩ ⟨ σ ′ ∣ \rho_R = \frac{1}{4}\sum_{\sigma\sigma'} \langle\varphi_h^{\sigma'}|\varphi_h^\sigma\rangle|\sigma\rangle\langle\sigma'| ρ R = 4 1 ∑ σ σ ′ ⟨ φ h σ ′ ∣ φ h σ ⟩ ∣ σ ⟩ ⟨ σ ′ ∣ O ( e − L / ξ ′ ) O(e^{-L/\xi'}) O ( e − L / ξ ′ ) h < h c h < h_c h < h c (23) , and so we can focus on the first term. Using logical-X X X
ρ R E − 1 4 1 R ⊗ ρ E = ∑ σ ≠ σ ′ ∑ e , e ′ Γ e e ′ σ σ ′ ∣ σ ⟩ ⟨ σ ′ ∣ ⊗ ∣ e ⟩ ⟨ e ′ ∣ , (94) \rho_{RE} - \frac{1}{4}\mathbb{1}_R \otimes\rho_E = \sum_{\sigma\ne\sigma'}\sum_{e,e'}\Gamma_{ee'}^{\sigma\sigma'}|\sigma\rangle\langle\sigma'| \otimes|e\rangle\langle e'|, \tag*{(94)} ρ R E − 4 1 1 R ⊗ ρ E = σ = σ ′ ∑ e , e ′ ∑ Γ e e ′ σ σ ′ ∣ σ ⟩ ⟨ σ ′ ∣ ⊗ ∣ e ⟩ ⟨ e ′ ∣ , (94) Γ e e ′ σ σ ′ ≡ p e p e ′ 4 ⟨ φ h σ ′ ∣ X e ′ X e ∣ φ h σ ⟩ , (95) \Gamma_{ee'}^{\sigma\sigma'} \equiv\frac{\sqrt{p_e p_{e'}}}{4}\langle\varphi_h^{\sigma'}|X_{e'}X_e|\varphi_h^\sigma\rangle, \tag*{(95)} Γ e e ′ σ σ ′ ≡ 4 p e p e ′ ⟨ φ h σ ′ ∣ X e ′ X e ∣ φ h σ ⟩ , (95) and so
∥ ρ R E − 1 4 1 R ⊗ ρ E ∥ 1 ≤ ∑ σ ≠ σ ′ ∥ ∑ e , e ′ Γ e e ′ σ σ ′ ∣ e ⟩ ⟨ e ′ ∣ ∥ 1 . (96) \left\lVert\rho_{RE} - \frac{1}{4}\mathbb{1}_R \otimes\rho_E \right\rVert_1 \le\sum_{\sigma\ne\sigma'}\left\lVert\sum_{e,e'}\Gamma_{ee'}^{\sigma\sigma'}|e\rangle\langle e'|\right\rVert_1. \tag*{(96)} ρ R E − 4 1 1 R ⊗ ρ E 1 ≤ σ = σ ′ ∑ e , e ′ ∑ Γ e e ′ σ σ ′ ∣ e ⟩ ⟨ e ′ ∣ 1 . (96) To bound the remaining trace norm, define for each z \mathbf{z} z
∣ E z σ ⟩ = ∑ e p e ⟨ z ∣ X e ∣ φ h σ ⟩ ∣ e ⟩ , (97) |E_z^\sigma\rangle= \sum_e\sqrt{p_e}\langle z|X_e|\varphi_h^\sigma\rangle|e\rangle, \tag*{(97)} ∣ E z σ ⟩ = e ∑ p e ⟨ z ∣ X e ∣ φ h σ ⟩ ∣ e ⟩ , (97) whose squared norm is ⟨ E z σ ∣ E z σ ⟩ = Q z σ \langle E_z^\sigma|E_z^\sigma\rangle= Q_z^\sigma ⟨ E z σ ∣ E z σ ⟩ = Q z σ 1 Q = ∑ z ∣ z ⟩ ⟨ z ∣ \mathbb{1}_Q=\sum_z|z\rangle\langle z| 1 Q = ∑ z ∣ z ⟩ ⟨ z ∣
∥ ∑ e , e ′ Γ e e ′ σ σ ′ ∣ e ⟩ ⟨ e ′ ∣ ∥ 1 ≤ 1 4 ∑ z ∥ ∣ E z σ ⟩ ⟨ E z σ ′ ∣ ∥ 1 = 1 4 ∑ z ⟨ E z σ ∣ E z σ ⟩ ⟨ E z σ ′ ∣ E z σ ′ ⟩ = 1 4 ∑ z Q z σ Q z σ ′ ≤ ε 4 . (98) \begin{aligned}
\left\lVert\sum_{e,e'}\Gamma_{ee'}^{\sigma\sigma'}|e\rangle\langle e'|\right\rVert_1
&\le\frac{1}{4}\sum_z\left\lVert|E_z^\sigma\rangle\langle E_z^{\sigma'}|\right\rVert_1\\
&= \frac{1}{4}\sum_z\sqrt{\langle E_z^\sigma|E_z^\sigma\rangle\langle E_z^{\sigma'}|E_z^{\sigma'}\rangle}\\
&= \frac{1}{4}\sum_z\sqrt{Q_z^\sigma Q_z^{\sigma'}}\\
&\le\frac{\varepsilon}{4}. \tag*{(98)}
\end{aligned} e , e ′ ∑ Γ e e ′ σ σ ′ ∣ e ⟩ ⟨ e ′ ∣ 1 ≤ 4 1 z ∑ ∣ E z σ ⟩ ⟨ E z σ ′ ∣ 1 = 4 1 z ∑ ⟨ E z σ ∣ E z σ ⟩ ⟨ E z σ ′ ∣ E z σ ′ ⟩ = 4 1 z ∑ Q z σ Q z σ ′ ≤ 4 ε . (98) Altogether, we find that
∥ ρ R E − ρ R ⊗ ρ E ∥ 1 ≤ 3 ε + O ( e − L / ξ ′ ) , (99) \left\lVert\rho_{RE}-\rho_R\otimes\rho_E\right\rVert_1\le3\varepsilon+O(e^{-L/\xi'}), \tag*{(99)} ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 ≤ 3 ε + O ( e − L / ξ ′ ) , (99) and therefore p p p p c i n t ( h ) p_c^{\mathrm{int}}(h) p c int ( h ) (92) implies that criterion 2 is satisfied with high probability, i.e., any encoded information can be recovered with high fidelity from the postmeasurement states. To this end, define the conditional probability distributions
Q z σ ( s ) ≡ 1 P s ∑ e p e ∣ ⟨ z ∣ Π ^ s X e ∣ φ h σ ⟩ ∣ 2 , (100) Q_z^\sigma(s) \equiv\frac{1}{P_s}\sum_e p_e\left|\langle z|\hat{\Pi}_sX_e|\varphi_h^\sigma\rangle\right|^2, \tag*{(100)} Q z σ ( s ) ≡ P s 1 e ∑ p e ⟨ z ∣ Π ^ s X e ∣ φ h σ ⟩ 2 , (100) which are simply the diagonal matrix elements of the postmeasurement decohered basis states ρ h σ ( s ) ∝ Π ^ s ρ h σ Π ^ s \rho_h^\sigma(s)\propto\hat{\Pi}_s\rho_h^\sigma\hat{\Pi}_s ρ h σ ( s ) ∝ Π ^ s ρ h σ Π ^ s
∑ s P s ∑ z Q z σ ( s ) Q z σ ′ ( s ) = ∑ z Q z σ Q z σ ′ ≤ ε , (101) \sum_s P_s\sum_z\sqrt{Q_z^\sigma(s)Q_z^{\sigma'}(s)}=\sum_z\sqrt{Q_z^\sigma Q_z^{\sigma'}}\le\varepsilon, \tag*{(101)} s ∑ P s z ∑ Q z σ ( s ) Q z σ ′ ( s ) = z ∑ Q z σ Q z σ ′ ≤ ε , (101) and so Markov’s inequality implies
∑ z Q z σ ( s ) Q z σ ′ ( s ) ≤ δ (102) \sum_z\sqrt{Q_z^\sigma(s)Q_z^{\sigma'}(s)}\le\delta \tag*{(102)} z ∑ Q z σ ( s ) Q z σ ′ ( s ) ≤ δ (102) with probability 1 − ( ε / δ ) 1-(\varepsilon/\delta) 1 − ( ε / δ ) ε → 0 \varepsilon\to0 ε → 0 L → ∞ L\to\infty L → ∞ δ \delta δ L → ∞ L\to\infty L → ∞ ε \varepsilon ε Q z σ ( s ) Q_z^\sigma(s) Q z σ ( s ) L → ∞ L\to\infty L → ∞
We can now construct an identical bound on the trace distance ∥ ρ R E ( s ) − ρ R ( s ) ⊗ ρ E ( s ) ∥ 1 \left\lVert\rho_{RE}(s)-\rho_R(s)\otimes\rho_E(s)\right\rVert_1 ∥ ρ R E ( s ) − ρ R ( s ) ⊗ ρ E ( s ) ∥ 1 ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 \left\lVert\rho_{RE}-\rho_R\otimes\rho_E\right\rVert_1 ∥ ρ R E − ρ R ⊗ ρ E ∥ 1
ρ R E ( s ) − 1 4 1 R ⊗ ρ E ( s ) = ∑ σ ≠ σ ′ ∑ e , e ′ Γ e e ′ σ σ ′ ( s ) ∣ σ ⟩ ⟨ σ ′ ∣ ⊗ ∣ e ⟩ ⟨ e ′ ∣ , (103) \rho_{RE}(s)-\frac{1}{4}\mathbb{1}_R\otimes\rho_E(s)=\sum_{\sigma\ne\sigma'}\sum_{e,e'}\Gamma_{ee'}^{\sigma\sigma'}(s)|\sigma\rangle\langle\sigma'|\otimes|e\rangle\langle e'|, \tag*{(103)} ρ R E ( s ) − 4 1 1 R ⊗ ρ E ( s ) = σ = σ ′ ∑ e , e ′ ∑ Γ e e ′ σ σ ′ ( s ) ∣ σ ⟩ ⟨ σ ′ ∣ ⊗ ∣ e ⟩ ⟨ e ′ ∣ , (103) Γ e e ′ σ σ ′ ( s ) ≡ p e p e ′ 4 P s ⟨ φ h σ ′ ∣ X e ′ Π ^ s X e ∣ φ h σ ⟩ , \Gamma_{ee'}^{\sigma\sigma'}(s)\equiv\frac{\sqrt{p_ep_{e'}}}{4P_s}\langle\varphi_h^{\sigma'}|X_{e'}\hat{\Pi}_sX_e|\varphi_h^\sigma\rangle, Γ e e ′ σ σ ′ ( s ) ≡ 4 P s p e p e ′ ⟨ φ h σ ′ ∣ X e ′ Π ^ s X e ∣ φ h σ ⟩ , and so
∥ ρ R E ( s ) − 1 4 1 R ⊗ ρ E ( s ) ∥ 1 ≤ ∑ σ ≠ σ ′ ∥ ∑ e , e ′ Γ e e ′ σ σ ′ ( s ) ∣ e ⟩ ⟨ e ′ ∣ ∥ 1 ≤ 3 δ , (104) \begin{aligned}
\left\lVert\rho_{RE}(s)-\frac{1}{4}\mathbb{1}_R\otimes\rho_E(s)\right\rVert_1
&\le\sum_{\sigma\ne\sigma'}\left\lVert\sum_{e,e'}\Gamma_{ee'}^{\sigma\sigma'}(s)|e\rangle\langle e'|\right\rVert_1\\
&\le3\delta, \tag*{(104)}
\end{aligned} ρ R E ( s ) − 4 1 1 R ⊗ ρ E ( s ) 1 ≤ σ = σ ′ ∑ e , e ′ ∑ Γ e e ′ σ σ ′ ( s ) ∣ e ⟩ ⟨ e ′ ∣ 1 ≤ 3 δ , (104) where the second bound is obtained via an identical calculation to that of Eq. (98) . Combined with Eq. (90) , we find
∥ ρ R E ( s ) − ρ R ( s ) ⊗ ρ E ( s ) ∥ 1 ≤ 3 δ + O ( e − L / ξ ′ ) (105) \left\lVert\rho_{RE}(s)-\rho_R(s)\otimes\rho_E(s)\right\rVert_1\le3\delta+O(e^{-L/\xi'}) \tag*{(105)} ∥ ρ R E ( s ) − ρ R ( s ) ⊗ ρ E ( s ) ∥ 1 ≤ 3 δ + O ( e − L / ξ ′ ) (105) with high probability, and so criterion 2 is satisfied. In summary, we have shown that the physical assumption Eq. (92) implies that stabilizer measurements can be used as part of an asymptotically optimal decoding strategy for the TFTC under bit-flip decoherence. While we have proven that Eq. (92) implies that p p p
∑ z Q z σ Q z σ ′ ≤ ∑ e e ′ p e p e ′ ∑ z ⟨ φ h σ ∣ X e ∣ z ⟩ ⟨ z ∣ X e ′ ∣ φ h σ ′ ⟩ = 4 ∑ e e ′ Γ e e ′ σ σ ′ \begin{aligned}
\sum_z \sqrt{Q_z^\sigma Q_z^{\sigma'}} \leq\sum_{ee'} \sqrt{p_e p_{e'}} \sum_z \langle\varphi_h^\sigma|X_e|z\rangle\langle z|X_{e'}|\varphi_h^{\sigma'}\rangle\\
&=4\sum_{ee'}\Gamma_{ee'}^{\sigma\sigma'}
\end{aligned} z ∑ Q z σ Q z σ ′ ≤ e e ′ ∑ p e p e ′ z ∑ ⟨ φ h σ ∣ X e ∣ z ⟩ ⟨ z ∣ X e ′ ∣ φ h σ ′ ⟩ = 4 e e ′ ∑ Γ e e ′ σ σ ′ ≤ 4 D E ∥ ∑ e e ′ Γ e e ′ σ σ ′ ∣ e ⟩ ⟨ e ′ ∣ ∥ 1 . (106) \leq4D_E\left\|\sum_{ee'}\Gamma_{ee'}^{\sigma\sigma'}|e\rangle\langle e'|\right\|_1 . \tag*{(106)} ≤ 4 D E e e ′ ∑ Γ e e ′ σ σ ′ ∣ e ⟩ ⟨ e ′ ∣ 1 . (106) where stoquasticity has implied that each matrix element ⟨ z ∣ X e ∣ φ h σ ⟩ \langle z|X_e|\varphi_h^\sigma\rangle ⟨ z ∣ X e ∣ φ h σ ⟩ ∥ A ∥ ℓ 1 = ∑ i j ∣ A i j ∣ ≤ D ∥ A ∥ 1 \|A\|_{\ell_1}=\sum_{ij}|A_{ij}|\leq D\|A\|_1 ∥ A ∥ ℓ 1 = ∑ ij ∣ A ij ∣ ≤ D ∥ A ∥ 1 D × D D\times D D × D A A A [104 ] , so that D E D_E D E E E E ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 \|\rho_{RE}-\rho_R\otimes\rho_E\|_1 ∥ ρ R E − ρ R ⊗ ρ E ∥ 1 O ( e − L / ξ ) O(e^{-L/\xi}) O ( e − L / ξ ) D E = 2 2 L 2 D_E=2^{2L^2} D E = 2 2 L 2
On the other hand, we expect on physical grounds that the inequality leading to the final line of Eq. (106) is very loose. Whereas the inequality is saturated when Γ e e ′ σ σ ′ \Gamma_{ee'}^{\sigma\sigma'} Γ e e ′ σ σ ′ Γ e e ′ σ σ ′ \Gamma_{ee'}^{\sigma\sigma'} Γ e e ′ σ σ ′ sparse .19 ∣ φ h σ ⟩ |\varphi_h^\sigma\rangle ∣ φ h σ ⟩ ⟨ φ h σ ′ ∣ X e ′ X e ∣ φ h σ ⟩ \langle\varphi_h^{\sigma'}|X_{e'}X_e|\varphi_h^\sigma\rangle ⟨ φ h σ ′ ∣ X e ′ X e ∣ φ h σ ⟩ X e X_e X e X e ′ X_{e'} X e ′ h = 0 h=0 h = 0 (106) can be constructed. It would be interesting to promote this intuition to a rigorous inequality in the h > 0 h>0 h > 0
Physically, we can think of the pure-state distributions ∣ ⟨ z ∣ φ h σ ⟩ ∣ 2 |\langle z|\varphi_h^\sigma\rangle|^2 ∣ ⟨ z ∣ φ h σ ⟩ ∣ 2 classical code, and the mixed-state distributions Q z σ Q_z^\sigma Q z σ (92) is satisfied. Our physical assumption is therefore that the quantum code defined by the code states ∣ φ h σ ⟩ |\varphi_h^\sigma\rangle ∣ φ h σ ⟩
the classical code defined by their matrix elements in the Pauli-Z Z Z ∣ φ h σ ⟩ |\varphi_h^\sigma\rangle ∣ φ h σ ⟩ X e X_e X e E ( ∣ φ h σ ⟩ ⟨ φ h σ ∣ ) \mathcal{E}(|\varphi_h^\sigma\rangle\langle\varphi_h^\sigma|) E ( ∣ φ h σ ⟩ ⟨ φ h σ ∣ )
Transverse-Field Ising Model under Bit-Flip Decoherence and Stabilizer Measurements
Thus far, we have focused on the TFTC (8) as a Hamiltonian deformation of the stabilizer toric code. However, our general formalism is widely applicable to a broad class of stoquastic Hamiltonian deformations of CSS stabilizer codes. As another concrete example, here we briefly investigate a simpler problem: the ( d + 1 ) (d+1) ( d + 1 ) [72 ] demonstrated that information encoded in pure ( 1 + 1 ) d (1+1)d ( 1 + 1 ) d ( d + 1 ) (d+1) ( d + 1 )
One approach to analyzing the TFIM is to follow an identical strategy as in the TFTC; i.e., utilize stoquasticity to describe code states via ITE, and then construct the optimal decoder using the quantum-classical mapping. However, as we shall see, the presence of a spontaneously broken global ( 0 (0 ( 0 1 1 1
Setup
The ( d + 1 ) (d+1) ( d + 1 ) 20
H g = − ∑ ⟨ v v ′ ⟩ Z v Z v ′ − g ∑ v X v . (107) H_g=-\sum_{\langle vv'\rangle}Z_vZ_{v'}-g\sum_vX_v . \tag*{(107)} H g = − ⟨ v v ′ ⟩ ∑ Z v Z v ′ − g v ∑ X v . (107) where v v v d d d N ≡ L d N \equiv L^d N ≡ L d g = 0 g = 0 g = 0 C 0 \mathcal{C}_0 C 0 H g = 0 H_{g=0} H g = 0 + 1 +1 + 1 Z v Z v ′ Z_v Z_{v'} Z v Z v ′ ∣ 0 0 ⟩ ≡ ∣ 0 ⟩ ⊗ N \lvert0_0\rangle\equiv\lvert0\rangle^{\otimes N} ∣ 0 0 ⟩ ≡ ∣ 0 ⟩ ⊗ N ∣ 1 0 ⟩ ≡ ∣ 1 ⟩ ⊗ N \lvert1_0\rangle\equiv\lvert1\rangle^{\otimes N} ∣ 1 0 ⟩ ≡ ∣ 1 ⟩ ⊗ N Z 2 \mathbb{Z}_2 Z 2 η ^ ≡ ∏ v X v \hat{\eta} \equiv\prod_v X_v η ^ ≡ ∏ v X v
For g g g g c g_c g c H g H_g H g Z 2 \mathbb{Z}_2 Z 2 H g H_g H g C g \mathcal{C}_g C g L d L^d L d C g \mathcal{C}_g C g (4) (up to exponentially small corrections) [86 ] , but only for Z 2 \mathbb{Z}_2 Z 2 O O O C g \mathcal{C}_g C g Z 2 \mathbb{Z}_2 Z 2 g < g c g < g_c g < g c
In more detail, let ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩ ∣ 1 g ⟩ \lvert1_g\rangle ∣ 1 g ⟩ 21 H g H_g H g η ^ ∣ 0 g ⟩ = ∣ 1 g ⟩ \hat{\eta}\lvert0_g\rangle= \lvert1_g\rangle η ^ ∣ 0 g ⟩ = ∣ 1 g ⟩
m ( g ) ≡ ⟨ 0 g ∣ m ^ ∣ 0 g ⟩ ≡ 1 N ∑ v ⟨ 0 g ∣ Z v ∣ 0 g ⟩ ∼ O ( 1 ) , (108) m(g) \equiv\langle0_g \lvert\hat{m} \rvert0_g\rangle\equiv\frac{1}{N}\sum_v \langle0_g \lvert Z_v \rvert0_g\rangle
\sim O(1), \tag*{(108)} m ( g ) ≡ ⟨ 0 g ∣ m ^ ∣ 0 g ⟩ ≡ N 1 v ∑ ⟨ 0 g ∣ Z v ∣ 0 g ⟩ ∼ O ( 1 ) , (108) and ⟨ 1 g ∣ m ^ ∣ 1 g ⟩ = − m ( g ) \langle1_g\lvert\hat{m}\rvert1_g\rangle=-m(g) ⟨ 1 g ∣ m ^ ∣ 1 g ⟩ = − m ( g ) ∣ ψ g ⟩ ∈ C g \lvert\psi_g\rangle\in\mathcal{C}_g ∣ ψ g ⟩ ∈ C g
∣ ψ g ⟩ = ∑ σ = 0 , 1 b σ ∣ σ g ⟩ , (109) \lvert\psi_g\rangle=\sum_{\sigma=0,1}b_\sigma\lvert\sigma_g\rangle, \tag*{(109)} ∣ ψ g ⟩ = σ = 0 , 1 ∑ b σ ∣ σ g ⟩ , (109) where b σ ∈ C b_\sigma\in\mathbb{C} b σ ∈ C Z 2 \mathbb{Z}_2 Z 2 C g \mathcal{C}_g C g Z 2 \mathbb{Z}_2 Z 2
Given a code state ∣ ψ g ⟩ ∈ C g \lvert\psi_g\rangle\in\mathcal{C}_g ∣ ψ g ⟩ ∈ C g
E v ( ρ ) = ( 1 − p ) ρ + p X v ρ X v (110) \mathcal{E}_v(\rho)=(1-p)\rho+pX_v\rho X_v \tag*{(110)} E v ( ρ ) = ( 1 − p ) ρ + p X v ρ X v (110) to each vertex v v v ρ g ≡ E ( ∣ ψ g ⟩ ⟨ ψ g ∣ ) \rho_g\equiv\mathcal{E}(\lvert\psi_g\rangle\langle\psi_g\rvert) ρ g ≡ E (∣ ψ g ⟩ ⟨ ψ g ∣) E ≡ ∏ v E v \mathcal{E}\equiv\prod_v\mathcal{E}_v E ≡ ∏ v E v
As in the TFTC, we will attempt to recover ∣ ψ g ⟩ \lvert\psi_g\rangle ∣ ψ g ⟩ ρ g \rho_g ρ g Z v Z v ′ Z_vZ_{v'} Z v Z v ′
Π ^ s = ∏ ⟨ v v ′ ⟩ ( 1 + ( − 1 ) s v v ′ Z v Z v ′ 2 ) , (111) \hat{\Pi}_{\boldsymbol{s}}=\prod_{\langle vv'\rangle}\left(\frac{1+(-1)^{s_{vv'}}Z_vZ_{v'}}{2}\right), \tag*{(111)} Π ^ s = ⟨ v v ′ ⟩ ∏ ( 2 1 + ( − 1 ) s v v ′ Z v Z v ′ ) , (111) where s ≡ { s v v ′ } \boldsymbol{s}\equiv\{s_{vv'}\} s ≡ { s v v ′ } s v v ′ = 1 s_{vv'}=1 s v v ′ = 1 s v v ′ = 0 s_{vv'}=0 s v v ′ = 0 v v v v ′ v' v ′ ρ g ( s ) ≡ Π ^ s ρ g Π ^ s / P s \rho_g(\boldsymbol{s})\equiv\hat{\Pi}_{\boldsymbol{s}}\rho_g\hat{\Pi}_{\boldsymbol{s}}/P_{\boldsymbol{s}} ρ g ( s ) ≡ Π ^ s ρ g Π ^ s / P s P s = tr [ Π ^ s ρ g ] P_{\boldsymbol{s}}=\operatorname{tr}[\hat{\Pi}_{\boldsymbol{s}}\rho_g] P s = tr [ Π ^ s ρ g ]
The state ρ g ( s ) \rho_g(\boldsymbol{s}) ρ g ( s ) C 0 \mathcal{C}_0 C 0 X X X X s X_{\boldsymbol{s}} X s X s Z v Z v ′ = ( − 1 ) s v v ′ Z v Z v ′ X s X_{\boldsymbol{s}}Z_vZ_{v'}=(-1)^{s_{vv'}}Z_vZ_{v'}X_{\boldsymbol{s}} X s Z v Z v ′ = ( − 1 ) s v v ′ Z v Z v ′ X s X s X_{\boldsymbol{s}} X s s \boldsymbol{s} s Z 2 \mathbb{Z}_2 Z 2 η ^ \hat{\eta} η ^ X s X_{\boldsymbol{s}} X s Z v Z v ′ = + 1 Z_vZ_{v'}=+1 Z v Z v ′ = + 1 C g \mathcal{C}_g C g U g ≡ ∑ σ ∣ σ g ⟩ ⟨ σ 0 ∣ U_g\equiv\sum_\sigma\lvert\sigma_g\rangle\langle\sigma_0\rvert U g ≡ ∑ σ ∣ σ g ⟩ ⟨ σ 0 ∣
Our precise claim is that the average decoding fidelity F ‾ \overline{\mathcal{F}} F g < g c g<g_c g < g c p < 1 2 p<\frac{1}{2} p < 2 1
F ‾ ( ∣ ψ g ⟩ ) = ∑ s P s ⟨ ψ g ∣ U g X s ρ g ( s ) X s † U g † ∣ ψ g ⟩ = 1 − O ( e − ( L / ξ ) d ) , (112) \begin{aligned}
\overline{\mathcal{F}}(\lvert\psi_g\rangle)=\sum_{\boldsymbol{s}}P_{\boldsymbol{s}}\langle\psi_g\lvert U_gX_{\boldsymbol{s}}\rho_g(\boldsymbol{s})X_{\boldsymbol{s}}^\dagger U_g^\dagger\lvert\psi_g\rangle\\
&=1-O(e^{-(L/\xi)^d}), \tag*{(112)}
\end{aligned} F (∣ ψ g ⟩) = s ∑ P s ⟨ ψ g ∣ U g X s ρ g ( s ) X s † U g † ∣ ψ g ⟩ = 1 − O ( e − ( L / ξ ) d ) , (112) for any ∣ ψ g ⟩ ∈ C g \lvert\psi_g\rangle\in\mathcal{C}_g ∣ ψ g ⟩ ∈ C g ξ \xi ξ g → g c g\to g_c g → g c (112) , first note that the bit-flip channel E \mathcal{E} E strong Z 2 \mathbb{Z}_2 Z 2
η ^ E ( ρ ) = E ( η ^ ρ ) , E ( ρ ) η ^ = E ( ρ η ^ ) (113) \hat{\eta}\mathcal{E}(\rho)=\mathcal{E}(\hat{\eta}\rho),\qquad\mathcal{E}(\rho)\hat{\eta}=\mathcal{E}(\rho\hat{\eta}) \tag*{(113)} η ^ E ( ρ ) = E ( η ^ ρ ) , E ( ρ ) η ^ = E ( ρ η ^ ) (113) for any ρ \rho ρ U g X s U_gX_{\boldsymbol{s}} U g X s Z 2 \mathbb{Z}_2 Z 2 § 3.11 ), it is therefore sufficient to focus on decoding the specific spin-polarized code state ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩
Magnetization in Decohered State
As noted in Eq. (108) , the spin-polarized state ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩ g < g c g<g_c g < g c g g g g c g_c g c m m m m ( g ) ∼ ( g c − g ) β m(g)\sim(g_c-g)^\beta m ( g ) ∼ ( g c − g ) β β \beta β ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩ m ^ \hat{m} m ^ 1 / N 1/N 1/ N
σ m 2 ( g ) = ⟨ 0 g ∣ m ^ 2 ∣ 0 g ⟩ − ⟨ 0 g ∣ m ^ ∣ 0 g ⟩ 2 = 1 N 2 ∑ v , v ′ [ ⟨ 0 g ∣ Z v Z v ′ ∣ 0 g ⟩ − ⟨ 0 g ∣ Z v ∣ 0 g ⟩ ⟨ 0 g ∣ Z v ′ ∣ 0 g ⟩ ] ∼ O ( 1 / N ) . (114) \begin{aligned}
\sigma_m^2(g)=\langle0_g\lvert\hat{m}^2\rvert0_g\rangle-\langle0_g\lvert\hat{m}\rvert0_g\rangle^2 \\
&=\frac{1}{N^2}\sum_{v,v'}\left[\langle0_g\lvert Z_vZ_{v'}\rvert0_g\rangle-\langle0_g\lvert Z_v\rvert0_g\rangle\langle0_g\lvert Z_{v'}\rvert0_g\rangle\right]
\sim O(1/N). \tag*{(114)}
\end{aligned} σ m 2 ( g ) = ⟨ 0 g ∣ m ^ 2 ∣ 0 g ⟩ − ⟨ 0 g ∣ m ^ ∣ 0 g ⟩ 2 = N 2 1 v , v ′ ∑ [ ⟨ 0 g ∣ Z v Z v ′ ∣ 0 g ⟩ − ⟨ 0 g ∣ Z v ∣ 0 g ⟩ ⟨ 0 g ∣ Z v ′ ∣ 0 g ⟩ ] ∼ O ( 1/ N ) . (114) Therefore, in the thermodynamic limit, essentially all of the wavefunction amplitude in ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩ m ( g ) m(g) m ( g ) 22 m ^ \hat{m} m ^ ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩ m ( g ) m(g) m ( g ) N N N
P [ ∣ m ^ − m ( g ) ∣ ≥ ε ] ∼ O ( e − ε 2 ( L / ξ ) d ) . (115) \mathbb{P}\left[\left\lvert\hat{m}-m(g)\right\rvert\ge\varepsilon\right]\sim\mathcal{O}\left(e^{-\varepsilon^2(L/\xi)^d}\right). \tag*{(115)} P [ ∣ m ^ − m ( g ) ∣ ≥ ε ] ∼ O ( e − ε 2 ( L / ξ ) d ) . (115) where ξ \xi ξ ρ g 0 ≡ E ( ∣ 0 g ⟩ ⟨ 0 g ∣ ) \rho_g^0\equiv\mathcal{E}(\lvert0_g\rangle\langle0_g\rvert) ρ g 0 ≡ E (∣ 0 g ⟩ ⟨ 0 g ∣)
m ( g , p ) ≡ tr [ m ^ ρ g 0 ] = ⟨ 0 g ∣ E ( m ^ ) ∣ 0 g ⟩ = ( 1 − 2 p ) m ( g , p = 0 ) , (116) m(g,p)\equiv\operatorname{tr}[\hat{m}\rho_g^0]=\langle0_g\lvert\mathcal{E}(\hat{m})\rvert0_g\rangle
=(1-2p)m(g,p=0), \tag*{(116)} m ( g , p ) ≡ tr [ m ^ ρ g 0 ] = ⟨ 0 g ∣ E ( m ^ )∣ 0 g ⟩ = ( 1 − 2 p ) m ( g , p = 0 ) , (116) where we have used the fact that the Kraus operators of E \mathcal{E} E E ( Z v ) = ( 1 − 2 p ) Z v \mathcal{E}(Z_v)=(1-2p)Z_v E ( Z v ) = ( 1 − 2 p ) Z v ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩ p < 1 2 p<\frac{1}{2} p < 2 1
σ m 2 ( g , p ) = ( 1 − 2 p ) 2 σ m 2 ( g , p = 0 ) + 1 − ( 1 − 2 p ) 2 N , (117) \sigma_m^2(g,p)=(1-2p)^2\sigma_m^2(g,p=0)+\frac{1-(1-2p)^2}{N}, \tag*{(117)} σ m 2 ( g , p ) = ( 1 − 2 p ) 2 σ m 2 ( g , p = 0 ) + N 1 − ( 1 − 2 p ) 2 , (117) and therefore remains O ( 1 / N ) \mathcal{O}(1/N) O ( 1/ N ) g < g c g<g_c g < g c p < 1 2 p<\frac{1}{2} p < 2 1 ρ g σ ≡ E ( ∣ σ g ⟩ ⟨ σ g ∣ ) \rho_g^\sigma\equiv\mathcal{E}(\lvert\sigma_g\rangle\langle\sigma_g\rvert) ρ g σ ≡ E (∣ σ g ⟩ ⟨ σ g ∣) p < 1 2 p<\frac{1}{2} p < 2 1
Decoding via Stabilizer Measurements and Majority Vote
Now, suppose that we measure the stabilizers Z v Z v ′ Z_vZ_{v'} Z v Z v ′ ρ g 0 ≡ E ( ∣ 0 g ⟩ ⟨ 0 g ∣ ) \rho_g^0\equiv\mathcal{E}(\lvert0_g\rangle\langle0_g\rvert) ρ g 0 ≡ E (∣ 0 g ⟩ ⟨ 0 g ∣) ρ g 0 ( s ) \rho_g^0(s) ρ g 0 ( s ) P s P_s P s
ρ g 0 ( s ) = X s ( ∑ σ , σ ′ = 0 , 1 M σ σ ′ ( s ) ∣ σ 0 ⟩ ⟨ σ 0 ′ ∣ ) X s , (118) \rho_g^0(s)=X_s\left(\sum_{\sigma,\sigma'=0,1}M_{\sigma\sigma'}(s)\lvert\sigma_0\rangle\langle\sigma'_0\rvert\right)X_s, \tag*{(118)} ρ g 0 ( s ) = X s σ , σ ′ = 0 , 1 ∑ M σ σ ′ ( s ) ∣ σ 0 ⟩ ⟨ σ 0 ′ ∣ X s , (118) where X s X_s X s X s ∣ 0 0 ⟩ X_s\lvert0_0\rangle X s ∣ 0 0 ⟩ X s ∣ 1 0 ⟩ X_s\lvert1_0\rangle X s ∣ 1 0 ⟩
If we were to subsequently measure Z v Z_v Z v m ^ \hat{m} m ^ X s X_s X s M σ σ ( s ) M_{\sigma\sigma}(s) M σ σ ( s ) ρ g 0 \rho_g^0 ρ g 0
M 00 ( s ) = P [ m ^ > 0 ∣ s ] , M 11 ( s ) = P [ m ^ < 0 ∣ s ] . (119) M_{00}(s)=\mathbb{P}[\hat{m}>0\mid s],\qquad M_{11}(s)=\mathbb{P}[\hat{m}<0\mid s]. \tag*{(119)} M 00 ( s ) = P [ m ^ > 0 ∣ s ] , M 11 ( s ) = P [ m ^ < 0 ∣ s ] . (119) Using the majority vote decoder X s X_s X s s s s
F s ( ∣ 0 g ⟩ ) = ⟨ 0 g ∣ X s ρ g 0 ( s ) X s ∣ 0 g ⟩ = M 00 ( s ) , (120) \mathcal{F}_s(\lvert0_g\rangle)=\langle0_g\lvert X_s\rho_g^0(s)X_s\rvert0_g\rangle=M_{00}(s), \tag*{(120)} F s (∣ 0 g ⟩) = ⟨ 0 g ∣ X s ρ g 0 ( s ) X s ∣ 0 g ⟩ = M 00 ( s ) , (120) and the average decoding fidelity is given by
F ‾ ( ∣ 0 g ⟩ ) = ∑ s P s M 00 ( s ) = P [ m ^ > 0 ] , (121) \overline{\mathcal{F}}(\lvert0_g\rangle)=\sum_sP_sM_{00}(s)=\mathbb{P}[\hat{m}>0], \tag*{(121)} F (∣ 0 g ⟩) = s ∑ P s M 00 ( s ) = P [ m ^ > 0 ] , (121) i.e., F ‾ \overline{\mathcal{F}} F unconditional probability of observing a positive magnetization in the state ρ g 0 \rho_g^0 ρ g 0 g < g c g<g_c g < g c p < 1 2 p<\frac{1}{2} p < 2 1
F ‾ ( ∣ 0 g ⟩ ) = 1 − O ( e − ( L / ξ ) d ) . (122) \overline{\mathcal{F}}(\lvert0_g\rangle)=1-\mathcal{O}\left(e^{-(L/\xi)^d}\right). \tag*{(122)} F (∣ 0 g ⟩) = 1 − O ( e − ( L / ξ ) d ) . (122) In turn, this implies that F s \mathcal{F}_s F s ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩ any code state; see Appendix § 3.11 ) with high fidelity for all g < g c g<g_c g < g c p < 1 2 p<\frac{1}{2} p < 2 1 p c = 1 2 p_c=\frac{1}{2} p c = 2 1
We note that, unlike our analysis of the TFTC, stoquasticity played essentially no role in the preceding analysis. The only requirements were (i) a pair of short-range entangled symmetry-broken ground states, related by the Z 2 \mathbb{Z}_2 Z 2 Z v Z_v Z v Z 2 \mathbb{Z}_2 Z 2 [96 ] , while property (ii) holds for generic 23 Z 2 \mathbb{Z}_2 Z 2 g < g c g<g_c g < g c Z 2 \mathbb{Z}_2 Z 2 H g = 0 H_{g=0} H g = 0
(1+1)d Transverse-Field Ising Model: Numerical Results
We now numerically demonstrate the preceding result in the (1+1)d case, where efficient numerics are available via the Gaussian fermion covariance matrix formalism. Figure 11 presents our main numerical results, which we discuss in the following. See Appendix § 3.15 for further details on our numerical protocol, and additional numerical plots.
Figure 11. Average decoding fidelity F ‾ \overline{F} F g g g p p p Z j Z j + 1 Z_jZ_{j+1} Z j Z j + 1 p p p g = 0.99 g=0.99 g = 0.99 L L L p < p c = 1 2 p<p_c=\frac{1}{2} p < p c = 2 1 F ‾ \overline{F} F L L L (124) , where for simplicity we set ϕ ( L ) = 2 F ‾ ( p = 0 , L ) − 1 \phi(L)=2\overline{F}(p=0,L)-1 ϕ ( L ) = 2 F ( p = 0 , L ) − 1 g g g p = 0.4 p=0.4 p = 0.4 L L L F ‾ \overline{F} F L L L g < g c g<g_c g < g c ( p , g ) (p,g) ( p , g ) L = 256 L=256 L = 256 F ‾ \overline{F} F ( g , p ) ∈ [ 0 , 1 ) × [ 0 , 1 2 ) (g,p)\in[0,1)\times[0,\frac{1}{2}) ( g , p ) ∈ [ 0 , 1 ) × [ 0 , 2 1 )
We begin by preparing a superposition of the two lowest-energy eigenstates of the (1+1)d TFIM, which yields the symmetry-broken ground state ∣ 0 g ⟩ |0_g\rangle ∣ 0 g ⟩ g < g c g<g_c g < g c g > g c g>g_c g > g c open boundary conditions for numerical convenience.
For each qubit j = 1 , … , L j=1,\ldots,L j = 1 , … , L X j X_j X j p p p 1 − p 1-p 1 − p e ≡ { e j } e\equiv\{e_j\} e ≡ { e j } e j = 1 e_j=1 e j = 1 e j = 0 e_j=0 e j = 0 j j j Z j Z j + 1 Z_jZ_{j+1} Z j Z j + 1 j = 1 , … , L − 1 j=1,\ldots,L-1 j = 1 , … , L − 1 s ≡ { s j } s\equiv\{s_j\} s ≡ { s j } X s X_s X s X j X_j X j j j j
Given the bit-flip realization e e e s s s ∣ 0 g ⟩ |0_g\rangle ∣ 0 g ⟩ ∣ ψ g ( s ∣ e ) ⟩ = b 0 ∣ 0 g ⟩ + b 1 ∣ 1 g ⟩ |\psi_g(s|e)\rangle=b_0|0_g\rangle+b_1|1_g\rangle ∣ ψ g ( s ∣ e )⟩ = b 0 ∣ 0 g ⟩ + b 1 ∣ 1 g ⟩ F s ∣ e = ∣ b 0 ∣ 2 \mathcal{F}_{s|e}=|b_0|^2 F s ∣ e = ∣ b 0 ∣ 2 Z 1 = + 1 Z_1=+1 Z 1 = + 1
F s ∣ e ( ∣ 0 g ⟩ ) = 1 2 [ 1 + ⟨ Z 1 ⟩ s , e ] . (123) \mathcal{F}_{s|e}(|0_g\rangle)=\frac{1}{2}\left[1+\langle Z_1\rangle_{s,e}\right]. \tag*{(123)} F s ∣ e ( ∣ 0 g ⟩) = 2 1 [ 1 + ⟨ Z 1 ⟩ s , e ] . (123) Note that this single-site observable can be computed without implementing the majority vote on every site; we need only to flip site 1 whenever it belongs to the minority domain. Finally, the average fidelity F ‾ \overline{F} F e e e s s s
Figure 11 (a) depicts the average decoding fidelity F ‾ ( ∣ 0 g ⟩ ) \overline{F}(|0_g\rangle) F ( ∣ 0 g ⟩) p p p g = 0.99 g=0.99 g = 0.99 g c = 1 g_c=1 g c = 1 Figure 18 for similar plots with other values of g g g g = 0.999 g=0.999 g = 0.999 p < p c = 1 2 p<p_c=\frac{1}{2} p < p c = 2 1 F ‾ \overline{F} F (122) . For p > 1 2 p > \frac{1}{2} p > 2 1 F ‾ ( 1 − p ) = 1 − F ‾ ( p ) \overline{F}(1-p)=1-\overline{F}(p) F ( 1 − p ) = 1 − F ( p ) F ‾ \overline{F} F
Figure 18. Average decoding fidelity F ‾ \overline{F} F Z j Z j + 1 Z_jZ_{j+1} Z j Z j + 1 p p p g g g
F ‾ ( p , L ) = 1 2 + ϕ ( L ) Φ ( ( p − p c ) L d / 2 ) , (124) \overline{F}(p,L)=\frac{1}{2}+\phi(L)\Phi\left((p-p_c)L^{d/2}\right), \tag*{(124)} F ( p , L ) = 2 1 + ϕ ( L ) Φ ( ( p − p c ) L d /2 ) , (124) where d = 1 d=1 d = 1 Φ ( x ) \Phi(x) Φ ( x ) ∓ 1 2 \mp\frac{1}{2} ∓ 2 1 x → ± ∞ x\to\pm\infty x → ± ∞ ϕ ( L ) = 1 − O ( e − L / ξ ) \phi(L)=1-O(e^{-L/\xi}) ϕ ( L ) = 1 − O ( e − L / ξ ) ϕ ( L ) = 2 F ‾ ( 0 , L ) − 1 \phi(L)=2\overline{F}(0,L)-1 ϕ ( L ) = 2 F ( 0 , L ) − 1 d / 2 d/2 d /2 (115) : for a fixed p < p c = 1 2 p<p_c=\frac{1}{2} p < p c = 2 1 m ( g , p ) m(g,p) m ( g , p ) ( p c − p ) (p_c-p) ( p c − p ) 1 / L d 1/L^d 1/ L d exp { − O ( ( p − p c ) 2 L d ) } \exp\{-O((p-p_c)^2L^d)\} exp { − O (( p − p c ) 2 L d )}
Figure 11 (b) depicts the same average decoding fidelity, now as a function of the transverse field strength g g g p = 0.4 p=0.4 p = 0.4 Figure 19 for other values of p p p F ‾ \overline{F} F L L L g < 1 g<1 g < 1 g > 1 g>1 g > 1 ∣ 00 ⟩ \lvert00\rangle ∣ 00 ⟩ O ( 1 / L ) O(1/\sqrt{L}) O ( 1/ L ) O ( 1 / L ) O(1/L) O ( 1/ L ) g → ∞ g\to\infty g → ∞ g > 1 g>1 g > 1 O ( 1 ) O(1) O ( 1 ) L L L 1 2 \frac{1}{2} 2 1
Figure 19. Average decoding fidelity F ‾ \overline{F} F Z j Z j + 1 Z_jZ_{j+1} Z j Z j + 1 g g g p p p L L L
Finally, Figure 11 (c) depicts the same average decoding fidelity for the fixed system size L = 256 L=256 L = 256 g g g p p p g < 1 g<1 g < 1 p < 1 2 p<\frac{1}{2} p < 2 1
Discussion
In this work, we have investigated QEC protocols for near-critical topological quantum codes, defined by the ground-state manifold of the transverse-field toric code (TFTC). For small transverse fields h h h Z 2 \mathbb{Z}_2 Z 2
We first considered stabilizer measurements of the pure (un-decohered) ground states of the TFTC, where we proved that the encoded information is preserved with high fidelity, with high probability in the measurement outcome, throughout the topological phase h < h c h<h_c h < h c
Building on the preceding result, we then investigated the optimal postmeasurement decoder for the TFTC subjected to independent bit-flip and phase-flip decoherence. Strikingly, the decoding threshold for bit-flip decoherence close to the quantum critical point is only slightly reduced from its value in the stabilizer limit (at least within our spacetime-isotropic numerical simulations), providing direct support for the field-theoretic prediction in Ref. [49]. On the other hand, the threshold for phase-flip decoherence is exactly equal to its value in the stabilizer toric code, in part because the transverse field does not generate fluctuations of the e e e
An interesting feature of our decoders is the scaling of its resource requirements as the critical point is approached: although we performed all numerical simulations on system sizes of fixed aspect ratio for simplicity, in principle the extent L τ L_\tau L τ ξ \xi ξ h = 0 h=0 h = 0 h h h 2 d 2d 2 d L τ ∼ ξ L_\tau\sim\xi L τ ∼ ξ ( ∣ h c − h ∣ ∼ L − 1 / ν ) (|h_c-h|\sim L^{-1/\nu}) ( ∣ h c − h ∣ ∼ L − 1/ ν )
Relation to Previous Works
The present work is connected to several different themes in recent literature. Most directly, it relates to recent studies of mixed-state topological order [9 , 10 , 12 , 13 , 17 , 18 , 19 , 39 , 105 , 106 , 107 ] , and mixed-state phases of matter more broadly [11 , 14 , 15 , 16 , 21 , 108 , 109 , 110 , 111 , 112 , 113 ] . While much of this literature has focused on mixed states obtained by decohering fixed-point states (e.g. stabilizer states), our setting provides a tractable example in which substantial analytic and numerical control persists beyond the fixed-point limit. Such studies help to establish the broader phenomenology of mixed-state phases beyond particular solvable examples. The interplay between pure-state quantum critical phenomena and information-theoretic decoherence-induced transitions is of particular interest, and deserves further study.
There are also similarities between the present work and previous results on measurements [114 , 115 , 116 , 117 ] and decoherence [11 , 118 ] performed on critical quantum ground states. As in that setting, both measurements and decoherence appear within the present formalism as a disordered defect along a codimension-one slice of Euclidean spacetime. From this perspective, it is tempting to analyze the effect of stabilizer measurements near the quantum critical point using a similar approach: namely, by weakening the measurement projector Π ^ s \hat{\Pi}_s Π ^ s Π ^ s ( μ ) ∝ exp { μ ∑ v ( − 1 ) s v A v + μ ∑ p ( − 1 ) s p B p } \hat{\Pi}_s(\mu) \propto\exp\left\{\mu\sum_v(-1)^{s_v}A_v+\mu\sum_p(-1)^{s_p}B_p\right\} Π ^ s ( μ ) ∝ exp { μ ∑ v ( − 1 ) s v A v + μ ∑ p ( − 1 ) s p B p } [49 ] , one cannot preclude the possibility of a measurement-induced transition at a finite value of μ \mu μ [65 , 66 , 67 , 68 , 69 , 70 , 71 ] , it would be interesting to analyze whether arbitrarily weak stabilizer measurements similarly destroy the information, or if there might exist a transition driven by the measurement strength.
Along different lines, Ref. [73 ] previously proposed a relation between the pure-state decoding problem analyzed in § 3 and the emergence of higher-form symmetries. In the simple case of the TFIM, the connection is particularly transparent. Specifically, the d d d d d d Z ˉ v \bar{Z}_v Z ˉ v [86 ] , and any Z 2 \mathbb{Z}_2 Z 2 Z v Z_v Z v Z ˉ v \bar{Z}_v Z ˉ v 1 N ∑ v Z v \frac{1}{N}\sum_v Z_v N 1 ∑ v Z v d d d
On the other hand, the same logic does not obviously carry over to the case of the TFTC. In this setting, the code space exhibits an emergent “logical-Z Z Z h < h c h<h_c h < h c X X X d d d Z Z Z L L L Z Z Z B p B_p B p [86 ] . From this perspective, the precise connection between emergent higher-form symmetries and the success of QEC via stabilizer measurement remains partially unresolved24 h = 0 h=0 h = 0 h < h c h<h_c h < h c Z Z Z [119 ] ?
There are also interesting potential connections to recent work on bottlenecks in the classical dynamics of Z 2 \mathbb{Z}_2 Z 2 [120 , 121 ] . In that setting, the large free energy cost of long flux loops discussed in Sec. III B leads to long memory lifetimes of the gauge theory under local dissipative dynamics. Interestingly, Refs. [120 , 121 ] were able to extend this result beyond the pure LGT to include a coupling to matter, which breaks the microscopic one-form symmetry relating the code states; in some sense, this symmetry re-emerges entropically within the topological phase, leading to well-defined code states separated by bottlenecks. It would be interesting to determine whether similar techniques can be applied to analyze QEC in the toric code perturbed by transverse and longitudinal fields (discussed in the following section), which is related to the LGT with matter by the quantum-classical mapping [122 ] .
Finally, it is useful to compare our numerical method with previous work on QEC and measurement-induced phenomena. There are technical similarities with Ref. [123 ] , which showed how quantum Monte Carlo (QMC) can be used to study measurement-induced collective phenomena in ground states. In that work as well as here, post-measurement ground states are represented as imaginary-time path integrals with additional constraints or operator insertions at a particular imaginary time. Key features of the WA implemented here and the directed loop algorithm in Ref. [123 ] are that they (i) satisfy detailed balance with respect to the post-measurement ensemble and (ii) remain ergodic in spite of the constraints associated with measurements. These features are shared with other recent applications of the WA in the context of QEC for stabilizer codes [79 , 100 ] and symmetry-enriched analogs [42 ] .
In developing our QEC protocols for the TFTC, many of the technical details were simplified greatly by the presence of an exact 1-form “logical-X” symmetry at all transverse field strengths. Given that the topological phase of the toric code is stable to all local perturbations [6 , 87 ] , it is interesting to ask whether “bare” stabilizer measurements remain an effective decoding strategy when this symmetry is microscopically broken. A model of particular interest is the Fradkin-Shenker (FS) model [53 , 54 , 64 , 89 , 122 , 124 ] , i.e., the toric code perturbed by both transverse and longitudinal fields:
H h = − ∑ v A v − ∑ p B p − h X ∑ ℓ X ℓ − h Z ∑ ℓ Z ℓ , (125) H_h = - \sum_v A_v - \sum_p B_p - h_X \sum_\ell X_\ell- h_Z \sum_\ell Z_\ell, \tag*{(125)} H h = − v ∑ A v − p ∑ B p − h X ℓ ∑ X ℓ − h Z ℓ ∑ Z ℓ , (125) where h = ( h X , h Z ) \mathbf{h}=(h_X,h_Z) h = ( h X , h Z ) h X h_X h X m m m B p = − 1 B_p=-1 B p = − 1 h Z h_Z h Z e e e h Z h_Z h Z Z Z Z X X X not in the stabilizer basis: any imaginary-time process in which an e e e m m m − 1 -1 − 1 e e e m m m
On one hand, the presence of h Z h_Z h Z h Z = 0 h_Z=0 h Z = 0 H h H_h H h [64 ] , and so it is possible to make several well-founded conjectures on the decoding problem in this model.
For example, it is straightforward to extend the construction of the sign-free ITE code states ∣ φ h σ ⟩ ∝ e − β H h / 2 ∣ φ 0 σ ⟩ \lvert\varphi_h^\sigma\rangle\propto e^{-\beta H_h/2}\lvert\varphi_0^\sigma\rangle ∣ φ h σ ⟩ ∝ e − β H h /2 ∣ φ 0 σ ⟩ h Z = 0 h_Z=0 h Z = 0 unitary map from the stabilizer code space C 0 \mathcal{C}_0 C 0 C h \mathcal{C}_h C h
⟨ φ h σ ∣ φ h σ ′ ⟩ = ⟨ φ 0 σ ∣ e − β H h ∣ φ 0 σ ′ ⟩ 1 4 tr [ Π 0 e − β H h ] = δ σ σ ′ + O ( e − L / ξ ) . (126) \langle\varphi_h^\sigma\vert\varphi_h^{\sigma'}\rangle= \frac{\langle\varphi_0^\sigma\vert e^{-\beta H_h}\vert\varphi_0^{\sigma'}\rangle}{\frac{1}{4}\operatorname{tr}\left[\Pi_0 e^{-\beta H_h}\right]} = \delta_{\sigma\sigma'} + O\left(e^{-L/\xi}\right). \tag*{(126)} ⟨ φ h σ ∣ φ h σ ′ ⟩ = 4 1 tr [ Π 0 e − β H h ] ⟨ φ 0 σ ∣ e − β H h ∣ φ 0 σ ′ ⟩ = δ σ σ ′ + O ( e − L / ξ ) . (126) Indeed, similar to the discussion of Sec. III B, the processes leading to nonzero off-diagonal matrix elements can be regarded as arising from imaginary spacetime processes where m m m e e e h h h L L L h X h_X h X h Z h_Z h Z m m m e e e m m m
Given that both e e e m m m A v A_v A v B p B_p B p τ = 0 \tau=0 τ = 0 C h \mathcal{C}_h C h h Z = 0 h_Z=0 h Z = 0 prove this to be the case using stoquasticity; see § 3 A. With nonzero h Z h_Z h Z (126) to be true. The key difference is that, whereas the postmeasurement basis states { ∣ φ h σ ( s ) ⟩ } \{\lvert\varphi_h^\sigma(s)\rangle\} {∣ φ h σ ( s )⟩} h Z h_Z h Z e e e e e e m m m H h H_h H h e e e m m m
In the absence of analytical support for the protection of information under stabilizer measurements (either with or without decoherence), one strategy is to construct an optimal postmeasurement decoder and numerically evaluate its performance. Here we arrive at another stumbling block: such an optimal decoder involves conditioning on both the e e e m m m e e e m m m [122 ] . Unfortunately, this representation does not (to our knowledge) admit a useful nonlocal Monte Carlo update like the WA which efficiently tunnels between the different worldline homology sectors. Local Monte Carlo updates like the Metropolis algorithm can be employed to simulate the model [91 , 122 , 125 ] , but such algorithms are likely to face severe bottlenecks for estimating the statistics of the different homology sectors.
In spite of these challenges, we expect that much more can be concretely said about the protection and recovery of quantum information in the Fradkin-Shenker model. For instance, perhaps it is possible to simultaneously utilize stoquasticity in both the Pauli-X X X Z Z Z
Open Questions and Future Directions
Much of the physical intuition, and efficient numerical methods, behind our QEC protocols arose from stoquasticity and the quantum-classical mapping. Nevertheless, one may reasonably ask if a more physical mechanism than stoquasticity can guarantee the protection of quantum information under stabilizer measurement. One specific example of a non-stoquastic deformation we expect to be decodable via stabilizer measurements is the toric code in a Pauli-Y Y Y H Y = H 0 − h Y ∑ ℓ Y ℓ H_Y = H_0 - h_Y \sum_{\ell} Y_\ell H Y = H 0 − h Y ∑ ℓ Y ℓ [126 , 127 , 128 ] . More broadly, although we have recently argued [49 ] that a finite intrinsic error threshold should remain robust to generic gap-preserving deformations of the toric code, we expect that such deformed codes should generically not be decodable via stabilizer measurements. It would be especially interesting to study the breakdown of our QEC protocol as one tunes away from a stoquastic limit.
A related open question concerns the optimality of stabilizer measurements for decoding bit-flip decoherence. Although we have (non-rigorously) argued that stabilizer measurements can be utilized as part of an optimal decoding strategy for the TFTC, it remains possible that the ‘intrinsic’ threshold is slightly larger than the threshold of our optimal postmeasurement decoder. In principle, this question can be resolved by (for example) comparing the coherent information I c ( R ⟩ Q ) I_c(R\rangle Q) I c ( R ⟩ Q ) ∑ s p s I c ( R ⟩ Q ∣ s ) \sum_s p_s I_c(R\rangle Q|s) ∑ s p s I c ( R ⟩ Q ∣ s ) [61 , 80 ] , we expect from the discussion of Sec. V E that both quantities exhibit a transition from their maximal value to zero at the same critical decoherence strength. If, on the contrary, the two quantities can be shown to exhibit different transition points, then stabilizer measurements cannot serve as an optimal decoding protocol.
Our work naturally invites further numerical studies in several directions. A straightforward extension of our numerical results on the isotropic LGT is to work directly in the limit of continuous imaginary time, where quantum Monte Carlo (QMC) via the WA [94 ] can be employed to more quantitatively explore decoherence thresholds in the TFTC. Another possibility is to work with a generalized stoquastic series expansion involving both Hamiltonian terms and local stabilizer measurement operators, as in Ref. [123 ] . In particular, it would be interesting to perform a more detailed finite-size scaling analysis of the decoherence-induced phase transitions at nonzero h h h 2 d 2d 2 d h < h c h < h_c h < h c h = h c h = h_c h = h c p = p c o p t ( h c ) p = p_c^{\mathrm{opt}}(h_c) p = p c opt ( h c ) 2 d 2d 2 d 3 d 3d 3 d [129 , 130 ] , one might ask if the present multicritical point also exhibits a higher replica symmetry.
The results of this work broadly open the door to further investigations of QEC in physically motivated non-stabilizer codes. While the TFTC and Fradkin-Shenker model are the most natural non-stabilizer topological codes from the perspective of classical statistical physics, recent work [131 ] has demonstrated that a large class of twisted quantum double models, as well as their Hamiltonian deformations, can be efficiently simulated using QMC. It would be interesting to investigate the protection and recovery of quantum information in such models, and to understand the conditions under which the associated post-measurement states permit sign-free QMC calculations. Given the deep connections between QEC and topologically ordered phases of matter, such studies may provide an insightful new perspective on topological phases away from their well-understood solvable limits.
Acknowledgments
We thank Ehud Altman, Ruihua Fan, Tim Hsieh, Michael Knap, Nathaniel Selub, and Simon Trebst for fruitful discussions, and Zac Tobias in particular for clarifying discussions on the worm algorithm. Z.W. acknowledges funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-2317110). S.J.G. was supported by the Gordon & Betty Moore Foundation, and in part by a Brown Investigator Award, a program of the Brown Institute for Basic Sciences at the California Institute of Technology. This research was done using services provided by the OSG Consortium [132 , 133 , 134 , 135 ] , which is supported by the National Science Foundation awards #2030508 and #2323298.
Use of Artificial Intelligence. —The authors acknowledge the use of artificial intelligence (AI) for finding additional references, assisting with proofs, analysis of numerical data, optimization of Monte Carlo parameters, and proofreading of this manuscript. No AI was used for writing code used in this work’s Monte Carlo simulations, or in writing any part of this manuscript.
Quantum-Classical Mapping
In this Appendix, we review the quantum-classical mapping for the transverse-field toric code H h H_h H h (8) ]. As discussed in Sec. II B, the ground-state physics of H h H_h H h ( 2 + 1 ) d (2+1)d ( 2 + 1 ) d Z 2 \mathbb{Z}_2 Z 2 (9) ], and so the quantum-classical mapping of the latter model also describes the ground-state physics of the former model. We focus on the latter model for technical simplicity25
The thermal partition function of the Z 2 \mathbb{Z}_2 Z 2
Z L G T ≡ tr [ P e − β H L G T ] , (127) Z_{\mathrm{LGT}} \equiv\operatorname{tr}\left[\mathcal{P}e^{-\beta H_{\mathrm{LGT}}}\right], \tag*{(127)} Z LGT ≡ tr [ P e − β H LGT ] , (127) where H L G T H_{\mathrm{LGT}} H LGT (9) , and P ≡ ∏ v ( 1 + A v 2 ) \mathcal{P} \equiv\prod_v \left(\frac{1+A_v}{2}\right) P ≡ ∏ v ( 2 1 + A v ) β → ∞ \beta\to\infty β → ∞ e − β H L G T e^{-\beta H_{\mathrm{LGT}}} e − β H LGT H L G T H_{\mathrm{LGT}} H LGT
As usual [51 , 55 , 56 , 81 ] , the quantum-classical mapping is derived by trotterizing β ≡ ε M \beta\equiv\varepsilon M β ≡ εM M M M 26 ε \varepsilon ε P \mathcal{P} P P \mathcal{P} P H L G T H_{\mathrm{LGT}} H LGT P 2 = P \mathcal{P}^2=\mathcal{P} P 2 = P M M M
Z L G T = tr [ ( P e − ε H L G T ) M ] = ∑ { U } ⟨ U 1 ∣ P e − ε H L G T ∣ U M ⟩ ⋯ ⟨ U 2 ∣ P e − ε H L G T ∣ U 1 ⟩ , (128) Z_{\mathrm{LGT}} = \operatorname{tr}\left[(\mathcal{P}e^{-\varepsilon H_{\mathrm{LGT}}})^M\right] = \sum_{\{U\}} \langle U_1|\mathcal{P}e^{-\varepsilon H_{\mathrm{LGT}}}|U_M\rangle\cdots\langle U_2|\mathcal{P}e^{-\varepsilon H_{\mathrm{LGT}}}|U_1\rangle, \tag*{(128)} Z LGT = tr [ ( P e − ε H LGT ) M ] = { U } ∑ ⟨ U 1 ∣ P e − ε H LGT ∣ U M ⟩ ⋯ ⟨ U 2 ∣ P e − ε H LGT ∣ U 1 ⟩ , (128) where U m ≡ { U ℓ , m } U_m \equiv\{U_{\ell,m}\} U m ≡ { U ℓ , m } Z Z Z m m m U ℓ , m = ± 1 U_{\ell,m}=\pm1 U ℓ , m = ± 1 ℓ \ell ℓ ∑ { U } ≡ ∑ U 1 ⋯ ∑ U M \sum_{\{U\}} \equiv\sum_{U_1}\cdots\sum_{U_M} ∑ { U } ≡ ∑ U 1 ⋯ ∑ U M ε \varepsilon ε Z Z Z X X X
⟨ U m ∣ P e − ε H L G T ∣ U m − 1 ⟩ ≈ ⟨ U m ∣ e ε ∑ p B p P e ε h ∑ ℓ X ℓ ∣ U m − 1 ⟩ = e ε ∑ p B p ∏ ℓ ∈ p U ℓ , m ⟨ U m ∣ P e ε h ∑ ℓ X ℓ ∣ U m − 1 ⟩ . (129) \langle U_m|\mathcal{P}e^{-\varepsilon H_{\mathrm{LGT}}}|U_{m-1}\rangle\approx\langle U_m|e^{\varepsilon\sum_p B_p}\mathcal{P}e^{\varepsilon h\sum_\ell X_\ell}|U_{m-1}\rangle
= e^{\varepsilon\sum_p B_p\prod_{\ell\in p}U_{\ell,m}}\langle U_m|\mathcal{P}e^{\varepsilon h\sum_\ell X_\ell}|U_{m-1}\rangle. \tag*{(129)} ⟨ U m ∣ P e − ε H LGT ∣ U m − 1 ⟩ ≈ ⟨ U m ∣ e ε ∑ p B p P e ε h ∑ ℓ X ℓ ∣ U m − 1 ⟩ = e ε ∑ p B p ∏ ℓ ∈ p U ℓ , m ⟨ U m ∣ P e ε h ∑ ℓ X ℓ ∣ U m − 1 ⟩ . (129) From the Z Z Z U ℓ , m U_{\ell,m} U ℓ , m p p p m m m
To evaluate the remaining matrix element, which contains entirely X ℓ X_\ell X ℓ A v ∣ U m ⟩ A_v|U_m\rangle A v ∣ U m ⟩ U m U_m U m v v v U ℓ , m → − U ℓ , m U_{\ell,m}\to-U_{\ell,m} U ℓ , m → − U ℓ , m ℓ \ell ℓ v v v U v v ′ , m → s v U v v ′ , m s v ′ U_{vv',m}\to s_vU_{vv',m}s_{v'} U v v ′ , m → s v U v v ′ , m s v ′ s v = ± 1 s_v=\pm1 s v = ± 1 v v v U v v ′ , m ≡ U ℓ , m U_{vv',m}\equiv U_{\ell,m} U v v ′ , m ≡ U ℓ , m ℓ = ⟨ v v ′ ⟩ \ell=\langle vv'\rangle ℓ = ⟨ v v ′ ⟩ A v A_v A v s v = − 1 s_v=-1 s v = − 1 P \mathcal{P} P ⟨ U m ∣ \langle U_m| ⟨ U m ∣
⟨ U m ∣ P = ⟨ U m ∣ ∏ v ( 1 + A v 2 ) = 1 2 L 2 ∑ s ⟨ s U m s ∣ , (130) \langle U_m|\mathcal{P}=\langle U_m|\prod_v\left(\frac{1+A_v}{2}\right)=\frac{1}{2^{L^2}}\sum_s\langle sU_ms|, \tag*{(130)} ⟨ U m ∣ P = ⟨ U m ∣ v ∏ ( 2 1 + A v ) = 2 L 2 1 s ∑ ⟨ s U m s ∣ , (130) where s ≡ { s v } s\equiv\{s_v\} s ≡ { s v } s U m s ≡ { s v U v v ′ , m s v ′ } sU_ms\equiv\{s_vU_{vv',m}s_{v'}\} s U m s ≡ { s v U v v ′ , m s v ′ }
⟨ U m ∣ P e ε h ∑ ℓ X ℓ ∣ U m − 1 ⟩ = 1 2 L 2 ∑ s ∏ ⟨ v v ′ ⟩ ⟨ s v U v v ′ , m s v ′ ∣ e ε h X v v ′ ∣ U v v ′ , m − 1 ⟩ = 1 2 L 2 ∑ s ∏ ⟨ v v ′ ⟩ [ cosh ( ε h ) e J e J s v U v v ′ , m s v ′ U v v ′ , m − 1 ] . (131) \begin{aligned}
\langle U_m|\mathcal{P}e^{\varepsilon h\sum_\ell X_\ell}|U_{m-1}\rangle=\frac{1}{2^{L^2}}\sum_s\prod_{\langle vv'\rangle}\langle s_vU_{vv',m}s_{v'}|e^{\varepsilon hX_{vv'}}|U_{vv',m-1}\rangle\\
&=\frac{1}{2^{L^2}}\sum_s\prod_{\langle vv'\rangle}\left[\frac{\cosh(\varepsilon h)}{e^J}e^{Js_vU_{vv',m}s_{v'}U_{vv',m-1}}\right]. \tag*{(131)}
\end{aligned} ⟨ U m ∣ P e ε h ∑ ℓ X ℓ ∣ U m − 1 ⟩ = 2 L 2 1 s ∑ ⟨ v v ′ ⟩ ∏ ⟨ s v U v v ′ , m s v ′ ∣ e ε h X v v ′ ∣ U v v ′ , m − 1 ⟩ = 2 L 2 1 s ∑ ⟨ v v ′ ⟩ ∏ [ e J cosh ( ε h ) e J s v U v v ′ , m s v ′ U v v ′ , m − 1 ] . (131) with J = − 1 2 log tanh ( ε h ) J=-\frac{1}{2}\log\tanh(\varepsilon h) J = − 2 1 log tanh ( ε h ) X v v ′ = X ℓ X_{vv'}=X_{\ell} X v v ′ = X ℓ ℓ = ⟨ v v ′ ⟩ \ell=\langle vv'\rangle ℓ = ⟨ v v ′ ⟩
As a final step, let us regard the spins s v s_v s v s v ≡ U v , m − 1 2 s_v\equiv U_{v,m-\frac{1}{2}} s v ≡ U v , m − 2 1 ( v , m ) (v,m) ( v , m ) ( v , m − 1 ) (v,m-1) ( v , m − 1 )
Altogether, we obtain the following partition function (up to trivial constants):
Z L G T ∝ ∑ { U } exp { ε ∑ p , m ∏ ℓ ∈ ∂ p U ℓ , m + J ∑ ⟨ v v ′ ⟩ , m U v , m − 1 2 U v v ′ , m U v ′ , m − 1 2 U v v ′ , m − 1 } = ∑ { U } exp { ε ∑ [ i j k l ] ⊥ U i j U j k U k l U l i + J ∑ [ i j k l ] ∥ U i j U j k U k l U l i } ≡ ∑ { U } e − S L G T [ U ] . (132) \begin{aligned}
Z_{\mathrm{LGT}} &\propto\sum_{\{U\}}\exp\left\{\varepsilon\sum_{p,m}\prod_{\ell\in\partial p}U_{\ell,m}+J\sum_{\langle vv'\rangle,m}U_{v,m-\frac{1}{2}}U_{vv',m}U_{v',m-\frac{1}{2}}U_{vv',m-1}\right\}\\
&=\sum_{\{U\}}\exp\left\{\varepsilon\sum_{[ijkl]_{\perp}}U_{ij}U_{jk}U_{kl}U_{li}+J\sum_{[ijkl]_{\parallel}}U_{ij}U_{jk}U_{kl}U_{li}\right\}\\
&\equiv\sum_{\{U\}}e^{-S_{\mathrm{LGT}}[U]} . \tag*{(132)}
\end{aligned} Z LGT ∝ { U } ∑ exp ⎩ ⎨ ⎧ ε p , m ∑ ℓ ∈ ∂ p ∏ U ℓ , m + J ⟨ v v ′ ⟩ , m ∑ U v , m − 2 1 U v v ′ , m U v ′ , m − 2 1 U v v ′ , m − 1 ⎭ ⎬ ⎫ = { U } ∑ exp ⎩ ⎨ ⎧ ε [ ij k l ] ⊥ ∑ U ij U j k U k l U l i + J [ ij k l ] ∥ ∑ U ij U j k U k l U l i ⎭ ⎬ ⎫ ≡ { U } ∑ e − S LGT [ U ] . (132) where in the second expression, [ i j k l ] ⊥ [ijkl]_{\perp} [ ij k l ] ⊥ [ i j k l ] ∥ [ijkl]_{\parallel} [ ij k l ] ∥ i j k l ijkl ij k l Figure 3 (b). As expected, this is the extremely anisotropic limit of a classical 3d LGT partition function [50 , 51 ] . Note in particular that no gauge condition has to be enforced on the classical model; here we can regard gauge redundancy as nothing more special than a local symmetry acting at a point i = ( v , m ) i=(v,m) i = ( v , m ) U i j U_{ij} U ij
The preceding derivation was performed for the thermal partition function, i.e., with periodic boundary conditions in the imaginary time direction. It is simple to generalize the above mapping to evaluate matrix elements of the form ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ ∝ ⟨ ψ 0 σ ∣ e − β H L G T ∣ ψ 0 σ ′ ⟩ \langle\psi_{0}^{\sigma}|e^{-\beta H_h}|\psi_{0}^{\sigma'}\rangle\propto\langle\psi_{0}^{\sigma}|e^{-\beta H_{\mathrm{LGT}}}|\psi_{0}^{\sigma'}\rangle ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ ∝ ⟨ ψ 0 σ ∣ e − β H LGT ∣ ψ 0 σ ′ ⟩ ∣ ψ 0 σ ⟩ |\psi_{0}^{\sigma}\rangle ∣ ψ 0 σ ⟩ Z Z Z τ = ± β 2 \tau=\pm\frac{\beta}{2} τ = ± 2 β m m m τ = ± β 2 \tau=\pm\frac{\beta}{2} τ = ± 2 β ∣ ψ 0 σ ′ ⟩ |\psi_{0}^{\sigma'}\rangle ∣ ψ 0 σ ′ ⟩ ∣ ψ 0 σ ⟩ |\psi_{0}^{\sigma}\rangle ∣ ψ 0 σ ⟩ X X X contractible when σ = σ ′ \sigma=\sigma' σ = σ ′ non-contractible when σ ≠ σ ′ \sigma\ne\sigma' σ = σ ′
The required worldline homologies for each matrix element ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ \langle\psi_{0}^{\sigma}|e^{-\beta H_h}|\psi_{0}^{\sigma'}\rangle ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ S L G T S_{\mathrm{LGT}} S LGT 27
⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ = ∑ n = 0 ∞ h n ∑ ℓ 1 … ℓ n ∫ 0 ≤ τ 1 < … < τ n ≤ β d τ 1 … d τ n ⟨ ψ 0 σ ∣ e − ( β − τ n ) H 0 X ℓ n e − ( τ n − τ n − 1 ) H 0 … e − ( τ 2 − τ 1 ) H 0 X ℓ 1 e − τ 1 H 0 ∣ ψ 0 σ ′ ⟩ . (133) \langle\psi_{0}^{\sigma}|e^{-\beta H_h}|\psi_{0}^{\sigma'}\rangle=\sum_{n=0}^{\infty}h^n\sum_{\ell_1\ldots\ell_n}\int_{0\le\tau_1<\ldots<\tau_n\le\beta}d\tau_1\ldots d\tau_n\langle\psi_{0}^{\sigma}|e^{-(\beta-\tau_n)H_0}X_{\ell_n}e^{-(\tau_n-\tau_{n-1})H_0}\ldots e^{-(\tau_2-\tau_1)H_0}X_{\ell_1}e^{-\tau_1H_0}|\psi_{0}^{\sigma'}\rangle. \tag*{(133)} ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ = n = 0 ∑ ∞ h n ℓ 1 … ℓ n ∑ ∫ 0 ≤ τ 1 < … < τ n ≤ β d τ 1 … d τ n ⟨ ψ 0 σ ∣ e − ( β − τ n ) H 0 X ℓ n e − ( τ n − τ n − 1 ) H 0 … e − ( τ 2 − τ 1 ) H 0 X ℓ 1 e − τ 1 H 0 ∣ ψ 0 σ ′ ⟩ . (133) In the above perturbative expansion, each fixed order n n n { ( τ α , ℓ α ) } \{(\tau_\alpha,\ell_\alpha)\} {( τ α , ℓ α )} m m m X X X X ℓ n … X ℓ 1 X_{\ell_n}\ldots X_{\ell_1} X ℓ n … X ℓ 1 ∣ ψ 0 σ ′ ⟩ |\psi_{0}^{\sigma'}\rangle ∣ ψ 0 σ ′ ⟩ e − ( τ α − τ α − 1 ) H 0 e^{-(\tau_\alpha-\tau_{\alpha-1})H_0} e − ( τ α − τ α − 1 ) H 0 ⟨ ψ 0 σ ∣ X ℓ n … X ℓ 1 ∣ ψ 0 σ ′ ⟩ \langle\psi_{0}^{\sigma}|X_{\ell_n}\ldots X_{\ell_1}|\psi_{0}^{\sigma'}\rangle ⟨ ψ 0 σ ∣ X ℓ n … X ℓ 1 ∣ ψ 0 σ ′ ⟩ X ℓ n … X ℓ 1 X_{\ell_n}\ldots X_{\ell_1} X ℓ n … X ℓ 1 X X X ∣ ψ 0 σ ′ ⟩ |\psi_{0}^{\sigma'}\rangle ∣ ψ 0 σ ′ ⟩ ∣ ψ 0 σ ⟩ |\psi_{0}^{\sigma}\rangle ∣ ψ 0 σ ⟩
ITE states are approximately true code states
In this Appendix, we prove that stabilizer code states evolved by ITE lie approximately within the TFTC code space C h \mathcal{C}_h C h Δ \Delta Δ C h \mathcal{C}_h C h L → ∞ L\to\infty L → ∞ ∣ ψ 0 ⟩ ∈ C 0 |\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0 C h \mathcal{C}_h C h L L L O ( e − ρ L ) \mathcal{O}(e^{-\rho L}) O ( e − ρ L ) h < h c h<h_c h < h c h < h c h<h_c h < h c [97 ] .
We first fix some notation. Let ∣ χ h η ⟩ |\chi_h^\eta\rangle ∣ χ h η ⟩ exact four lowest-energy eigenstates of H h H_h H h h < h c h<h_c h < h c X X X X ‾ μ \overline{X}_\mu X μ η ≡ ( η x , η y ) \eta\equiv(\eta_x,\eta_y) η ≡ ( η x , η y ) η μ = ± \eta_\mu=\pm η μ = ± X ‾ μ ∣ χ h η ⟩ = η μ ∣ χ h η ⟩ \overline{X}_\mu|\chi_h^\eta\rangle=\eta_\mu|\chi_h^\eta\rangle X μ ∣ χ h η ⟩ = η μ ∣ χ h η ⟩ E g . s . η E_{\mathrm{g.s.}}^\eta E g.s. η H h ∣ χ h η ⟩ = E g . s . η ∣ χ h η ⟩ H_h|\chi_h^\eta\rangle=E_{\mathrm{g.s.}}^\eta|\chi_h^\eta\rangle H h ∣ χ h η ⟩ = E g.s. η ∣ χ h η ⟩ 28 [136 ] , the absolute ground state is ∣ χ h ( + , + ) ⟩ |\chi_h^{(+,+)}\rangle ∣ χ h ( + , + ) ⟩ E g . s . ( + , + ) E_{\mathrm{g.s.}}^{(+,+)} E g.s. ( + , + ) H h → H h − E g . s . ( + , + ) H_h\to H_h-E_{\mathrm{g.s.}}^{(+,+)} H h → H h − E g.s. ( + , + ) Δ \Delta Δ [86 ] proved that (for sufficiently large L L L δ \delta δ L L L
δ ≡ max η E g . s . η ≤ O ( e − L / ξ ) , (134) \delta\equiv\max_\eta E_{\mathrm{g.s.}}^\eta\leq O(e^{-L/\xi}), \tag*{(134)} δ ≡ η max E g.s. η ≤ O ( e − L / ξ ) , (134) where ξ \xi ξ Δ → 0 \Delta\to0 Δ → 0
For each stabilizer code state ∣ ψ 0 ⟩ ∈ C 0 |\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0 ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ (21) , repeated here for convenience:
∣ φ h ⟩ ≡ e − β H h / 2 ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ . (135) |\varphi_h\rangle\equiv\frac{e^{-\beta H_h/2}|\psi_0\rangle}{\sqrt{\langle\psi_0|e^{-\beta H_h}|\psi_0\rangle}}. \tag*{(135)} ∣ φ h ⟩ ≡ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ e − β H h /2 ∣ ψ 0 ⟩ . (135) Our goal in this section is to show that ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ C h \mathcal{C}_h C h
∑ η ∣ ⟨ χ h η ∣ φ h ⟩ ∣ 2 ≥ 1 − O ( e − L / ξ ) . (136) \sum_\eta|\langle\chi_h^\eta|\varphi_h\rangle|^2\geq1-O(e^{-L/\xi}). \tag*{(136)} η ∑ ∣ ⟨ χ h η ∣ φ h ⟩ ∣ 2 ≥ 1 − O ( e − L / ξ ) . (136) To this end, let us denote the remaining excited energy eigenstates of H h H_h H h h h h { ∣ n ⟩ } n = 1 ∞ \{|n\rangle\}_{n=1}^{\infty} { ∣ n ⟩ } n = 1 ∞ E n E_n E n Δ = E 1 ≤ E 2 ≤ … \Delta=E_1\leq E_2\leq\ldots Δ = E 1 ≤ E 2 ≤ … (136) as
∑ η ∣ ⟨ χ h η ∣ φ h ⟩ ∣ 2 = ∑ η ∣ ⟨ χ h η ∣ e − β H h / 2 ∣ ψ 0 ⟩ ∣ 2 ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ = ∑ η e − β E g . s . η ∣ ⟨ χ h η ∣ ψ 0 ⟩ ∣ 2 ∑ η e − β E g . s . η ∣ ⟨ χ h η ∣ ψ 0 ⟩ ∣ 2 + ∑ n ≥ 1 e − β E n ∣ ⟨ n ∣ ψ 0 ⟩ ∣ 2 ≥ e − β δ Ω Ω + e − β Δ ( 1 − Ω ) = e − β δ 1 + e − β Δ ( Ω − 1 − 1 ) . (137) \begin{aligned}
\sum_\eta|\langle\chi_h^\eta|\varphi_h\rangle|^2
&=\frac{\sum_\eta|\langle\chi_h^\eta|e^{-\beta H_h/2}|\psi_0\rangle|^2}{\langle\psi_0|e^{-\beta H_h}|\psi_0\rangle}\\
&=\frac{\sum_\eta e^{-\beta E_{\mathrm{g.s.}}^\eta}|\langle\chi_h^\eta|\psi_0\rangle|^2}{\sum_\eta e^{-\beta E_{\mathrm{g.s.}}^\eta}|\langle\chi_h^\eta|\psi_0\rangle|^2+\sum_{n\geq1}e^{-\beta E_n}|\langle n|\psi_0\rangle|^2}\\
&\geq\frac{e^{-\beta\delta}\Omega}{\Omega+e^{-\beta\Delta}(1-\Omega)}\\
&=\frac{e^{-\beta\delta}}{1+e^{-\beta\Delta}(\Omega^{-1}-1)}. \tag*{(137)}
\end{aligned} η ∑ ∣ ⟨ χ h η ∣ φ h ⟩ ∣ 2 = ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ ∑ η ∣ ⟨ χ h η ∣ e − β H h /2 ∣ ψ 0 ⟩ ∣ 2 = ∑ η e − β E g.s. η ∣ ⟨ χ h η ∣ ψ 0 ⟩ ∣ 2 + ∑ n ≥ 1 e − β E n ∣ ⟨ n ∣ ψ 0 ⟩ ∣ 2 ∑ η e − β E g.s. η ∣ ⟨ χ h η ∣ ψ 0 ⟩ ∣ 2 ≥ Ω + e − β Δ ( 1 − Ω ) e − β δ Ω = 1 + e − β Δ ( Ω − 1 − 1 ) e − β δ . (137) where we have defined Ω \Omega Ω ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ C h \mathcal{C}_h C h
Ω ≡ ∑ η ∣ ⟨ χ h η ∣ ψ 0 ⟩ ∣ 2 . (138) \Omega\equiv\sum_\eta|\langle\chi_h^\eta|\psi_0\rangle|^2. \tag*{(138)} Ω ≡ η ∑ ∣ ⟨ χ h η ∣ ψ 0 ⟩ ∣ 2 . (138) So long as Ω \Omega Ω β \beta β e − β Δ Ω − 1 e^{-\beta\Delta}\Omega^{-1} e − β Δ Ω − 1 e − β δ e^{-\beta\delta} e − β δ δ ≤ O ( e − L / ξ ) \delta\leq O(e^{-L/\xi}) δ ≤ O ( e − L / ξ ) e − β δ ≥ 1 − O ( e − L / ξ ) e^{-\beta\delta}\geq1-O(e^{-L/\xi}) e − β δ ≥ 1 − O ( e − L / ξ ) β ∼ O ( poly ( L ) ) \beta\sim O(\operatorname{poly}(L)) β ∼ O ( poly ( L )) Ω \Omega Ω L L L
Ω ≥ O ( e − poly ( L ) ) , (139) \Omega\geq O(e^{-\operatorname{poly}(L)}), \tag*{(139)} Ω ≥ O ( e − poly ( L ) ) , (139) and in particular that Ω ∼ e − α h L 2 \Omega\sim e^{-\alpha_hL^2} Ω ∼ e − α h L 2 h < h c h<h_c h < h c [97 ] . With this assumption, Eq. (136) is obtained by choosing β \beta β
Δ − 1 log Ω − 1 ≪ β ≪ δ − 1 . (140) \Delta^{-1}\log\Omega^{-1}\ll\beta\ll\delta^{-1}. \tag*{(140)} Δ − 1 log Ω − 1 ≪ β ≪ δ − 1 . (140) Decoding Fidelities of Nearby States
In the main text, we have focused on decoding algorithms for the ITE states ∣ ϕ h ⟩ \lvert\phi_h\rangle ∣ ϕ h ⟩ (21) . Since these states form a four-dimensional vector space and lie within the true code space C h \mathcal{C}_h C h C h \mathcal{C}_h C h ∣ ψ h ⟩ ∈ C h \lvert\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h ∣ ϕ h ⟩ \lvert\phi_h\rangle ∣ ϕ h ⟩ ∣ ⟨ ϕ h ∣ ψ h ⟩ ∣ ≥ 1 − O ( e − L / ξ ) \lvert\langle\phi_h\vert\psi_h\rangle\rvert\ge1-O(e^{-L/\xi}) ∣⟨ ϕ h ∣ ψ h ⟩∣ ≥ 1 − O ( e − L / ξ ) ∣ ψ h ⟩ = U h ∣ ψ 0 ⟩ \lvert\psi_h\rangle=U_h\lvert\psi_0\rangle ∣ ψ h ⟩ = U h ∣ ψ 0 ⟩ ∣ ϕ h ⟩ ∝ e − β H h / 2 ∣ ψ 0 ⟩ \lvert\phi_h\rangle\propto e^{-\beta H_h/2}\lvert\psi_0\rangle ∣ ϕ h ⟩ ∝ e − β H h /2 ∣ ψ 0 ⟩ ∣ ψ h ⟩ \lvert\psi_h\rangle ∣ ψ h ⟩
In this Appendix, we will show that any QEC protocol succeeds with high fidelity on the true code space if and only if it succeeds on the ITE states. This result is quite natural, since the ITE states and true code states are essentially indistinguishable from each other. If the QEC protocol were to succeed on one class of states and not the other, it could be used as a channel which distinguishes between the two states.
We will handle the cases of § 3 (without Pauli decoherence) and § 3.2 (with Pauli decoherence) simultaneously; indeed, the former case is a special case of the latter, with decoherence strength p = 0 p=0 p = 0 (13) ]. First recall that the average decoding fidelity F ‾ ( ∣ ψ h ⟩ ) \overline{\mathcal{F}}(\lvert\psi_h\rangle) F (∣ ψ h ⟩) ∣ ψ h ⟩ ∈ C h \lvert\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h
F ‾ ( ∣ ψ h ⟩ ) = ∑ s P s ⟨ ψ h ∣ V s ρ h ( s ) V s † ∣ ψ h ⟩ = ∑ s ⟨ ψ h ∣ V s Π ^ s ρ h Π ^ s V s † ∣ ψ h ⟩ = ⟨ ψ h ∣ M ( ρ h ) ∣ ψ h ⟩ = F ( ∣ ψ h ⟩ ⟨ ψ h ∣ , M ( ρ h ) ) , (141) \begin{aligned}
\overline{\mathcal{F}}(\lvert\psi_h\rangle)=\sum_s P_s\langle\psi_h\vert V_s\rho_h(s)V_s^\dagger\vert\psi_h\rangle\\
&=\sum_s\langle\psi_h\vert V_s\widehat{\Pi}_s\rho_h\widehat{\Pi}_sV_s^\dagger\vert\psi_h\rangle\\
&=\langle\psi_h\vert\mathcal{M}(\rho_h)\vert\psi_h\rangle\\
&=\mathcal{F}(\lvert\psi_h\rangle\langle\psi_h\rvert,\mathcal{M}(\rho_h)), \tag*{(141)}
\end{aligned} F (∣ ψ h ⟩) = s ∑ P s ⟨ ψ h ∣ V s ρ h ( s ) V s † ∣ ψ h ⟩ = s ∑ ⟨ ψ h ∣ V s Π s ρ h Π s V s † ∣ ψ h ⟩ = ⟨ ψ h ∣ M ( ρ h ) ∣ ψ h ⟩ = F (∣ ψ h ⟩ ⟨ ψ h ∣ , M ( ρ h )) , (141) where ρ h ≡ E ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) \rho_h\equiv\mathcal{E}(\lvert\psi_h\rangle\langle\psi_h\rvert) ρ h ≡ E (∣ ψ h ⟩ ⟨ ψ h ∣) ρ h ( s ) ≡ Π ^ s ρ h Π ^ s / P s \rho_h(s)\equiv\widehat{\Pi}_s\rho_h\widehat{\Pi}_s/P_s ρ h ( s ) ≡ Π s ρ h Π s / P s decoding channel ,
M ( ρ ) ≡ ∑ s V s Π ^ s ρ Π ^ s V s † , (142) \mathcal{M}(\rho)\equiv\sum_s V_s\widehat{\Pi}_s\rho\widehat{\Pi}_sV_s^\dagger, \tag*{(142)} M ( ρ ) ≡ s ∑ V s Π s ρ Π s V s † , (142) and in the final line we have defined the Uhlmann fidelity F ( ρ , ω ) \mathcal{F}(\rho,\omega) F ( ρ , ω ) ρ \rho ρ ω \omega ω [80 ] ,
F ( ρ , ω ) = ( tr ρ ω ρ ) 2 . (143) \mathcal{F}(\rho,\omega)=\left(\operatorname{tr}\sqrt{\sqrt{\rho}\omega\sqrt{\rho}}\right)^2. \tag*{(143)} F ( ρ , ω ) = ( tr ρ ω ρ ) 2 . (143) The Uhlmann fidelity satisfies two important properties for our purposes. First, it satisfies a quantum data processing inequality [61 , 80 ] : for any quantum channel N \mathcal{N} N
F ( N ( ρ ) , N ( ω ) ) ≥ F ( ρ , ω ) , (144) \mathcal{F}(\mathcal{N}(\rho),\mathcal{N}(\omega))\ge\mathcal{F}(\rho,\omega), \tag*{(144)} F ( N ( ρ ) , N ( ω )) ≥ F ( ρ , ω ) , (144) i.e., two states ρ \rho ρ ω \omega ω A ( ρ , ω ) ≡ cos − 1 [ F ( ρ , ω ) ] A(\rho,\omega)\equiv\cos^{-1}[\sqrt{\mathcal{F}(\rho,\omega)}] A ( ρ , ω ) ≡ cos − 1 [ F ( ρ , ω ) ] A ( ρ , ω ) ≤ A ( ρ , η ) + A ( η , ω ) A(\rho,\omega)\le A(\rho,\eta)+A(\eta,\omega) A ( ρ , ω ) ≤ A ( ρ , η ) + A ( η , ω ) ρ , ω , η \rho,\omega,\eta ρ , ω , η [80 ] . Note that since cos − 1 ( x ) \cos^{-1}(x) cos − 1 ( x ) x ∈ [ 0 , 1 ] x\in[0,1] x ∈ [ 0 , 1 ] A ( N ( ρ ) , N ( ω ) ) ≤ A ( ρ , ω ) A(\mathcal{N}(\rho),\mathcal{N}(\omega))\le A(\rho,\omega) A ( N ( ρ ) , N ( ω )) ≤ A ( ρ , ω )
With these properties in mind, we can derive a lower bound on the average decoding fidelity for true code states ∣ ψ h ⟩ ∈ C h \lvert\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h ∣ ϕ h ⟩ \lvert\phi_h\rangle ∣ ϕ h ⟩ N ≡ M ∘ E \mathcal{N}\equiv\mathcal{M}\circ\mathcal{E} N ≡ M ∘ E E \mathcal{E} E M \mathcal{M} M
A ( ∣ ψ h ⟩ ⟨ ψ h ∣ , N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) ) ≤ A ( ∣ ψ h ⟩ ⟨ ψ h ∣ , ∣ ϕ h ⟩ ⟨ ϕ h ∣ ) + A ( ∣ ϕ h ⟩ ⟨ ϕ h ∣ , N ( ∣ ϕ h ⟩ ⟨ ϕ h ∣ ) ) + A ( N ( ∣ ϕ h ⟩ ⟨ ϕ h ∣ ) , N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) ) A(\lvert\psi_h\rangle\langle\psi_h\rvert,\mathcal{N}(\lvert\psi_h\rangle\langle\psi_h\rvert))\le A(\lvert\psi_h\rangle\langle\psi_h\rvert,\lvert\phi_h\rangle\langle\phi_h\rvert)+A(\lvert\phi_h\rangle\langle\phi_h\rvert,\mathcal{N}(\lvert\phi_h\rangle\langle\phi_h\rvert))+A(\mathcal{N}(\lvert\phi_h\rangle\langle\phi_h\rvert),\mathcal{N}(\lvert\psi_h\rangle\langle\psi_h\rvert)) A (∣ ψ h ⟩ ⟨ ψ h ∣ , N (∣ ψ h ⟩ ⟨ ψ h ∣)) ≤ A (∣ ψ h ⟩ ⟨ ψ h ∣ , ∣ ϕ h ⟩ ⟨ ϕ h ∣) + A (∣ ϕ h ⟩ ⟨ ϕ h ∣ , N (∣ ϕ h ⟩ ⟨ ϕ h ∣)) + A ( N (∣ ϕ h ⟩ ⟨ ϕ h ∣) , N (∣ ψ h ⟩ ⟨ ψ h ∣)) ≤ 2 A ( ∣ ψ h ⟩ ⟨ ψ h ∣ , ∣ ϕ h ⟩ ⟨ ϕ h ∣ ) + A ( ∣ ϕ h ⟩ ⟨ ϕ h ∣ , N ( ∣ ϕ h ⟩ ⟨ ϕ h ∣ ) ) = 2 cos − 1 ( ∣ ⟨ ϕ h ∣ ψ h ⟩ ∣ ) + A ( ∣ ϕ h ⟩ ⟨ ϕ h ∣ , N ( ∣ ϕ h ⟩ ⟨ ϕ h ∣ ) ) , (145) \begin{aligned}
\le2A(\lvert\psi_h\rangle\langle\psi_h\rvert,\lvert\phi_h\rangle\langle\phi_h\rvert)+A(\lvert\phi_h\rangle\langle\phi_h\rvert,\mathcal{N}(\lvert\phi_h\rangle\langle\phi_h\rvert)) \\
&=2\cos^{-1}(\lvert\langle\phi_h\vert\psi_h\rangle\rvert)+A(\lvert\phi_h\rangle\langle\phi_h\rvert,\mathcal{N}(\lvert\phi_h\rangle\langle\phi_h\rvert)), \tag*{(145)}
\end{aligned} ≤ 2 A (∣ ψ h ⟩ ⟨ ψ h ∣ , ∣ ϕ h ⟩ ⟨ ϕ h ∣) + A (∣ ϕ h ⟩ ⟨ ϕ h ∣ , N (∣ ϕ h ⟩ ⟨ ϕ h ∣)) = 2 cos − 1 (∣⟨ ϕ h ∣ ψ h ⟩∣) + A (∣ ϕ h ⟩ ⟨ ϕ h ∣ , N (∣ ϕ h ⟩ ⟨ ϕ h ∣)) , (145) where we have used the quantum data processing inequality to obtain the second expression. By computing the cosine of both sides, straightforward algebra29
F ‾ ( ∣ ψ h ⟩ ) ≥ F ‾ ( ∣ ϕ h ⟩ ) − 4 1 − ∣ ⟨ ϕ h ∣ ψ h ⟩ ∣ 2 . (146) \overline{\mathcal{F}}(\lvert\psi_h\rangle)\ge\overline{\mathcal{F}}(\lvert\phi_h\rangle)-4\sqrt{1-\lvert\langle\phi_h\vert\psi_h\rangle\rvert^2}. \tag*{(146)} F (∣ ψ h ⟩) ≥ F (∣ ϕ h ⟩) − 4 1 − ∣⟨ ϕ h ∣ ψ h ⟩ ∣ 2 . (146) We can similarly perform the same calculation with ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩
∣ F ‾ ( ∣ ψ h ⟩ ) − F ‾ ( ∣ φ h ⟩ ) ∣ ≤ 4 1 − ∣ ⟨ φ h ∣ ψ h ⟩ ∣ 2 . (147) \left|\overline{F}(|\psi_h\rangle)-\overline{F}(|\varphi_h\rangle)\right|\leq4\sqrt{1-\left|\langle\varphi_h|\psi_h\rangle\right|^2}. \tag*{(147)} F ( ∣ ψ h ⟩) − F ( ∣ φ h ⟩) ≤ 4 1 − ∣ ⟨ φ h ∣ ψ h ⟩ ∣ 2 . (147) As discussed in Sec. III B, we can approximate any true code state ∣ ψ h ⟩ ∈ C h |\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ O ( e − L / ξ ) \mathcal{O}(e^{-L/\xi}) O ( e − L / ξ ) ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩
Equivalence of Imaginary Time Evolution and Adiabatic Unitary Evolution
Given a stabilizer state ∣ ψ 0 ⟩ ∈ C 0 |\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0 ∣ ψ h ⟩ ∈ C h |\psi_h\rangle\in\mathcal{C}_h ∣ ψ h ⟩ ∈ C h ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ H h ( t ) H_{h(t)} H h ( t ) h ( t ) h(t) h ( t ) h ( 0 ) = 0 h(0)=0 h ( 0 ) = 0 h ( T ) = h h(T)=h h ( T ) = h T T T
∣ ψ h ⟩ ≡ U h ∣ ψ 0 ⟩ = T exp { − i ∫ 0 T d t H h ( t ) } ∣ ψ 0 ⟩ , (148) |\psi_h\rangle\equiv U_h|\psi_0\rangle=\mathcal{T}\exp\left\{-i\int_0^T\mathrm{d}t\,H_{h(t)}\right\}|\psi_0\rangle, \tag*{(148)} ∣ ψ h ⟩ ≡ U h ∣ ψ 0 ⟩ = T exp { − i ∫ 0 T d t H h ( t ) } ∣ ψ 0 ⟩ , (148) where T \mathcal{T} T [137 ] then guarantees that ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ C h \mathcal{C}_h C h T T T Δ − 2 \Delta^{-2} Δ − 2 T T T e − i ∫ 0 T d t E η g . s . ( t ) e^{-i\int_0^T\mathrm{d}t\,E_{\eta}^{\mathrm{g.s.}}(t)} e − i ∫ 0 T d t E η g.s. ( t ) [137 ] , we will consider the limit Δ − 2 poly ( L ) ≪ T ≪ exp ( L ) \Delta^{-2}\operatorname{poly}(L)\ll T\ll\exp(L) Δ − 2 poly ( L ) ≪ T ≪ exp ( L )
While conceptually appealing, the preceding approach has the drawback that the states ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ project into C h \mathcal{C}_h C h
∣ φ h ⟩ ≡ e − β H h / 2 ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ , (149) |\varphi_h\rangle\equiv\frac{e^{-\beta H_h/2}|\psi_0\rangle}{\sqrt{\langle\psi_0|e^{-\beta H_h}|\psi_0\rangle}}, \tag*{(149)} ∣ φ h ⟩ ≡ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ e − β H h /2 ∣ ψ 0 ⟩ , (149) where β ∼ poly ( L ) \beta\sim\operatorname{poly}(L) β ∼ poly ( L ) C h \mathcal{C}_h C h e − β H h e^{-\beta H_h} e − β H h C 0 \mathcal{C}_0 C 0 C h \mathcal{C}_h C h h < h c h<h_c h < h c
In this Appendix, we will argue that these two approaches are essentially equivalent throughout the topological phase; that is,
∣ ψ h ⟩ = e − i ϕ ( h , T ) ∣ φ h ⟩ + O ( e − L / ξ ) , (150) |\psi_h\rangle=e^{-i\phi(h,T)}|\varphi_h\rangle+\mathcal{O}(e^{-L/\xi}), \tag*{(150)} ∣ ψ h ⟩ = e − i ϕ ( h , T ) ∣ φ h ⟩ + O ( e − L / ξ ) , (150) where ϕ ( h , T ) \phi(h,T) ϕ ( h , T ) m m m h < h c h<h_c h < h c ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ C h \mathcal{C}_h C h C h \mathcal{C}_h C h C h ( t ) \mathcal{C}_{h(t)} C h ( t ) ∣ ψ h a ⟩ |\psi_h^a\rangle ∣ ψ h a ⟩ C h \mathcal{C}_h C h h h h ∣ ψ h ⟩ |\psi_h\rangle ∣ ψ h ⟩ [138 ] :
∣ ψ h ⟩ = e − i ϕ ( h , T ) ∑ a b a ( h ) ∣ ψ h a ⟩ , b a ( h ) = ∑ a ′ [ P exp { i ∫ 0 h d h ′ A ( h ′ ) } ] a a ′ b a ′ ( 0 ) , (151) |\psi_h\rangle=e^{-i\phi(h,T)}\sum_a b_a(h)|\psi_h^a\rangle,\qquad b_a(h)=\sum_{a'}\left[\mathcal{P}\exp\left\{i\int_0^h\mathrm{d}h'\,A(h')\right\}\right]_{aa'}b_{a'}(0), \tag*{(151)} ∣ ψ h ⟩ = e − i ϕ ( h , T ) a ∑ b a ( h ) ∣ ψ h a ⟩ , b a ( h ) = a ′ ∑ [ P exp { i ∫ 0 h d h ′ A ( h ′ ) } ] a a ′ b a ′ ( 0 ) , (151) where P \mathcal{P} P A ( h ) A(h) A ( h )
[ A ( h ) ] a a ′ ≡ ⟨ ψ h a ∣ i ∂ ∂ h ∣ ψ h a ′ ⟩ . (152) [A(h)]_{aa'}\equiv\langle\psi_h^a|i\frac{\partial}{\partial h}|\psi_h^{a'}\rangle. \tag*{(152)} [ A ( h ) ] a a ′ ≡ ⟨ ψ h a ∣ i ∂ h ∂ ∣ ψ h a ′ ⟩ . (152) Our goal is therefore to compute A ( h ) A(h) A ( h ) ∣ φ h a ⟩ |\varphi_h^a\rangle ∣ φ h a ⟩ (22) do not perfectly lie within C h \mathcal{C}_h C h exact orthonormal basis of C h \mathcal{C}_h C h ∣ φ ~ h σ ⟩ \lvert\widetilde{\varphi}_h^\sigma\rangle ∣ φ h σ ⟩ C h \mathcal{C}_h C h [139 ] . Both of these procedures modify the states by exponentially small corrections, i.e., ⟨ φ h σ ∣ φ ~ h σ ′ ⟩ ≥ 1 − O ( e − L / ξ ) \langle\varphi_h^\sigma\vert\widetilde{\varphi}_h^{\sigma'}\rangle\ge1-O(e^{-L/\xi}) ⟨ φ h σ ∣ φ h σ ′ ⟩ ≥ 1 − O ( e − L / ξ ) h h h h ( t ) h(t) h ( t ) C h ( t ) \mathcal{C}_{h(t)} C h ( t )
[ A ( h ) ] σ σ ′ = ⟨ φ ~ h σ ∣ i ∂ ∂ h ∣ φ ~ h σ ′ ⟩ = ⟨ φ h σ ∣ i ∂ ∂ h ∣ φ h σ ′ ⟩ + O ( e − L / ξ ) . (153) [A(h)]_{\sigma\sigma'}=\langle\widetilde{\varphi}_h^\sigma\vert i\frac{\partial}{\partial h}\vert\widetilde{\varphi}_h^{\sigma'}\rangle=\langle\varphi_h^\sigma\vert i\frac{\partial}{\partial h}\vert\varphi_h^{\sigma'}\rangle+O(e^{-L/\xi}). \tag*{(153)} [ A ( h ) ] σ σ ′ = ⟨ φ h σ ∣ i ∂ h ∂ ∣ φ h σ ′ ⟩ = ⟨ φ h σ ∣ i ∂ h ∂ ∣ φ h σ ′ ⟩ + O ( e − L / ξ ) . (153) To compute the first term, we note the following identity30
∂ ∂ h e − β H h / 2 = ∫ 0 β / 2 d τ e − τ H h ( ∑ ℓ X ℓ ) e τ H h e − β H h / 2 = ∫ 0 β / 2 d τ ∑ ℓ X ℓ ( − τ ) e − β H h / 2 , (154) \frac{\partial}{\partial h}e^{-\beta H_h/2}=\int_0^{\beta/2}d\tau\,e^{-\tau H_h}\left(\sum_{\ell}X_\ell\right)e^{\tau H_h}e^{-\beta H_h/2}=\int_0^{\beta/2}d\tau\sum_{\ell}X_\ell(-\tau)e^{-\beta H_h/2}, \tag*{(154)} ∂ h ∂ e − β H h /2 = ∫ 0 β /2 d τ e − τ H h ( ℓ ∑ X ℓ ) e τ H h e − β H h /2 = ∫ 0 β /2 d τ ℓ ∑ X ℓ ( − τ ) e − β H h /2 , (154) where X ℓ ( − τ ) ≡ e − τ H h X ℓ e τ H h X_\ell(-\tau)\equiv e^{-\tau H_h}X_\ell e^{\tau H_h} X ℓ ( − τ ) ≡ e − τ H h X ℓ e τ H h (22) of the ITE states ∣ φ h σ ⟩ \lvert\varphi_h^\sigma\rangle ∣ φ h σ ⟩
⟨ φ h σ ∣ ∂ ∂ h ∣ φ h σ ′ ⟩ = ∫ 0 β / 2 d τ ∑ ℓ ⟨ φ h σ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ − 1 2 ∫ − β / 2 β / 2 d τ ∑ ℓ ⟨ φ h σ ′ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ ⟨ φ h σ ∣ φ h σ ′ ⟩ = ∫ 0 β / 2 d τ ∑ ℓ { ⟨ φ h σ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ − ⟨ φ h σ ′ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ ⟨ φ h σ ∣ φ h σ ′ ⟩ } . (155) \begin{aligned}
\langle\varphi_h^\sigma\vert\frac{\partial}{\partial h}\vert\varphi_h^{\sigma'}\rangle
&=\int_0^{\beta/2}d\tau\sum_\ell\langle\varphi_h^\sigma\vert X_\ell(-\tau)\vert\varphi_h^{\sigma'}\rangle-\frac{1}{2}\int_{-\beta/2}^{\beta/2}d\tau\sum_\ell\langle\varphi_h^{\sigma'}\vert X_\ell(-\tau)\vert\varphi_h^{\sigma'}\rangle\langle\varphi_h^\sigma\vert\varphi_h^{\sigma'}\rangle\\
&=\int_0^{\beta/2}d\tau\sum_\ell\left\{\langle\varphi_h^\sigma\vert X_\ell(-\tau)\vert\varphi_h^{\sigma'}\rangle-\langle\varphi_h^{\sigma'}\vert X_\ell(-\tau)\vert\varphi_h^{\sigma'}\rangle\langle\varphi_h^\sigma\vert\varphi_h^{\sigma'}\rangle\right\}. \tag*{(155)}
\end{aligned} ⟨ φ h σ ∣ ∂ h ∂ ∣ φ h σ ′ ⟩ = ∫ 0 β /2 d τ ℓ ∑ ⟨ φ h σ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ − 2 1 ∫ − β /2 β /2 d τ ℓ ∑ ⟨ φ h σ ′ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ ⟨ φ h σ ∣ φ h σ ′ ⟩ = ∫ 0 β /2 d τ ℓ ∑ { ⟨ φ h σ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ − ⟨ φ h σ ′ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ ⟨ φ h σ ∣ φ h σ ′ ⟩ } . (155) where the second term in both lines comes from differentiating the denominator Z h Z_h Z h σ = σ ′ \sigma=\sigma' σ = σ ′ σ ≠ σ ′ \sigma\ne\sigma' σ = σ ′ O ( e − L / ξ ) O(e^{-L/\xi}) O ( e − L / ξ ) (23) , while the first term in brackets can be expressed as
⟨ φ h σ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ = 1 Z h ⟨ ψ 0 σ ∣ e − ( τ + β / 2 ) H h X ℓ e ( τ − β / 2 ) H h ∣ ψ 0 σ ′ ⟩ . (156) \langle\varphi_h^\sigma\vert X_\ell(-\tau)\vert\varphi_h^{\sigma'}\rangle=\frac{1}{Z_h}\langle\psi_0^\sigma\vert e^{-(\tau+\beta/2)H_h}X_\ell e^{(\tau-\beta/2)H_h}\vert\psi_0^{\sigma'}\rangle. \tag*{(156)} ⟨ φ h σ ∣ X ℓ ( − τ ) ∣ φ h σ ′ ⟩ = Z h 1 ⟨ ψ 0 σ ∣ e − ( τ + β /2 ) H h X ℓ e ( τ − β /2 ) H h ∣ ψ 0 σ ′ ⟩ . (156) We can interpret the numerator as an LGT partition function in which, at imaginary time − τ -\tau − τ m m m ℓ \ell ℓ m m m ⟨ φ h σ ∣ φ h σ ′ ⟩ ∝ ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ \langle\varphi_h^\sigma\vert\varphi_h^{\sigma'}\rangle\propto\langle\psi_0^\sigma\vert e^{-\beta H_h}\vert\psi_0^{\sigma'}\rangle ⟨ φ h σ ∣ φ h σ ′ ⟩ ∝ ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ′ ⟩ X ℓ X_\ell X ℓ − τ -\tau − τ L L L h < h c h<h_c h < h c A ( h ) A(h) A ( h ) L L L b σ ( h ) = b σ ( 0 ) + O ( e − L / ξ ) b_\sigma(h)=b_\sigma(0)+O(e^{-L/\xi}) b σ ( h ) = b σ ( 0 ) + O ( e − L / ξ ) ∣ ψ 0 ⟩ = ∑ σ b σ ( 0 ) ∣ ψ 0 σ ⟩ \lvert\psi_0\rangle=\sum_\sigma b_\sigma(0)\lvert\psi_0^\sigma\rangle ∣ ψ 0 ⟩ = ∑ σ b σ ( 0 ) ∣ ψ 0 σ ⟩
∣ ψ h ⟩ = e − i ϕ ( h , T ) ∑ σ b σ ( 0 ) ∣ φ h σ ⟩ + O ( e − L / ξ ) = e − i ϕ ( h , T ) e − β H h / 2 ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ + O ( e − L / ξ ) , (157) \lvert\psi_h\rangle=e^{-i\phi(h,T)}\sum_\sigma b_\sigma(0)\lvert\varphi_h^\sigma\rangle+O(e^{-L/\xi})=e^{-i\phi(h,T)}\frac{e^{-\beta H_h/2}\lvert\psi_0\rangle}{\sqrt{\langle\psi_0\vert e^{-\beta H_h}\vert\psi_0\rangle}}+O(e^{-L/\xi}), \tag*{(157)} ∣ ψ h ⟩ = e − i ϕ ( h , T ) σ ∑ b σ ( 0 ) ∣ φ h σ ⟩ + O ( e − L / ξ ) = e − i ϕ ( h , T ) ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ e − β H h /2 ∣ ψ 0 ⟩ + O ( e − L / ξ ) , (157) as desired.
Pruning
In this Appendix, we discuss an alternative method to that of Sec. III B for promoting ITE into an exactly unitary transformation. For reasons to become clear, we refer to this method as “pruning”.
First recall that the ITE basis states ∣ φ h σ ⟩ \lvert\varphi_h^\sigma\rangle ∣ φ h σ ⟩ Z Z Z
∣ φ h σ ⟩ ≡ 1 Z h e − β H h / 2 ∣ ψ 0 σ ⟩ , Z h ≡ ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ⟩ . (158) \lvert\varphi_h^\sigma\rangle\equiv\frac{1}{\sqrt{Z_h}}e^{-\beta H_h/2}\lvert\psi_0^\sigma\rangle,\qquad Z_h\equiv\langle\psi_0^\sigma\vert e^{-\beta H_h}\vert\psi_0^\sigma\rangle. \tag*{(158)} ∣ φ h σ ⟩ ≡ Z h 1 e − β H h /2 ∣ ψ 0 σ ⟩ , Z h ≡ ⟨ ψ 0 σ ∣ e − β H h ∣ ψ 0 σ ⟩ . (158) As described in Sec. III B, so long as h < h c h<h_c h < h c L L L (23) ]. Furthermore, because H h H_h H h § 3 A). It follows from these two observations together that the four states ∣ φ h σ ⟩ \lvert\varphi_h^\sigma\rangle ∣ φ h σ ⟩
∂ ∂ x e M ( x ) = ∫ 0 1 d s e s M ( x ) ∂ M ∂ x e ( 1 − s ) M ( x ) , (159) \frac{\partial}{\partial x}e^{M(x)}=\int_0^1ds\,e^{sM(x)}\frac{\partial M}{\partial x}e^{(1-s)M(x)}, \tag*{(159)} ∂ x ∂ e M ( x ) = ∫ 0 1 d s e s M ( x ) ∂ x ∂ M e ( 1 − s ) M ( x ) , (159) which can be derived by writing e M ( x ) = lim n → ∞ [ 1 + 1 n M ( x ) ] n e^{M(x)}=\lim_{n\to\infty}\left[1+\frac{1}{n}M(x)\right]^n e M ( x ) = lim n → ∞ [ 1 + n 1 M ( x ) ] n
Explicitly, divide the full set of computational basis states z ≡ { z ℓ } \mathbf{z} \equiv\{z_\ell\} z ≡ { z ℓ } S h σ \mathcal{S}_h^\sigma S h σ ∣ φ h σ ⟩ \lvert\varphi_h^\sigma\rangle ∣ φ h σ ⟩
S h σ = { z : argmax σ ′ ⟨ z ∣ φ h σ ′ ⟩ = σ } , (160) \mathcal{S}_h^\sigma= \left\{ \mathbf{z} : \operatorname{argmax}_{\sigma'} \langle\mathbf{z} \lvert\varphi_h^{\sigma'} \rangle= \sigma\right\}, \tag*{(160)} S h σ = { z : argmax σ ′ ⟨ z ∣ φ h σ ′ ⟩ = σ } , (160) i.e., z ∈ S h σ \mathbf{z} \in\mathcal{S}_h^\sigma z ∈ S h σ ⟨ z ∣ φ h σ ⟩ > ⟨ z ∣ φ h σ ′ ⟩ \langle\mathbf{z} \lvert\varphi_h^\sigma\rangle> \langle\mathbf{z} \lvert\varphi_h^{\sigma'} \rangle ⟨ z ∣ φ h σ ⟩ > ⟨ z ∣ φ h σ ′ ⟩ σ ′ ≠ σ \sigma' \ne\sigma σ ′ = σ ⟨ z ∣ φ h σ ⟩ = ⟨ z ∣ φ h σ ′ ⟩ \langle\mathbf{z} \lvert\varphi_h^\sigma\rangle= \langle\mathbf{z} \lvert\varphi_h^{\sigma'} \rangle ⟨ z ∣ φ h σ ⟩ = ⟨ z ∣ φ h σ ′ ⟩ σ ≠ σ ′ \sigma\ne\sigma' σ = σ ′ z \mathbf{z} z ∣ P ~ h σ ⟩ \lvert\widetilde{P}_h^\sigma\rangle ∣ P h σ ⟩
⟨ z ∣ P ~ h σ ⟩ = { ⟨ z ∣ φ h σ ⟩ , z ∈ S h σ , 0 , z ∉ S h σ , (161) \langle\mathbf{z} \lvert\widetilde{P}_h^\sigma\rangle=
\begin{cases}
\langle\mathbf{z} \lvert\varphi_h^\sigma\rangle, & \mathbf{z} \in\mathcal{S}_h^\sigma, \\
0, & \mathbf{z} \notin\mathcal{S}_h^\sigma,
\end{cases} \tag*{(161)} ⟨ z ∣ P h σ ⟩ = { ⟨ z ∣ φ h σ ⟩ , 0 , z ∈ S h σ , z ∈ / S h σ , (161) i.e., ∣ P ~ h σ ⟩ \lvert\widetilde{P}_h^\sigma\rangle ∣ P h σ ⟩ ∣ φ h σ ⟩ \lvert\varphi_h^\sigma\rangle ∣ φ h σ ⟩ S h σ \mathcal{S}_h^\sigma S h σ ∣ P ~ h σ ⟩ \lvert\widetilde{P}_h^\sigma\rangle ∣ P h σ ⟩ ∣ P h σ ⟩ \lvert P_h^\sigma\rangle ∣ P h σ ⟩
∣ P h σ ⟩ ≡ ∣ P ~ h σ ⟩ ∥ ∣ P ~ h σ ⟩ ∥ . (162) \lvert P_h^\sigma\rangle\equiv\frac{\lvert\widetilde{P}_h^\sigma\rangle}{\left\|\lvert\widetilde{P}_h^\sigma\rangle\right\|}. \tag*{(162)} ∣ P h σ ⟩ ≡ ∣ P h σ ⟩ ∣ P h σ ⟩ . (162) We now show that, throughout the topological phase h < h c h < h_c h < h c ∣ P h σ ⟩ \lvert P_h^\sigma\rangle ∣ P h σ ⟩ ∣ φ h σ ⟩ \lvert\varphi_h^\sigma\rangle ∣ φ h σ ⟩ ∣ P h σ ⟩ \lvert P_h^\sigma\rangle ∣ P h σ ⟩ ∣ φ h σ ⟩ \lvert\varphi_h^\sigma\rangle ∣ φ h σ ⟩ ∥ ∣ P ~ h σ ⟩ ∥ \left\|\lvert\widetilde{P}_h^\sigma\rangle\right\| ∣ P h σ ⟩ ∣ P ~ h σ ⟩ \lvert\widetilde{P}_h^\sigma\rangle ∣ P h σ ⟩ ∣ P ~ h σ ⟩ \lvert\widetilde{P}_h^\sigma\rangle ∣ P h σ ⟩ (23) as follows:
1 − ⟨ P ~ h σ ∣ P ~ h σ ⟩ = ∑ z ∉ S h σ ∣ ⟨ z ∣ φ h σ ⟩ ∣ 2 ≤ ∑ z ∉ S h σ ∑ σ ′ : σ ′ ≠ σ ⟨ φ h σ ∣ z ⟩ ⟨ z ∣ φ h σ ′ ⟩ ≤ ∑ σ ′ : σ ′ ≠ σ ⟨ φ h σ ∣ φ h σ ′ ⟩ ≤ O ( e − L / ξ ) . (163) \begin{aligned}
1-\langle\widetilde{P}_h^\sigma\lvert\widetilde{P}_h^\sigma\rangle
&= \sum_{\mathbf{z}\notin\mathcal{S}_h^\sigma}\left|\langle\mathbf{z}\lvert\varphi_h^\sigma\rangle\right|^2 \\
&\leq\sum_{\mathbf{z}\notin\mathcal{S}_h^\sigma}\sum_{\sigma':\sigma'\ne\sigma}\langle\varphi_h^\sigma\lvert\mathbf{z}\rangle\langle\mathbf{z}\lvert\varphi_h^{\sigma'}\rangle\\
&\leq\sum_{\sigma':\sigma'\ne\sigma}\langle\varphi_h^\sigma\lvert\varphi_h^{\sigma'}\rangle\\
&\leq O(e^{-L/\xi}). \tag*{(163)}
\end{aligned} 1 − ⟨ P h σ ∣ P h σ ⟩ = z ∈ / S h σ ∑ ∣ ⟨ z ∣ φ h σ ⟩ ∣ 2 ≤ z ∈ / S h σ ∑ σ ′ : σ ′ = σ ∑ ⟨ φ h σ ∣ z ⟩ ⟨ z ∣ φ h σ ′ ⟩ ≤ σ ′ : σ ′ = σ ∑ ⟨ φ h σ ∣ φ h σ ′ ⟩ ≤ O ( e − L / ξ ) . (163) In the first line, we have used the fact that ∣ φ h σ ⟩ \lvert\varphi_h^\sigma\rangle ∣ φ h σ ⟩ z ∉ S h σ \mathbf{z}\notin\mathcal{S}_h^\sigma z ∈ / S h σ σ ′ \sigma' σ ′ ⟨ z ∣ φ h σ ⟩ ≤ ⟨ z ∣ φ h σ ′ ⟩ \langle\mathbf{z}\lvert\varphi_h^\sigma\rangle\leq\langle\mathbf{z}\lvert\varphi_h^{\sigma'}\rangle ⟨ z ∣ φ h σ ⟩ ≤ ⟨ z ∣ φ h σ ′ ⟩ z \mathbf{z} z (23) . This proves that the pruned code states have high overlap with the original ITE code states,
⟨ φ h σ ∣ P h σ ⟩ ≥ 1 − O ( e − L / ξ ) , (164) \langle\varphi_h^\sigma\lvert P_h^\sigma\rangle\geq1-O(e^{-L/\xi}), \tag*{(164)} ⟨ φ h σ ∣ P h σ ⟩ ≥ 1 − O ( e − L / ξ ) , (164) which in turn implies that the pruned states ∣ P h σ ⟩ \lvert P_h^\sigma\rangle ∣ P h σ ⟩ C h C_h C h C h C_h C h ⟨ z ∣ φ h σ ⟩ \langle\mathbf{z}\lvert\varphi_h^\sigma\rangle ⟨ z ∣ φ h σ ⟩
An interesting fact about the pruned code states is that they remain exactly orthonormal upon measuring the stabilizers B p B_p B p ∣ P h σ ⟩ \lvert P_h^\sigma\rangle ∣ P h σ ⟩
⟨ P h σ ∣ Π ^ s ∣ P h σ ′ ⟩ = P s δ σ σ ′ (165) \langle P_h^\sigma\lvert\widehat{\Pi}_s\lvert P_h^{\sigma'}\rangle=P_s\delta_{\sigma\sigma'} \tag*{(165)} ⟨ P h σ ∣ Π s ∣ P h σ ′ ⟩ = P s δ σ σ ′ (165) exactly, where P s = ⟨ P h σ ∣ Π ^ s ∣ P h σ ⟩ P_s=\langle P_h^\sigma\lvert\widehat{\Pi}_s\lvert P_h^\sigma\rangle P s = ⟨ P h σ ∣ Π s ∣ P h σ ⟩ P s P_s P s Π ^ s \widehat{\Pi}_s Π s ∣ P h ⟩ ≡ ∑ σ c σ ∣ P h σ ⟩ \lvert\mathcal{P}_h\rangle\equiv\sum_\sigma c_\sigma\lvert P_h^\sigma\rangle ∣ P h ⟩ ≡ ∑ σ c σ ∣ P h σ ⟩ P s P_s P s s s s ∣ P h ⟩ \lvert P_h\rangle ∣ P h ⟩
Similarly, once decoherence is added, the probability P s P_s P s s s s E ( ∣ P h ⟩ ⟨ P h ∣ ) \mathcal{E}(|\mathcal{P}_h\rangle\langle\mathcal{P}_h|) E ( ∣ P h ⟩ ⟨ P h ∣ )
⟨ P h σ ∣ E ( Π ^ s ) ∣ P h σ ′ ⟩ = P s δ σ σ ′ , (166) \langle\mathcal{P}_h^\sigma|\mathcal{E}(\hat{\Pi}_s)|\mathcal{P}_h^{\sigma'}\rangle=P_s\delta_{\sigma\sigma'}, \tag*{(166)} ⟨ P h σ ∣ E ( Π ^ s ) ∣ P h σ ′ ⟩ = P s δ σ σ ′ , (166) similar to the pure-state case. An immediate consequence is that stabilizer measurements of a bit-flip-decohered pruned state ∣ P h ⟩ |\mathcal{P}_h\rangle ∣ P h ⟩ ∣ ψ 0 ⟩ = ∑ σ b σ ∣ ψ 0 σ ⟩ |\psi_0\rangle=\sum_\sigma b_\sigma|\psi_0^\sigma\rangle ∣ ψ 0 ⟩ = ∑ σ b σ ∣ ψ 0 σ ⟩
Π ^ s E ( ∣ P h ⟩ ⟨ P h ∣ ) Π ^ s P s = ∑ e P e ∣ s X s ∣ e ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X s ∣ e , (167) \frac{\hat{\Pi}_s\mathcal{E}(|\mathcal{P}_h\rangle\langle\mathcal{P}_h|)\hat{\Pi}_s}{P_s}=\sum_e P_{e|s}X_{s|e}|\psi_0\rangle\langle\psi_0|X_{s|e}, \tag*{(167)} P s Π ^ s E ( ∣ P h ⟩ ⟨ P h ∣ ) Π ^ s = e ∑ P e ∣ s X s ∣ e ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X s ∣ e , (167) in contrast to the approximate action of the incoherent channel M s \mathcal{M}_s M s (73) . Here P e ∣ s = p e ⟨ P h ∣ X e Π ^ s X e ∣ P h ⟩ / P s P_{e|s}=p_e\langle\mathcal{P}_h|X_e\hat{\Pi}_sX_e|\mathcal{P}_h\rangle/P_s P e ∣ s = p e ⟨ P h ∣ X e Π ^ s X e ∣ P h ⟩ / P s e e e s s s X s ∣ e X_{s|e} X s ∣ e Π ^ s X e ∣ P h ⟩ ∝ X s ∣ e ∣ ψ 0 ⟩ \hat{\Pi}_sX_e|\mathcal{P}_h\rangle\propto X_{s|e}|\psi_0\rangle Π ^ s X e ∣ P h ⟩ ∝ X s ∣ e ∣ ψ 0 ⟩
Decoding Fidelity and Strong Symmetry
Throughout this work, our quantitative measure of successful QEC has been the measurement-averaged decoding fidelity,
F ‾ ( ∣ ψ h ⟩ ) = ⟨ ψ h ∣ N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) ∣ ψ h ⟩ . (168) \overline{F}(|\psi_h\rangle)=\langle\psi_h|\mathcal{N}(|\psi_h\rangle\langle\psi_h|)|\psi_h\rangle. \tag*{(168)} F ( ∣ ψ h ⟩) = ⟨ ψ h ∣ N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) ∣ ψ h ⟩ . (168) Here N ≡ M ∘ E \mathcal{N}\equiv\mathcal{M}\circ\mathcal{E} N ≡ M ∘ E E \mathcal{E} E M \mathcal{M} M (142) ], the latter of which involves measuring stabilizers A v A_v A v B p B_p B p C 0 \mathcal{C}_0 C 0 C h \mathcal{C}_h C h E \mathcal{E} E N \mathcal{N} N X ‾ μ \overline{X}_\mu X μ
X ‾ μ N ( ρ ) = N ( X ‾ μ ρ ) , N ( ρ ) X ‾ μ = N ( ρ X ‾ μ ) (169) \overline{X}_\mu\mathcal{N}(\rho)=\mathcal{N}(\overline{X}_\mu\rho),\qquad\mathcal{N}(\rho)\overline{X}_\mu=\mathcal{N}(\rho\overline{X}_\mu) \tag*{(169)} X μ N ( ρ ) = N ( X μ ρ ) , N ( ρ ) X μ = N ( ρ X μ ) (169) for any ρ \rho ρ ∣ φ h ( 0 , 0 ) ⟩ ∝ e − β H h / 2 ∣ ψ 0 ( 0 , 0 ) ⟩ |\varphi_h^{(0,0)}\rangle\propto e^{-\beta H_h/2}|\psi_0^{(0,0)}\rangle ∣ φ h ( 0 , 0 ) ⟩ ∝ e − β H h /2 ∣ ψ 0 ( 0 , 0 ) ⟩ ∣ ψ 0 ( 0 , 0 ) ⟩ |\psi_0^{(0,0)}\rangle ∣ ψ 0 ( 0 , 0 ) ⟩
To be precise, let31 ∣ ψ h σ ⟩ ≡ U h ∣ ψ 0 σ ⟩ |\psi_h^\sigma\rangle\equiv U_h|\psi_0^\sigma\rangle ∣ ψ h σ ⟩ ≡ U h ∣ ψ 0 σ ⟩ C h \mathcal{C}_h C h X ‾ σ ∣ ψ h σ ′ ⟩ = ∣ ψ h σ ⊕ σ ′ ⟩ \overline{X}^\sigma|\psi_h^{\sigma'}\rangle=|\psi_h^{\sigma\oplus\sigma'}\rangle X σ ∣ ψ h σ ′ ⟩ = ∣ ψ h σ ⊕ σ ′ ⟩ F ‾ ( ∣ ψ h ( 0 , 0 ) ⟩ ) ≥ 1 − ϵ \overline{F}(|\psi_h^{(0,0)}\rangle)\ge1-\epsilon F ( ∣ ψ h ( 0 , 0 ) ⟩) ≥ 1 − ϵ
min ∣ ψ h ⟩ ∈ C h F ‾ ( ∣ ψ h ⟩ ) ≥ 1 − 8 ϵ . (170) \min_{|\psi_h\rangle\in\mathcal{C}_h}\overline{F}(|\psi_h\rangle)\ge1-8\sqrt{\epsilon}. \tag*{(170)} ∣ ψ h ⟩ ∈ C h min F ( ∣ ψ h ⟩) ≥ 1 − 8 ϵ . (170) Consequently, an arbitrary TFTC state ∣ ψ h ⟩ = ∑ σ b σ ∣ ψ h σ ⟩ |\psi_h\rangle=\sum_\sigma b_\sigma|\psi_h^\sigma\rangle ∣ ψ h ⟩ = ∑ σ b σ ∣ ψ h σ ⟩ ∣ ψ h ( 0 , 0 ) ⟩ |\psi_h^{(0,0)}\rangle ∣ ψ h ( 0 , 0 ) ⟩ ∣ φ h ( 0 , 0 ) ⟩ |\varphi_h^{(0,0)}\rangle ∣ φ h ( 0 , 0 ) ⟩ ∣ ψ h ( 0 , 0 ) ⟩ |\psi_h^{(0,0)}\rangle ∣ ψ h ( 0 , 0 ) ⟩
To prove Eq. (170) , consider the entanglement fidelity F e F_e F e
F e = ⟨ Ψ Q R ∣ N ( ∣ Ψ Q R ⟩ ⟨ Ψ Q R ∣ ) ∣ Ψ Q R ⟩ , (171) F_e=\langle\Psi_{QR}|\mathcal{N}(|\Psi_{QR}\rangle\langle\Psi_{QR}|)|\Psi_{QR}\rangle, \tag*{(171)} F e = ⟨ Ψ QR ∣ N ( ∣ Ψ QR ⟩ ⟨ Ψ QR ∣ ) ∣ Ψ QR ⟩ , (171) where ∣ Ψ Q R ⟩ = 1 4 ∑ σ ∣ ψ h σ ⟩ Q ⊗ ∣ σ ⟩ R |\Psi_{QR}\rangle=\frac{1}{4}\sum_\sigma|\psi_h^\sigma\rangle_Q\otimes|\sigma\rangle_R ∣ Ψ QR ⟩ = 4 1 ∑ σ ∣ ψ h σ ⟩ Q ⊗ ∣ σ ⟩ R Q Q Q R R R N \mathcal{N} N Q Q Q N \mathcal{N} N
F e = 1 4 2 ∑ σ , σ ′ ⟨ ψ h σ ∣ N ( ∣ ψ h σ ⟩ ⟨ ψ h σ ′ ∣ ) ∣ ψ h σ ′ ⟩ = ⟨ ψ h ( 0 , 0 ) ∣ N ( ∣ ψ h ( 0 , 0 ) ⟩ ⟨ ψ h ( 0 , 0 ) ∣ ) ∣ ψ h ( 0 , 0 ) ⟩ = F ‾ ( ∣ ψ h ( 0 , 0 ) ⟩ ) , (172) F_e=\frac{1}{4^2}\sum_{\sigma,\sigma'}\langle\psi_h^\sigma|\mathcal{N}(|\psi_h^\sigma\rangle\langle\psi_h^{\sigma'}|)|\psi_h^{\sigma'}\rangle=\langle\psi_h^{(0,0)}|\mathcal{N}(|\psi_h^{(0,0)}\rangle\langle\psi_h^{(0,0)}|)|\psi_h^{(0,0)}\rangle=\overline{F}(|\psi_h^{(0,0)}\rangle), \tag*{(172)} F e = 4 2 1 σ , σ ′ ∑ ⟨ ψ h σ ∣ N ( ∣ ψ h σ ⟩ ⟨ ψ h σ ′ ∣ ) ∣ ψ h σ ′ ⟩ = ⟨ ψ h ( 0 , 0 ) ∣ N ( ∣ ψ h ( 0 , 0 ) ⟩ ⟨ ψ h ( 0 , 0 ) ∣ ) ∣ ψ h ( 0 , 0 ) ⟩ = F ( ∣ ψ h ( 0 , 0 ) ⟩) , (172) i.e., the entanglement fidelity is equal to the decoding fidelity of the single state ∣ ψ ( 0 , 0 ) ⟩ |\psi^{(0,0)}\rangle ∣ ψ ( 0 , 0 ) ⟩ 1 − ε 1-\varepsilon 1 − ε [80 ] gives
∥ N ( ∣ Ψ Q R ⟩ ⟨ Ψ Q R ∣ ) − ∣ Ψ Q R ⟩ ⟨ Ψ Q R ∣ ∥ 1 ≤ 2 1 − ⟨ ψ ( 0 , 0 ) ∣ N ( ∣ ψ ( 0 , 0 ) ⟩ ⟨ ψ ( 0 , 0 ) ∣ ) ∣ ψ ( 0 , 0 ) ⟩ ≤ 2 ε . (173) \left\|\mathcal{N}(|\Psi_{QR}\rangle\langle\Psi_{QR}|)-|\Psi_{QR}\rangle\langle\Psi_{QR}|\right\|_{1} \le2\sqrt{1-\langle\psi^{(0,0)}|\mathcal{N}(|\psi^{(0,0)}\rangle\langle\psi^{(0,0)}|)|\psi^{(0,0)}\rangle} \le2\sqrt{\varepsilon}. \tag*{(173)} ∥ N ( ∣ Ψ QR ⟩ ⟨ Ψ QR ∣ ) − ∣ Ψ QR ⟩ ⟨ Ψ QR ∣ ∥ 1 ≤ 2 1 − ⟨ ψ ( 0 , 0 ) ∣ N ( ∣ ψ ( 0 , 0 ) ⟩ ⟨ ψ ( 0 , 0 ) ∣ ) ∣ ψ ( 0 , 0 ) ⟩ ≤ 2 ε . (173) This further implies that the channel N \mathcal{N} N 32
∥ N − id ∥ ⋄ ≤ ( dim C h ) ∥ N ( ∣ Ψ Q R ⟩ ⟨ Ψ Q R ∣ ) − ∣ Ψ Q R ⟩ ⟨ Ψ Q R ∣ ∥ 1 ≤ 8 ε . (174) \|\mathcal{N}-\operatorname{id}\|_{\diamond} \le(\dim\mathcal{C}_{h})\left\|\mathcal{N}(|\Psi_{QR}\rangle\langle\Psi_{QR}|)-|\Psi_{QR}\rangle\langle\Psi_{QR}|\right\|_{1} \le8\sqrt{\varepsilon}. \tag*{(174)} ∥ N − id ∥ ⋄ ≤ ( dim C h ) ∥ N ( ∣ Ψ QR ⟩ ⟨ Ψ QR ∣ ) − ∣ Ψ QR ⟩ ⟨ Ψ QR ∣ ∥ 1 ≤ 8 ε . (174) Correspondingly, N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) \mathcal{N}(|\psi_h\rangle\langle\psi_h|) N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) ∣ ψ h ⟩ ⟨ ψ h ∣ |\psi_h\rangle\langle\psi_h| ∣ ψ h ⟩ ⟨ ψ h ∣
∥ N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) − ∣ ψ h ⟩ ⟨ ψ h ∣ ∥ 1 ≤ ∥ N − id ∥ ⋄ ≤ 8 ε . (175) \left\|\mathcal{N}(|\psi_h\rangle\langle\psi_h|)-|\psi_h\rangle\langle\psi_h|\right\|_{1} \le\|\mathcal{N}-\operatorname{id}\|_{\diamond} \le8\sqrt{\varepsilon}. \tag*{(175)} ∥ N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) − ∣ ψ h ⟩ ⟨ ψ h ∣ ∥ 1 ≤ ∥ N − id ∥ ⋄ ≤ 8 ε . (175) Finally, one more application of the Fuchs-van de Graaf inequality yields
F ( ∣ ψ h ⟩ ) = ⟨ ψ h ∣ N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) ∣ ψ h ⟩ ≥ ( 1 − 1 2 ∥ N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) − ∣ ψ h ⟩ ⟨ ψ h ∣ ∥ 1 ) 2 ≥ 1 − 8 ε . (176) F(|\psi_h\rangle)=\langle\psi_h|\mathcal{N}(|\psi_h\rangle\langle\psi_h|)|\psi_h\rangle\ge\left(1-\frac{1}{2}\left\|\mathcal{N}(|\psi_h\rangle\langle\psi_h|)-|\psi_h\rangle\langle\psi_h|\right\|_{1}\right)^2 \ge1-8\sqrt{\varepsilon}. \tag*{(176)} F ( ∣ ψ h ⟩) = ⟨ ψ h ∣ N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) ∣ ψ h ⟩ ≥ ( 1 − 2 1 ∥ N ( ∣ ψ h ⟩ ⟨ ψ h ∣ ) − ∣ ψ h ⟩ ⟨ ψ h ∣ ∥ 1 ) 2 ≥ 1 − 8 ε . (176) Decoding Coherent Errors
In this Appendix, we prove a necessary and sufficient condition for coherent errors to be correctable. These conditions are valid for general coherent errors on a general quantum error-correcting code, so the present discussion is abstracted away from the specific case of the toric code.
Let { ∣ ψ 0 a ⟩ } \{|\psi_0^a\rangle\} { ∣ ψ 0 a ⟩} C \mathcal{C} C ∣ ψ U a ⟩ ≡ U ∣ ψ 0 a ⟩ |\psi_U^a\rangle\equiv U|\psi_0^a\rangle ∣ ψ U a ⟩ ≡ U ∣ ψ 0 a ⟩ ∣ ψ 0 a ⟩ |\psi_0^a\rangle ∣ ψ 0 a ⟩ U U U P s a ≡ ⟨ ψ U a ∣ Π ^ s ∣ ψ U a ⟩ P_s^a \equiv\langle\psi_U^a|\hat{\Pi}_s|\psi_U^a\rangle P s a ≡ ⟨ ψ U a ∣ Π ^ s ∣ ψ U a ⟩ s s s ∣ ψ U a ⟩ |\psi_U^a\rangle ∣ ψ U a ⟩ ∣ ψ U a ( s ) ⟩ = 1 P s a Π ^ s ∣ ψ U a ⟩ |\psi_U^a(s)\rangle= \frac{1}{\sqrt{P_s^a}}\hat{\Pi}_s|\psi_U^a\rangle ∣ ψ U a ( s )⟩ = P s a 1 Π ^ s ∣ ψ U a ⟩ Π ^ s \hat{\Pi}_s Π ^ s ∑ s Π ^ s = 1 \sum_s \hat{\Pi}_s=1 ∑ s Π ^ s = 1 s s s
All of the measurement probabilities P s a P_s^a P s a P s a = P s P_s^a=P_s P s a = P s a a a
All of the states ∣ ψ U a ( s ) ⟩ |\psi_U^a(s)\rangle ∣ ψ U a ( s )⟩ ⟨ ψ U a ( s ) ∣ ψ U b ( s ) ⟩ = δ a b \langle\psi_U^a(s)|\psi_U^b(s)\rangle=\delta_{ab} ⟨ ψ U a ( s ) ∣ ψ U b ( s )⟩ = δ ab
If desired, these conditions can be written together in a basis-independent manner as follows: for any two code states ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ ∣ ψ 0 ′ ⟩ ∈ C |\psi'_0\rangle\in\mathcal{C} ∣ ψ 0 ′ ⟩ ∈ C
⟨ ψ 0 ∣ U † Π ^ s U ∣ ψ 0 ′ ⟩ = P s ⟨ ψ 0 ∣ ψ 0 ′ ⟩ . (177) \langle\psi_0|U^\dagger\hat{\Pi}_sU|\psi'_0\rangle=P_s\langle\psi_0|\psi'_0\rangle. \tag*{(177)} ⟨ ψ 0 ∣ U † Π ^ s U ∣ ψ 0 ′ ⟩ = P s ⟨ ψ 0 ∣ ψ 0 ′ ⟩ . (177) Note that this is simply the Knill-Laflamme condition [80 , 84 , 85 ] for an error channel with Kraus operators Π ^ s U \hat{\Pi}_sU Π ^ s U
The proof of this statement is rather trivial. First, let us denote the principal system as Q Q Q R R R Q Q Q
∣ Ψ Q R ⟩ ≡ 1 D ∑ a = 1 D ∣ ψ 0 a ⟩ Q ⊗ ∣ a ⟩ R , (178) |\Psi_{QR}\rangle\equiv\frac{1}{\sqrt{D}}\sum_{a=1}^{D}|\psi_0^a\rangle_Q\otimes|a\rangle_R, \tag*{(178)} ∣ Ψ QR ⟩ ≡ D 1 a = 1 ∑ D ∣ ψ 0 a ⟩ Q ⊗ ∣ a ⟩ R , (178) where D = dim C D=\dim\mathcal{C} D = dim C s s s
∣ Ψ Q R ( s ) ⟩ = 1 D P s ∑ a = 1 D P s a ∣ ψ U a ( s ) ⟩ Q ⊗ ∣ a ⟩ R , (179) |\Psi_{QR}(s)\rangle=\frac{1}{\sqrt{DP_s}}\sum_{a=1}^{D}\sqrt{P_s^a}|\psi_U^a(s)\rangle_Q\otimes|a\rangle_R, \tag*{(179)} ∣ Ψ QR ( s )⟩ = D P s 1 a = 1 ∑ D P s a ∣ ψ U a ( s ) ⟩ Q ⊗ ∣ a ⟩ R , (179) where P s = 1 D ∑ a = 1 D P s a P_s=\frac{1}{D}\sum_{a=1}^{D}P_s^a P s = D 1 ∑ a = 1 D P s a s s s ∣ Ψ Q R ⟩ |\Psi_{QR}\rangle ∣ Ψ QR ⟩
Any two purifications of a mixed state with the same dimension differ at most by a unitary on the purifying system [80 ] . Viewing Q Q Q R R R V s V_s V s ∣ Ψ Q R ( s ) ⟩ \lvert\Psi_{QR}(s)\rangle ∣ Ψ QR ( s )⟩ ∣ Ψ Q R ⟩ \lvert\Psi_{QR}\rangle ∣ Ψ QR ⟩ Q Q Q R R R R R R
ρ R ( s ) ≡ tr Q ∣ Ψ Q R ( s ) ⟩ ⟨ Ψ Q R ( s ) ∣ = 1 D P s ∑ a , b = 1 D ( P s a P s b ⟨ ψ U b ( s ) ∣ ψ U a ( s ) ⟩ ) ∣ a ⟩ ⟨ b ∣ R . (180) \rho_R(s) \equiv\operatorname{tr}_Q \lvert\Psi_{QR}(s)\rangle\langle\Psi_{QR}(s)\rvert= \frac{1}{D P_s}\sum_{a,b=1}^{D}\left(\sqrt{P_s^a P_s^b}\langle\psi_U^b(s)\vert\psi_U^a(s)\rangle\right)\lvert a\rangle\langle b\rvert_R . \tag*{(180)} ρ R ( s ) ≡ tr Q ∣ Ψ QR ( s )⟩ ⟨ Ψ QR ( s )∣ = D P s 1 a , b = 1 ∑ D ( P s a P s b ⟨ ψ U b ( s ) ∣ ψ U a ( s )⟩ ) ∣ a ⟩ ⟨ b ∣ R . (180) This state is maximally mixed on R R R R R R ⟨ ψ U a ( s ) ∣ ψ U b ( s ) ⟩ = δ a b \langle\psi_U^a(s)\vert\psi_U^b(s)\rangle= \delta_{ab} ⟨ ψ U a ( s ) ∣ ψ U b ( s )⟩ = δ ab P s a = P s P_s^a = P_s P s a = P s a a a V s V_s V s
Indeed, it is now simple to explicitly construct the desired recovery unitary:
V s = ∑ a = 1 D ∣ ψ 0 a ⟩ ⟨ ψ U a ( s ) ∣ , (181) V_s = \sum_{a=1}^{D}\lvert\psi_0^a\rangle\langle\psi_U^a(s)\rvert, \tag*{(181)} V s = a = 1 ∑ D ∣ ψ 0 a ⟩ ⟨ ψ U a ( s )∣ , (181) which acts unitarily on the space of postmeasurement states by condition 2. If we consider an arbitrary initial state ∣ ψ 0 ⟩ = ∑ a = 1 D c a ∣ ψ 0 a ⟩ \lvert\psi_0\rangle= \sum_{a=1}^{D} c_a\lvert\psi_0^a\rangle ∣ ψ 0 ⟩ = ∑ a = 1 D c a ∣ ψ 0 a ⟩
∣ ψ U ( s ) ⟩ ≡ 1 P s Π ^ s U ∣ ψ 0 ⟩ = ∑ a = 1 D c a ∣ ψ U a ( s ) ⟩ = V s † ∣ ψ 0 ⟩ , (182) \lvert\psi_U(s)\rangle\equiv\frac{1}{\sqrt{P_s}}\hat{\Pi}_s U\lvert\psi_0\rangle= \sum_{a=1}^{D}c_a\lvert\psi_U^a(s)\rangle= V_s^\dagger\lvert\psi_0\rangle, \tag*{(182)} ∣ ψ U ( s )⟩ ≡ P s 1 Π ^ s U ∣ ψ 0 ⟩ = a = 1 ∑ D c a ∣ ψ U a ( s )⟩ = V s † ∣ ψ 0 ⟩ , (182) i.e., the measurement has effectively implemented the unitary V s † V_s^\dagger V s † ∣ ψ 0 ⟩ \lvert\psi_0\rangle ∣ ψ 0 ⟩ V s V_s V s
More generally, suppose that Eq. (177) is only approximately satisfied, in the sense that
∣ ⟨ ψ 0 ∣ U † Π ^ s U ∣ ψ 0 ′ ⟩ − Q s ⟨ ψ 0 ∣ ψ 0 ′ ⟩ Q s ∣ ≤ ε (183) \left|\frac{\langle\psi_0\vert U^\dagger\hat{\Pi}_sU\vert\psi'_0\rangle-Q_s\langle\psi_0\vert\psi'_0\rangle}{Q_s}\right|\le\varepsilon \tag*{(183)} Q s ⟨ ψ 0 ∣ U † Π ^ s U ∣ ψ 0 ′ ⟩ − Q s ⟨ ψ 0 ∣ ψ 0 ′ ⟩ ≤ ε (183) for some Q s > 0 Q_s>0 Q s > 0 ∣ ψ 0 ′ ⟩ = ∣ ψ 0 ⟩ \lvert\psi'_0\rangle=\lvert\psi_0\rangle ∣ ψ 0 ′ ⟩ = ∣ ψ 0 ⟩ P s P_s P s s s s ( 1 − ε ) Q s (1-\varepsilon)Q_s ( 1 − ε ) Q s ( 1 + ε ) Q s (1+\varepsilon)Q_s ( 1 + ε ) Q s P s P_s P s Q s Q_s Q s relative error, for any code state ∣ ψ 0 ⟩ ∈ C 0 \lvert\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0
To construct a decoding unitary in this setting, we perform a polar decomposition of the operator Π ^ s U Π ^ 0 \hat{\Pi}_sU\hat{\Pi}_0 Π ^ s U Π ^ 0 Π ^ 0 \hat{\Pi}_0 Π ^ 0 C \mathcal{C} C
1 Q s Π ^ s U Π ^ 0 = V s I s 1 / 2 , I s ≡ Π ^ 0 U † Π ^ s U Π ^ 0 Q s , (184) \frac{1}{\sqrt{Q_s}}\hat{\Pi}_sU\hat{\Pi}_0=V_s\mathcal{I}_s^{1/2},\qquad\mathcal{I}_s\equiv\frac{\hat{\Pi}_0U^\dagger\hat{\Pi}_sU\hat{\Pi}_0}{Q_s}, \tag*{(184)} Q s 1 Π ^ s U Π ^ 0 = V s I s 1/2 , I s ≡ Q s Π ^ 0 U † Π ^ s U Π ^ 0 , (184) where V s V_s V s C \mathcal{C} C s s s I s = Π ^ 0 \mathcal{I}_s=\hat{\Pi}_0 I s = Π ^ 0 V s = V s † V_s=V_s^\dagger V s = V s † I s \mathcal{I}_s I s 1 − ε 1-\varepsilon 1 − ε 1 + ε 1+\varepsilon 1 + ε V s † V_s^\dagger V s †
F s ( ∣ ψ 0 ⟩ ) ≡ ∣ ⟨ ψ 0 ∣ V s † ∣ ψ U ( s ) ⟩ ∣ 2 = ∣ ⟨ ψ 0 ∣ V s † Π ^ s U ∣ ψ 0 ⟩ ∣ 2 ⟨ ψ 0 ∣ U † Π ^ s U ∣ ψ 0 ⟩ = ∣ ⟨ ψ 0 ∣ I s 1 / 2 ∣ ψ 0 ⟩ ∣ 2 ⟨ ψ 0 ∣ I s ∣ ψ 0 ⟩ ≥ 1 − ε 1 + ε ≥ 1 − 2 ε . (185) \mathcal{F}_s(\lvert\psi_0\rangle)\equiv\left|\langle\psi_0\vert V_s^\dagger\lvert\psi_U(s)\rangle\right|^2=\frac{\left|\langle\psi_0\vert V_s^\dagger\hat{\Pi}_sU\vert\psi_0\rangle\right|^2}{\langle\psi_0\vert U^\dagger\hat{\Pi}_sU\vert\psi_0\rangle}=\frac{\left|\langle\psi_0\vert\mathcal{I}_s^{1/2}\vert\psi_0\rangle\right|^2}{\langle\psi_0\vert\mathcal{I}_s\vert\psi_0\rangle}\ge\frac{1-\varepsilon}{1+\varepsilon}\ge1-2\varepsilon. \tag*{(185)} F s (∣ ψ 0 ⟩) ≡ ⟨ ψ 0 ∣ V s † ∣ ψ U ( s )⟩ 2 = ⟨ ψ 0 ∣ U † Π ^ s U ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ V s † Π ^ s U ∣ ψ 0 ⟩ 2 = ⟨ ψ 0 ∣ I s ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ I s 1/2 ∣ ψ 0 ⟩ 2 ≥ 1 + ε 1 − ε ≥ 1 − 2 ε . (185) Therefore, the approximate satisfaction of Eq. (177) (relative to the size of P s P_s P s
In the context of stabilizer codes, an important caveat to this theorem is that the recovery V s V_s V s incoherent Pauli decoherence is applied to a stabilizer code at a rate below its error threshold, the original encoded state can always be recovered with high fidelity using syndrome measurement followed by a Pauli unitary. However, when a coherent error is applied, it is possible in principle that a nontrivial coherent unitary rotation is induced on the code space, and error correction by Pauli operations alone is insufficient. In the toric code context, this is to say that incoherent errors performed below the threshold can always be corrected purely by pairing anyons; for coherent errors, it is possible in principle that the encoded information remains intact following syndrome measurement, but the original encoded state cannot be recovered simply by pairing anyons together. This situation is known to arise in the case of the coherent error U = ∏ ℓ e i θ X ℓ U=\prod_\ell e^{i\theta X_\ell} U = ∏ ℓ e i θ X ℓ θ > θ c \theta>\theta_c θ > θ c [65 ] . In the context of the present work, we have shown in Sec. VB that this situation does not arise for the TFTC due to stoquasticity.
The Worm Algorithm
In this Appendix, we discuss the worm algorithm (WA) [75 , 76 , 77 , 78 ] as an efficient Monte Carlo implementation of the optimal postmeasurement decoder in both of § 3 and § 3.2 . A useful ancillary result is that the worm algorithm serves as an efficient optimal decoder for the stabilizer toric code under independent bit-flip and phase-flip decoherence, as recently discussed in Ref. [79].
The Worm Algorithm and Z2 Lattice Gauge Theory
To start, we describe how the WA is used to efficiently simulate the low-temperature expansion of the classical Z 2 \mathbb{Z}_2 Z 2 3 d 3d 3 d Z 2 \mathbb{Z}_2 Z 2 m m m (52) as follows:
Z ~ λ ≡ ∑ { U } e − S ~ L G T [ U ] ∝ ∑ Closed Loops L e − 2 ∣ L ∣ / λ , (186) \widetilde{Z}_{\lambda} \equiv\sum_{\{U\}} e^{-\widetilde{S}_{\mathrm{LGT}}[U]} \propto\sum_{\text{Closed Loops }\mathcal{L}} e^{-2|\mathcal{L}|/\lambda}, \tag*{(186)} Z λ ≡ { U } ∑ e − S LGT [ U ] ∝ Closed Loops L ∑ e − 2∣ L ∣/ λ , (186) where L \mathcal{L} L Z 2 \mathbb{Z}_2 Z 2 3 d 3d 3 d L \mathcal{L} L S ~ L G T \widetilde{S}_{\mathrm{LGT}} S LGT Z h ≡ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ Z_h \equiv\langle\psi_0|e^{-\beta H_h}|\psi_0\rangle Z h ≡ ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ⟩ L \mathcal{L} L τ = ± β / 2 \tau= \pm\beta/2 τ = ± β /2
Given an initial loop configuration L \mathcal{L} L L ′ \mathcal{L}' L ′ [76 ] :
Pick a random site i ~ \tilde{i} i ~ h \mathrm{h} h t \mathrm{t} t i ~ \tilde{i} i ~
Allow the head variable to perform a random walk through the dual lattice by iterating the following steps: (a) Pick a random direction μ ∈ { ± x , ± y , ± τ } \mu\in\{\pm x, \pm y, \pm\tau\} μ ∈ { ± x , ± y , ± τ } μ \mu μ e μ e_\mu e μ h \mathrm{h} h h + e μ \mathrm{h} + e_\mu h + e μ e − 2 / λ e^{-2/\lambda} e − 2/ λ h \mathrm{h} h h + e μ \mathrm{h} + e_\mu h + e μ h ← h + e μ \mathrm{h} \leftarrow\mathrm{h} + e_\mu h ← h + e μ h \mathrm{h} h h + e μ \mathrm{h} + e_\mu h + e μ h ← h + e μ \mathrm{h} \leftarrow\mathrm{h} + e_\mu h ← h + e μ h = t \mathrm{h} = \mathrm{t} h = t
In this way, L ′ \mathcal{L}' L ′ L \mathcal{L} L 33
P [ L → L ′ ] P [ L ′ → L ] = e − 2 ∣ L ′ ∣ / λ e − 2 ∣ L ∣ / λ , (187) \frac{\mathbb{P}[\mathcal{L} \to\mathcal{L}']}{\mathbb{P}[\mathcal{L}' \to\mathcal{L}]} = \frac{e^{-2|\mathcal{L}'|/\lambda}}{e^{-2|\mathcal{L}|/\lambda}}, \tag*{(187)} P [ L ′ → L ] P [ L → L ′ ] = e − 2∣ L ∣/ λ e − 2∣ L ′ ∣/ λ , (187) and therefore the Markov process associated with this algorithm converges to the Gibbs distribution.
The WA is extraordinarily efficient within the topological phase λ < λ c \lambda< \lambda_c λ < λ c [76 , 77 ] . In the trivial phase λ > λ c \lambda> \lambda_c λ > λ c O ( L 3 ) O(L^3) O ( L 3 ) [78 ] , and the same result extends to the 3 d 3d 3 d
In § 3 C, we utilized the WA to numerically implement the optimal decoder for the TFTC under stabilizer measurements. We first use the WA to sample a loop configuration L \mathcal{L} L s ≡ { s p } s \equiv\{s_p\} s ≡ { s p } τ = 0 \tau= 0 τ = 0 conditional probability distribution by forbidding worms from passing through the τ = 0 \tau= 0 τ = 0 τ ≠ 0 \tau\ne0 τ = 0
In an actual decoding experiment, the observer does not have access to an initially sampled loop configuration giving rise to the syndrome s s s any initial loop configuration consistent with s s s
The Worm Algorithm for the Stabilizer Toric Code under Bit-Flip Decoherence
The WA can also be used as an efficient implementation of the optimal decoder for the stabilizer toric code under bit-flip decoherence [42 , 79 ] . Since this will serve as an intermediate step towards the optimal postmeasurement decoding of the TFTC under bit-flip decoherence, we start by explaining the WA decoder in the stabilizer case.
Let us first briefly review optimal decoding of i.i.d. bit-flip decoherence in the stabilizer toric code [7 ] . Starting from a code state ∣ ψ 0 ⟩ ∈ C 0 |\psi_0\rangle\in\mathcal{C}_0 ∣ ψ 0 ⟩ ∈ C 0 (66) , resulting in the state
ρ ≡ E ( ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ) = ∑ e p e X e ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X e . (188) \rho\equiv\mathcal{E}(|\psi_0\rangle\langle\psi_0|) = \sum_e p_e X_e |\psi_0\rangle\langle\psi_0| X_e. \tag*{(188)} ρ ≡ E ( ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ) = e ∑ p e X e ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X e . (188) We can visualize each bit-flip error e e e m m m ∂ e ≡ { ∂ e p } \partial e \equiv\{\partial e_p\} ∂ e ≡ { ∂ e p } B p B_p B p s ≡ { s p } s \equiv\{s_p\} s ≡ { s p }
ρ ( s ) ≡ Π ^ s ρ Π ^ s P s = 1 P s ∑ e : ∂ e = s p e X e ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X e , (189) \rho(s) \equiv\frac{\hat{\Pi}_s \rho\hat{\Pi}_s}{P_s} = \frac{1}{P_s}\sum_{e:\partial e=s}p_eX_e|\psi_0\rangle\langle\psi_0|X_e, \tag*{(189)} ρ ( s ) ≡ P s Π ^ s ρ Π ^ s = P s 1 e : ∂ e = s ∑ p e X e ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X e , (189) where P s P_s P s s s s
P s ≡ ∑ e : ∂ e = s p e = ( 1 − p ) N ∑ e : ∂ e = s ( p 1 − p ) ∣ e ∣ . (190) P_s \equiv\sum_{e:\partial e=s}p_e = (1-p)^N\sum_{e:\partial e=s}\left(\frac{p}{1-p}\right)^{|e|}. \tag*{(190)} P s ≡ e : ∂ e = s ∑ p e = ( 1 − p ) N e : ∂ e = s ∑ ( 1 − p p ) ∣ e ∣ . (190) The set of all errors e e e ∂ e = s \partial e=s ∂ e = s homology classes σ ∣ s \sigma|s σ ∣ s X s X_s X s e ∈ σ ∣ s e\in\sigma|s e ∈ σ ∣ s ∂ e = s \partial e=s ∂ e = s X e X s ∣ ψ 0 ⟩ = X σ ∣ ψ 0 ⟩ X_eX_s|\psi_0\rangle=X^\sigma|\psi_0\rangle X e X s ∣ ψ 0 ⟩ = X σ ∣ ψ 0 ⟩ X e X s X_eX_s X e X s X σ X^\sigma X σ A v A_v A v e e e e ′ e' e ′ contractible closed loops in the dual lattice. On the other hand, two non-homologous errors e , e ′ e,e' e , e ′
Since homologous errors act identically on code states ∣ ψ 0 ⟩ |\psi_0\rangle ∣ ψ 0 ⟩ ρ ( s ) \rho(s) ρ ( s )
ρ ( s ) = 1 P s X s ( ∑ σ ∣ s Q σ ( s ) X σ ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X σ ) X s . (191) \rho(s)=\frac{1}{P_s}X_s\left(\sum_{\sigma|s}Q_\sigma(s)X^\sigma|\psi_0\rangle\langle\psi_0|X^\sigma\right)X_s. \tag*{(191)} ρ ( s ) = P s 1 X s σ ∣ s ∑ Q σ ( s ) X σ ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ X σ X s . (191) where Q σ ( s ) Q_\sigma(s) Q σ ( s ) σ ∣ s \sigma|s σ ∣ s
Q σ ( s ) ≡ ∑ e ∈ σ ∣ s p e = ( 1 − p ) N ∑ e ∈ σ ∣ s ( p 1 − p ) ∣ e ∣ . (192) Q_\sigma(s)\equiv\sum_{e\in\sigma|s}p_e=(1-p)^N\sum_{e\in\sigma|s}\left(\frac{p}{1-p}\right)^{|e|}. \tag*{(192)} Q σ ( s ) ≡ e ∈ σ ∣ s ∑ p e = ( 1 − p ) N e ∈ σ ∣ s ∑ ( 1 − p p ) ∣ e ∣ . (192) Note that ∑ σ Q σ ( s ) = P s \sum_\sigma Q_\sigma(s)=P_s ∑ σ Q σ ( s ) = P s X s X_s X s a r g m a x σ Q σ ( s ) = ( 0 , 0 ) \mathop{\mathrm{argmax}}_\sigma Q_\sigma(s)=(0,0) argmax σ Q σ ( s ) = ( 0 , 0 ) X s X_s X s
F s = ⟨ ψ 0 ∣ X s ρ ( s ) X s ∣ ψ 0 ⟩ = Q ( 0 , 0 ) ( s ) P s = 1 P s max σ Q σ ( s ) . (193) \mathcal{F}_s=\langle\psi_0|X_s\rho(s)X_s|\psi_0\rangle=\frac{Q_{(0,0)}(s)}{P_s}=\frac{1}{P_s}\max_\sigma Q_\sigma(s). \tag*{(193)} F s = ⟨ ψ 0 ∣ X s ρ ( s ) X s ∣ ψ 0 ⟩ = P s Q ( 0 , 0 ) ( s ) = P s 1 σ max Q σ ( s ) . (193) Figure 12. Optimal decoding fidelity for the stabilizer toric code subjected to i.i.d. bit-flip decoherence, implemented via the WA. We obtain an excellent crossing at p = p c N i s h i ≈ 0.109 p = p_c^{\mathrm{Nishi}} \approx0.109 p = p c Nishi ≈ 0.109 ν N i s h i ≈ 1.534 \nu_{\mathrm{Nishi}} \approx1.534 ν Nishi ≈ 1.534 [101 , 102 ] .
Therefore, optimal decoding succeeds with high fidelity if, for the sampled syndrome s s s e e e s s s
In order to implement the optimal decoder, it is necessary to compute the four probabilities Q σ ( s ) Q_\sigma(s) Q σ ( s ) s s s [7 ] , which can be computed exactly using free fermion methods or approximately via tensor network contraction [141 ] . Here we will describe how the ratios Q σ ( s ) / P s Q_\sigma(s)/P_s Q σ ( s ) / P s s s s e e e s s s ∂ e = s \partial e = s ∂ e = s e ′ e' e ′ s s s e e e
Pick a random plaquette q q q q q q p p p p p p h h h t t t q q q
Allow the head variable to perform a random walk by iterating the following steps:
(a) Pick a random direction μ ∈ { ± x , ± y } \mu\in\{\pm x, \pm y\} μ ∈ { ± x , ± y }
(b) If the dual lattice link connecting h h h h + e μ h + e_\mu h + e μ p / ( 1 − p ) p/(1-p) p / ( 1 − p ) h ← h + e μ h \leftarrow h + e_\mu h ← h + e μ
(c) Otherwise, if the dual lattice link connecting h h h h + e μ h + e_\mu h + e μ h ← h + e μ h \leftarrow h + e_\mu h ← h + e μ
(d) If h = t h = t h = t
It is clear that this algorithm preserves the syndrome s s s e e e ∂ e = s \partial e = s ∂ e = s [ p / ( 1 − p ) ] ∣ e ∣ [p/(1-p)]^{|e|} [ p / ( 1 − p ) ] ∣ e ∣ Q σ ( s ) Q_\sigma(s) Q σ ( s )
Figure 12 depicts our numerical verification that the WA can be used as an efficient implementation of the optimal decoder for the stabilizer toric code under i.i.d. bit-flip decoherence. As is well-known [7 ] , the critical error threshold and its nearby critical phenomena are described by the 2d random-bond Ising model on its Nishimori line. We observe an excellent crossing at the latter model’s known critical point p c N i s h i p_c^{\mathrm{Nishi}} p c Nishi ν N i s h i ≈ 1.534 \nu_{\mathrm{Nishi}} \approx1.534 ν Nishi ≈ 1.534 [101 ] , we observe an excellent finite-size scaling collapse.
The Worm Algorithm for the Transverse-Field Toric Code under Bit-Flip Decoherence
Finally, we describe our algorithm for the optimal postmeasurement decoding of the TFTC under bit-flip decoherence. As described in § 3.2 C, our goal is effectively to sample any worldlines L \mathcal{L} L e e e
Z ~ λ , p ( s ) ≡ ∑ L , e ∣ s t ∣ L ∣ w ∣ e ∣ (194) \widetilde{Z}_{\lambda,p}(s) \equiv\sum_{\mathcal{L},e\mid s} t^{|\mathcal{L}|}w^{|e|} \tag*{(194)} Z λ , p ( s ) ≡ L , e ∣ s ∑ t ∣ L ∣ w ∣ e ∣ (194) where t ≡ e − 2 / λ t \equiv e^{-2/\lambda} t ≡ e − 2/ λ w ≡ p / ( 1 − p ) w \equiv p/(1-p) w ≡ p / ( 1 − p )
First, we update L → L ′ \mathcal{L} \to\mathcal{L}' L → L ′ s L s_{\mathcal{L}} s L L \mathcal{L} L s = s L ⊕ s e s=s_{\mathcal{L}}\oplus s_e s = s L ⊕ s e
Second, we update e → e ′ e \to e' e → e ′ s e s_e s e e e e s s s
The above two updates are not ergodic alone, since they do not mix between all the different pairs of syndromes ( s L , s e ) (s_{\mathcal{L}},s_e) ( s L , s e ) s s s e ℓ e_{\ell} e ℓ ℓ \ell ℓ ℓ \ell ℓ L ′ ′ \mathcal{L}'' L ′′ e ′ ′ e'' e ′′
min { 1 , t ∣ L ′ ′ ∣ w ∣ e ′ ′ ∣ t ∣ L ′ ∣ w ∣ e ′ ∣ } (195) \min\left\{1,\frac{t^{|\mathcal{L}''|}w^{|e''|}}{t^{|\mathcal{L}'|}w^{|e'|}}\right\} \tag*{(195)} min { 1 , t ∣ L ′ ∣ w ∣ e ′ ∣ t ∣ L ′′ ∣ w ∣ e ′′ ∣ } (195) which can be computed locally from the single bit-flip e ℓ e_{\ell} e ℓ ℓ \ell ℓ ℓ \ell ℓ
Additional Numerical Results
In this Appendix, we collect a number of additional numerical results.
Winding Number Statistics
In § 3 A, we asserted that non-contractible m m m ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ′ ⟩ \langle\psi_0|e^{-\beta H_h}|\psi_0'\rangle ⟨ ψ 0 ∣ e − β H h ∣ ψ 0 ′ ⟩ h < h c h<h_c h < h c
Z ~ λ W i n d i n g ≡ ∑ C l o s e d L o o p s L e − 2 ∣ L ∣ / λ (196) \widetilde{Z}_{\lambda}^{\mathrm{Winding}} \equiv\sum_{\mathrm{Closed\ Loops}\ \mathcal{L}} e^{-2|\mathcal{L}|/\lambda} \tag*{(196)} Z λ Winding ≡ Closed Loops L ∑ e − 2∣ L ∣/ λ (196) where each L \mathcal{L} L L × L × L L\times L\times L L × L × L L \mathcal{L} L Z ~ λ W i n d i n g \widetilde{Z}_{\lambda}^{\mathrm{Winding}} Z λ Winding β I s i n g = tanh − 1 ( e − 2 / λ ) \beta_{\mathrm{Ising}}=\tanh^{-1}(e^{-2/\lambda}) β Ising = tanh − 1 ( e − 2/ λ )
As usual, we study Z ~ λ W i n d i n g \widetilde{Z}_{\lambda}^{\mathrm{Winding}} Z λ Winding W μ W_{\mu} W μ L \mathcal{L} L μ ∈ { x , y , z } \mu\in\{x,y,z\} μ ∈ { x , y , z } Figure 13 (a) depicts the average of the three wrapping probabilities, 1 3 ∑ μ = x , y , z W μ \frac{1}{3}\sum_{\mu=x,y,z}W_{\mu} 3 1 ∑ μ = x , y , z W μ λ \lambda λ Figure 13 (b) depicts the same quantity as a function of L L L λ \lambda λ λ c \lambda_c λ c
Figure 13. Wrapping probabilities in the partition function Z ~ W i n d i n g \widetilde{Z}^{\mathrm{Winding}} Z Winding λ \lambda λ L L L λ < λ c \lambda< \lambda_c λ < λ c λ > λ c \lambda> \lambda_c λ > λ c 1 / 2 1/2 1/2 L L L O ( e − L / ξ ) O(e^{-L/\xi}) O ( e − L / ξ ) ξ \xi ξ λ → λ c \lambda\to\lambda_c λ → λ c ∣ φ h ⟩ |\varphi_h\rangle ∣ φ h ⟩ (21) ] are orthonormal up to exponentially small corrections for all h < h c h < h_c h < h c
As expected, the wrapping probability tends to zero exponentially quickly in the linear system size throughout the topological phase λ < λ c \lambda<\lambda_c λ < λ c 1 / 2 1/2 1/2 λ > λ c \lambda>\lambda_c λ > λ c m m m λ \lambda λ λ c ≈ 1.313 \lambda_c\approx1.313 λ c ≈ 1.313 ν ≈ 0.630 \nu\approx0.630 ν ≈ 0.630 [98 ] .
An identical result is expected to hold for the extremely anisotropic limit of this model, which implies that the ITE logical-Z Z Z ∣ φ h 0 ⟩ |\varphi_h^0\rangle ∣ φ h 0 ⟩ (22) ] are orthonormal up to exponentially small corrections for all h < h c h<h_c h < h c
Additional Numerical Plots for the Transverse-Field Toric Code under Bit-Flip Decoherence
Here we collect a few additional plots for the optimal postmeasurement decoding fidelity of the TFTC subjected to bit-flip decoherence.
Figure 14 depicts the average optimal postmeasurement decoding fidelity F ‾ I s o \overline{F}^{\mathrm{Iso}} F Iso λ = ( 0.98 λ c , 0.996 λ c , 1.0 λ c ) \lambda= (0.98\lambda_c, 0.996\lambda_c, 1.0\lambda_c) λ = ( 0.98 λ c , 0.996 λ c , 1.0 λ c ) λ \lambda λ (51) ], computed as described in § 3.2 C. In each case, we see a clear crossing with an excellent finite-size scaling collapse, indicating error thresholds at p 1 ≈ 0.102 p_1 \approx0.102 p 1 ≈ 0.102 p 2 ≈ 0.0975 p_2 \approx0.0975 p 2 ≈ 0.0975 p 3 ≈ 0.0931 p_3 \approx0.0931 p 3 ≈ 0.0931 ν N i s h i \nu_{\mathrm{Nishi}} ν Nishi p c o p t ( λ c ) p_c^{\mathrm{opt}}(\lambda_c) p c opt ( λ c ) p c o p t ( λ = 0 ) p_c^{\mathrm{opt}}(\lambda= 0) p c opt ( λ = 0 ) ν \nu ν ν \nu ν
Remarkably, Figure 14 (c) shows that our decoding protocol exhibits a nonzero error threshold even at λ = λ c \lambda= \lambda_c λ = λ c λ < λ c \lambda< \lambda_c λ < λ c
Figure 14. Average optimal postmeasurement decoding fidelity F ‾ I s o \overline{F}^{\mathrm{Iso}} F Iso p p p λ \lambda λ L L L ν N i s h i ≈ 1.534 \nu_{\mathrm{Nishi}} \approx1.534 ν Nishi ≈ 1.534 [101 ] , and the error thresholds p 1 ≈ 0.102 p_1 \approx0.102 p 1 ≈ 0.102 p 2 ≈ 0.0975 p_2 \approx0.0975 p 2 ≈ 0.0975 p 3 ≈ 0.0931 p_3 \approx0.0931 p 3 ≈ 0.0931
Figure 15 depicts the optimal postmeasurement decoding fidelity at several representative values of the bit-flip strength p p p λ \lambda λ p p p p = 0.03 p = 0.03 p = 0.03 Figure 15 (a), the decoding fidelity is essentially indistinguishable from the p = 0 p = 0 p = 0 § 3 C. In particular, there is a clear crossing at λ = λ c \lambda= \lambda_c λ = λ c ν ≈ 0.630 \nu\approx0.630 ν ≈ 0.630 [98 ] .
Figure 15. Average optimal postmeasurement decoding fidelity F ‾ I s o \overline{F}^{\mathrm{Iso}} F Iso λ \lambda λ p p p L L L p = 0.03 p = 0.03 p = 0.03 F ‾ I s o \overline{F}^{\mathrm{Iso}} F Iso p = 0 p = 0 p = 0 Figure 7 . Inset: finite-size scaling collapse using the 3d Ising exponent ν ≈ 0.630 \nu\approx0.630 ν ≈ 0.630 [98 ] . (b) At p = 0.06 p = 0.06 p = 0.06 λ c \lambda_c λ c p = 0.1 p = 0.1 p = 0.1 Figure 10 ], the decoding transition arises at some λ < λ c \lambda< \lambda_c λ < λ c
At intermediate values of p p p p = 0.06 p = 0.06 p = 0.06 Figure 15 (b), we see that there is a strong finite-size drift in the location of the crossing, and in particular the crossings between adjacent system sizes arise at λ > λ c \lambda> \lambda_c λ > λ c λ c \lambda_c λ c λ = λ c \lambda= \lambda_c λ = λ c
Finally, Figure 15 (c) depicts the optimal postmeasurement decoding fidelity at a value of p p p p c o p t ( λ c ) = p 3 ≈ 0.0931 p_c^{\mathrm{opt}}(\lambda_c) = p_3 \approx0.0931 p c opt ( λ c ) = p 3 ≈ 0.0931 p c N i s h ≈ 0.109 p_c^{\mathrm{Nish}} \approx0.109 p c Nish ≈ 0.109 λ \lambda λ λ c \lambda_c λ c λ \lambda λ
Minimal-Weight Perfect Matching
Minimal-weight perfect matching (MWPM) is a widely utilized and highly efficient suboptimal decoding algorithm [7 , 35 , 142 , 143 ] . In the stabilizer toric code subjected to bit-flip decoherence, it exhibits near-optimal performance [35 ] . In this Appendix, we will investigate MWPM as a suboptimal postmeasurement decoder for the TFTC, both without and with bit-flip decoherence.
As in the optimal decoder, we first sample a bulk worldline configuration L \mathcal{L} L e \mathbf{e} e p p p s L s_{\mathcal{L}} s L s e s_e s e s = s L ⊕ s e s = s_{\mathcal{L}} \oplus s_e s = s L ⊕ s e [143 ] to construct a MWPM recovery for the joint syndrome s s s ( L , e ) (\mathcal{L},e) ( L , e ) § 3.2 B), and is unsuccessful otherwise. The MWPM decoding fidelity F ‾ M W P M \overline{F}^{\mathrm{MWPM}} F MWPM
Figure 16 (a) depicts the MWPM decoding fidelity F ‾ M W P M \overline{F}^{\mathrm{MWPM}} F MWPM p p p λ = 0.998 λ c \lambda= 0.998\lambda_c λ = 0.998 λ c (51) ], just barely within the topological phase. We observe a clear crossing at the threshold p 4 ≡ p c M W P M ( 0.998 λ c ) ≈ 0.0904 p_4 \equiv p_c^{\mathrm{MWPM}}(0.998\lambda_c) \approx0.0904 p 4 ≡ p c MWPM ( 0.998 λ c ) ≈ 0.0904 ν M W P M ≈ 1.46 \nu_{\mathrm{MWPM}} \approx1.46 ν MWPM ≈ 1.46 λ = 0 \lambda= 0 λ = 0 [35 ] . The threshold p 4 p_4 p 4 p c M W P M ( λ = 0 ) ≈ 0.103 p_c^{\mathrm{MWPM}}(\lambda= 0) \approx0.103 p c MWPM ( λ = 0 ) ≈ 0.103 [35 ] , a noticeable but insubstantial reduction.
Figure 16. MWPM decoding fidelity F ‾ M W P M \overline{F}^{\mathrm{MWPM}} F MWPM λ \lambda λ p p p L L L λ = 0.998 λ c \lambda= 0.998\lambda_c λ = 0.998 λ c p 4 ≈ 0.0904 p_4 \approx0.0904 p 4 ≈ 0.0904 ν = ν M W P M = 1.46 \nu= \nu_{\mathrm{MWPM}} = 1.46 ν = ν MWPM = 1.46 [35 ] . (b) At the bulk critical point λ = λ c \lambda= \lambda_c λ = λ c p 5 ≈ 0.0874 p_5 \approx0.0874 p 5 ≈ 0.0874 λ > λ c \lambda> \lambda_c λ > λ c
Figure 16 (b) depicts the MWPM decoding fidelity directly at the critical point λ = λ c \lambda= \lambda_c λ = λ c p 5 ≡ p c M W P M ( λ c ) ≈ 0.0874 p_5 \equiv p_c^{\mathrm{MWPM}}(\lambda_c) \approx0.0874 p 5 ≡ p c MWPM ( λ c ) ≈ 0.0874 ν M W P M \nu_{\mathrm{MWPM}} ν MWPM ν \nu ν
Figure 16 (c) depicts the MWPM decoding fidelity at λ = 1.002 λ c \lambda= 1.002\lambda_c λ = 1.002 λ c L L L F ‾ M W P M \overline{F}^{\mathrm{MWPM}} F MWPM L L L
Figure 17 (a) depicts the MWPM decoding fidelity in the absence of bit-flip decoherence, p = 0 p = 0 p = 0 λ \lambda λ § 3 , depicted in Figure 7 . In particular, MWPM appears to yield a decoding fidelity exponentially close to unity throughout the topological phase λ < λ c \lambda< \lambda_c λ < λ c p = 0 p = 0 p = 0
Figure 17 (b) depicts the MWPM decoding fidelity at the fixed decoherence strength p = 0.08 p = 0.08 p = 0.08 λ \lambda λ p c M W P M ( λ ) p_c^{\mathrm{MWPM}}(\lambda) p c MWPM ( λ ) Figure 10 (c). We see that F ‾ M W P M \overline{F}^{\mathrm{MWPM}} F MWPM λ < λ c \lambda< \lambda_c λ < λ c p = 0 p = 0 p = 0 λ c \lambda_c λ c λ = λ c \lambda= \lambda_c λ = λ c λ c \lambda_c λ c p p p
Finally, Figure 17 (c) depicts the MWPM decoding fidelity at decoherence strength p = 0.1 p = 0.1 p = 0.1 F ‾ M W P M \overline{F}^{\mathrm{MWPM}} F MWPM L L L λ \lambda λ λ \lambda λ p = 0.1 p = 0.1 p = 0.1 p c M W P M ( λ = 0 ) ≈ 0.103 p_c^{\mathrm{MWPM}}(\lambda= 0) \approx0.103 p c MWPM ( λ = 0 ) ≈ 0.103 [35 ] . We have checked that our numerical method reproduces this value in the λ = 0 \lambda= 0 λ = 0 λ = 0.98 λ c \lambda= 0.98\lambda_c λ = 0.98 λ c p c M W P M ( λ = 0.98 λ c ) ≈ 0.0968 p_c^{\mathrm{MWPM}}(\lambda= 0.98\lambda_c) \approx0.0968 p c MWPM ( λ = 0.98 λ c ) ≈ 0.0968 p = 0.1 p = 0.1 p = 0.1 Figure 17 (c).
Free-Fermion Numerics for Decoding the Transverse-Field Ising Model
In this Appendix, we provide details for our numerical investigation of the 1d TFIM under bit-flip decoherence.
Setup
We consider the 1d TFIM, with open boundary conditions for numerical convenience. The Hamiltonian is
H = − g ∑ j = 1 L X j − ∑ j = 1 L − 1 Z j Z j + 1 = − i g ∑ j = 1 L γ 2 j − 1 γ 2 j − i ∑ j = 1 L − 1 γ 2 j γ 2 j + 1 , (197) H = -g \sum_{j=1}^{L} X_j - \sum_{j=1}^{L-1} Z_j Z_{j+1}
= -\mathrm{i}g \sum_{j=1}^{L} \gamma_{2j-1}\gamma_{2j} - \mathrm{i} \sum_{j=1}^{L-1} \gamma_{2j}\gamma_{2j+1}, \tag*{(197)} H = − g j = 1 ∑ L X j − j = 1 ∑ L − 1 Z j Z j + 1 = − i g j = 1 ∑ L γ 2 j − 1 γ 2 j − i j = 1 ∑ L − 1 γ 2 j γ 2 j + 1 , (197) where in the second line, we have performed a Jordan-Wigner transformation using the Majorana operators
γ 2 j − 1 ≡ [ ∏ i = 1 j − 1 X i ] Z j , γ 2 j ≡ [ ∏ i = 1 j − 1 X i ] Y j , (198) \gamma_{2j-1} \equiv\left[\prod_{i=1}^{j-1} X_i\right] Z_j,\qquad\gamma_{2j} \equiv\left[\prod_{i=1}^{j-1} X_i\right] Y_j, \tag*{(198)} γ 2 j − 1 ≡ [ i = 1 ∏ j − 1 X i ] Z j , γ 2 j ≡ [ i = 1 ∏ j − 1 X i ] Y j , (198) which satisfy the anticommutation relations { γ j , γ j ′ } = 2 δ j j ′ \{\gamma_j,\gamma_{j'}\}=2\delta_{jj'} { γ j , γ j ′ } = 2 δ j j ′ i γ 2 j − 1 γ 2 j = X j i\gamma_{2j-1}\gamma_{2j}=X_j i γ 2 j − 1 γ 2 j = X j i γ 2 j γ 2 j + 1 = Z j Z j + 1 i\gamma_{2j}\gamma_{2j+1}=Z_jZ_{j+1} i γ 2 j γ 2 j + 1 = Z j Z j + 1 H H H
H = i 4 ∑ i , j = 1 2 L γ i A i j γ j (199) H=\frac{i}{4}\sum_{i,j=1}^{2L}\gamma_i A_{ij}\gamma_j \tag*{(199)} H = 4 i i , j = 1 ∑ 2 L γ i A ij γ j (199) where A A A G i j ≡ i ⟨ γ i γ j ⟩ − i δ i j G_{ij}\equiv i\langle\gamma_i\gamma_j\rangle-i\delta_{ij} G ij ≡ i ⟨ γ i γ j ⟩ − i δ ij A A A [144 ] ; see Appendix A of Ref. [115] for a review. After an orthogonal transformation, H H H
H = − i 2 ∑ α = 1 L ϵ α η 2 α − 1 η 2 α (200) H=-\frac{i}{2}\sum_{\alpha=1}^{L}\epsilon_\alpha\eta_{2\alpha-1}\eta_{2\alpha} \tag*{(200)} H = − 2 i α = 1 ∑ L ϵ α η 2 α − 1 η 2 α (200) where ϵ α > 0 \epsilon_\alpha>0 ϵ α > 0 ϵ 1 ≤ ϵ 2 ≤ ⋯ \epsilon_1\le\epsilon_2\le\cdots ϵ 1 ≤ ϵ 2 ≤ ⋯ η α = ∑ j R α j γ j \eta_\alpha=\sum_jR_{\alpha j}\gamma_j η α = ∑ j R α j γ j γ j \gamma_j γ j R R R g < 1 g<1 g < 1 ϵ 1 \epsilon_1 ϵ 1 L L L ϵ α \epsilon_\alpha ϵ α η ℓ ≡ η 1 \eta_\ell\equiv\eta_1 η ℓ ≡ η 1 η r ≡ η 2 \eta_r\equiv\eta_2 η r ≡ η 2 i η ℓ η r = + 1 i\eta_\ell\eta_r=+1 i η ℓ η r = + 1 ∣ + g ⟩ ≡ 1 2 ( ∣ 0 g ⟩ + ∣ 1 g ⟩ ) |+_g\rangle\equiv\frac{1}{\sqrt{2}}(|0_g\rangle+|1_g\rangle) ∣ + g ⟩ ≡ 2 1 ( ∣ 0 g ⟩ + ∣ 1 g ⟩) i η ℓ η r = − 1 i\eta_\ell\eta_r=-1 i η ℓ η r = − 1 ∣ − g ⟩ ≡ 1 2 ( ∣ 0 g ⟩ − ∣ 1 g ⟩ ) |-_g\rangle\equiv\frac{1}{\sqrt{2}}(|0_g\rangle-|1_g\rangle) ∣ − g ⟩ ≡ 2 1 ( ∣ 0 g ⟩ − ∣ 1 g ⟩) i η ℓ η r = ± 1 i\eta_\ell\eta_r=\pm1 i η ℓ η r = ± 1 34 i η 1 η 2 = ± 1 i\eta_1\eta_2=\pm1 i η 1 η 2 = ± 1 g ≥ g c g\ge g_c g ≥ g c g g g g c g_c g c
Numerical Protocol
Our goal is to compute the average decoding fidelity F ‾ ( ∣ 0 g ⟩ ) \overline{F}(|0_g\rangle) F ( ∣ 0 g ⟩)
F ‾ ( ∣ 0 g ⟩ ) ≡ ⟨ 0 g ∣ M ∘ E ( ∣ 0 g ⟩ ⟨ 0 g ∣ ) ∣ 0 g ⟩ (201) \overline{F}(|0_g\rangle)\equiv\langle0_g|\mathcal{M}\circ\mathcal{E}(|0_g\rangle\langle0_g|)|0_g\rangle \tag*{(201)} F ( ∣ 0 g ⟩) ≡ ⟨ 0 g ∣ M ∘ E ( ∣ 0 g ⟩ ⟨ 0 g ∣ ) ∣ 0 g ⟩ (201) where E ( ⋅ ) ≡ ∑ e p e X e ( ⋅ ) X e \mathcal{E}(\cdot)\equiv\sum_e p_eX_e(\cdot)X_e E ( ⋅ ) ≡ ∑ e p e X e ( ⋅ ) X e X e ≡ ∏ j X j e j X_e\equiv\prod_jX_j^{e_j} X e ≡ ∏ j X j e j p e ≡ ( 1 − p ) L − ∣ e ∣ p ∣ e ∣ p_e\equiv(1-p)^{L-|e|}p^{|e|} p e ≡ ( 1 − p ) L − ∣ e ∣ p ∣ e ∣ M ( ⋅ ) ≡ ∑ s V s Π ^ s ( ⋅ ) Π ^ s V s † \mathcal{M}(\cdot)\equiv\sum_sV_s\hat{\Pi}_s(\cdot)\hat{\Pi}_sV_s^\dagger M ( ⋅ ) ≡ ∑ s V s Π ^ s ( ⋅ ) Π ^ s V s † Π ^ s ∝ ∏ j [ 1 + ( − 1 ) s j Z j Z j + 1 ] \hat{\Pi}_s\propto\prod_j[1+(-1)^{s_j}Z_jZ_{j+1}] Π ^ s ∝ ∏ j [ 1 + ( − 1 ) s j Z j Z j + 1 ] V s = U g X s V_s=U_gX_s V s = U g X s X s X_s X s U g U_g U g § 3.11 , the decoding fidelity for the symmetry-broken state ∣ 0 g ⟩ |0_g\rangle ∣ 0 g ⟩ ∣ ψ g ⟩ = ∑ σ b σ ∣ σ g ⟩ |\psi_g\rangle=\sum_\sigma b_\sigma|\sigma_g\rangle ∣ ψ g ⟩ = ∑ σ b σ ∣ σ g ⟩
To compute this quantity, we will randomly sample a bit-flip error e e e p e p_e p e s s s P s ∣ e = ⟨ 0 g ∣ X e Π ^ s X e ∣ 0 g ⟩ P_{s|e}=\langle0_g|X_e\hat{\Pi}_sX_e|0_g\rangle P s ∣ e = ⟨ 0 g ∣ X e Π ^ s X e ∣ 0 g ⟩ e e e s s s
F s ∣ e ( ∣ 0 g ⟩ ) = ⟨ 0 0 ∣ X s [ 1 P s ∣ e Π ^ s ( X e ∣ 0 g ⟩ ⟨ 0 g ∣ X e ) Π ^ s ] X s ∣ 0 0 ⟩ (202) \mathcal{F}_{s|e}(|0_g\rangle)=\langle0_0|X_s\left[\frac{1}{P_{s|e}}\hat{\Pi}_s(X_e|0_g\rangle\langle0_g|X_e)\hat{\Pi}_s\right]X_s|0_0\rangle \tag*{(202)} F s ∣ e ( ∣ 0 g ⟩) = ⟨ 0 0 ∣ X s [ P s ∣ e 1 Π ^ s ( X e ∣ 0 g ⟩ ⟨ 0 g ∣ X e ) Π ^ s ] X s ∣ 0 0 ⟩ (202) where the quantity in square brackets denotes the postmeasurement state after measuring the stabilizers Z j Z j + 1 Z_jZ_{j+1} Z j Z j + 1 e e e s s s F ‾ \overline{F} F F s ∣ e ( ∣ 0 g ⟩ ) \mathcal{F}_{s|e}(|0_g\rangle) F s ∣ e ( ∣ 0 g ⟩)
First, we require a representation of the state ∣ 0 g ⟩ |0_g\rangle ∣ 0 g ⟩ ∣ 0 g ⟩ = 1 2 ( ∣ + g ⟩ + ∣ − g ⟩ ) |0_g\rangle=\frac{1}{\sqrt{2}}(|+_g\rangle+|-_g\rangle) ∣ 0 g ⟩ = 2 1 ( ∣ + g ⟩ + ∣ − g ⟩) χ ℓ , χ r \chi_\ell,\chi_r χ ℓ , χ r X 0 = i χ r χ ℓ X_0=i\chi_r\chi_\ell X 0 = i χ r χ ℓ 35
∣ 0 ~ g ⟩ = 1 2 ( ∣ + 0 ⟩ ∣ + g ⟩ + ∣ − 0 ⟩ ∣ − g ⟩ ) = exp { − π 4 i χ r η ℓ } ∣ + 0 ⟩ ∣ + g ⟩ (203) |\widetilde{0}_g\rangle=\frac{1}{\sqrt{2}}(|+_0\rangle|+_g\rangle+|-_0\rangle|-_g\rangle)=\exp\left\{-\frac{\pi}{4}i\chi_r\eta_\ell\right\}|+_0\rangle|+_g\rangle \tag*{(203)} ∣ 0 g ⟩ = 2 1 ( ∣ + 0 ⟩ ∣ + g ⟩ + ∣ − 0 ⟩ ∣ − g ⟩) = exp { − 4 π i χ r η ℓ } ∣ + 0 ⟩ ∣ + g ⟩ (203) by imposing the Majorana pairing i χ ℓ η ℓ = i χ r η r = + 1 i\chi_{\ell}\eta_{\ell}=i\chi_{r}\eta_{r}=+1 i χ ℓ η ℓ = i χ r η r = + 1 ∣ 0 ~ g ⟩ \lvert\widetilde{0}_g\rangle ∣ 0 g ⟩ ∣ + ⟩ 0 ∣ + ⟩ g \lvert+\rangle_0\lvert+\rangle_g ∣ + ⟩ 0 ∣ + ⟩ g exp { π 4 χ r η ℓ } \exp\{\frac{\pi}{4}\chi_r\eta_{\ell}\} exp { 4 π χ r η ℓ } ∣ 0 ~ g ⟩ \lvert\widetilde{0}_g\rangle ∣ 0 g ⟩ ⟨ 0 ~ g ∣ O ∣ 0 ~ g ⟩ = ⟨ 0 g ∣ O ∣ 0 g ⟩ \langle\widetilde{0}_g\lvert O\lvert\widetilde{0}_g\rangle=\langle0_g\lvert O\lvert0_g\rangle ⟨ 0 g ∣ O ∣ 0 g ⟩ = ⟨ 0 g ∣ O ∣ 0 g ⟩ O O O ⟨ 0 ~ g ∣ Z 0 O ∣ 0 ~ g ⟩ = ⟨ 0 g ∣ O ∣ 0 g ⟩ \langle\widetilde{0}_g\lvert Z_0O\lvert\widetilde{0}_g\rangle=\langle0_g\lvert O\lvert0_g\rangle ⟨ 0 g ∣ Z 0 O ∣ 0 g ⟩ = ⟨ 0 g ∣ O ∣ 0 g ⟩ O O O 36 ∣ 0 g ⟩ \lvert0_g\rangle ∣ 0 g ⟩ ∣ 0 ~ g ⟩ \lvert\widetilde{0}_g\rangle ∣ 0 g ⟩ 2 ( L + 1 ) 2(L+1) 2 ( L + 1 ) 37
⟨ 0 g ∣ Z 1 ∣ 0 g ⟩ = ⟨ 0 ~ g ∣ Z 0 Z 1 ∣ 0 ~ g ⟩ = ⟨ 0 ~ g ∣ i χ ℓ γ 1 ∣ 0 ~ g ⟩ = 1 2 ∑ α { η α , γ 1 } ⟨ 0 ~ g ∣ i χ ℓ η α ∣ 0 ~ g ⟩ = 1 2 { η ℓ , γ 1 } , (204) \langle0_g\lvert Z_1\lvert0_g\rangle=\langle\widetilde{0}_g\lvert Z_0Z_1\lvert\widetilde{0}_g\rangle=\langle\widetilde{0}_g\lvert i\chi_{\ell}\gamma_1\lvert\widetilde{0}_g\rangle=\frac{1}{2}\sum_{\alpha}\{\eta_{\alpha},\gamma_1\}\langle\widetilde{0}_g\lvert i\chi_{\ell}\eta_{\alpha}\lvert\widetilde{0}_g\rangle=\frac{1}{2}\{\eta_{\ell},\gamma_1\}, \tag*{(204)} ⟨ 0 g ∣ Z 1 ∣ 0 g ⟩ = ⟨ 0 g ∣ Z 0 Z 1 ∣ 0 g ⟩ = ⟨ 0 g ∣ i χ ℓ γ 1 ∣ 0 g ⟩ = 2 1 α ∑ { η α , γ 1 } ⟨ 0 g ∣ i χ ℓ η α ∣ 0 g ⟩ = 2 1 { η ℓ , γ 1 } , (204) where we’ve noted that γ j = ∑ α R α j η α \gamma_j=\sum_{\alpha}R_{\alpha j}\eta_{\alpha} γ j = ∑ α R α j η α R α j = 1 2 { η α , γ j } R_{\alpha j}=\frac{1}{2}\{\eta_{\alpha},\gamma_j\} R α j = 2 1 { η α , γ j }
Next, we sample a bit-flip error X e X_e X e e j = 0 , 1 e_j=0,1 e j = 0 , 1 X k X_k X k k k k
G i j ↦ ( − 1 ) δ i , 2 k − 1 + δ i , 2 k + δ j , 2 k − 1 + δ j , 2 k G i j . (205) G_{ij}\mapsto(-1)^{\delta_{i,2k-1}+\delta_{i,2k}+\delta_{j,2k-1}+\delta_{j,2k}}G_{ij}. \tag*{(205)} G ij ↦ ( − 1 ) δ i , 2 k − 1 + δ i , 2 k + δ j , 2 k − 1 + δ j , 2 k G ij . (205) Subsequently, we measure each stabilizer Z k Z k + 1 = i γ 2 k γ 2 k + 1 Z_kZ_{k+1}=i\gamma_{2k}\gamma_{2k+1} Z k Z k + 1 = i γ 2 k γ 2 k + 1 s k = 0 , 1 s_k=0,1 s k = 0 , 1 1 2 [ 1 + ( − 1 ) s k G 2 k , 2 k + 1 ] \frac{1}{2}[1+(-1)^{s_k}G_{2k,2k+1}] 2 1 [ 1 + ( − 1 ) s k G 2 k , 2 k + 1 ]
G i j ↦ G i j + G i , 2 k + 1 G j , 2 k − G i , 2 k G j , 2 k + 1 ( − 1 ) s k + G 2 k , 2 k + 1 , (206) G_{ij}\mapsto G_{ij}+\frac{G_{i,2k+1}G_{j,2k}-G_{i,2k}G_{j,2k+1}}{(-1)^{s_k}+G_{2k,2k+1}}, \tag*{(206)} G ij ↦ G ij + ( − 1 ) s k + G 2 k , 2 k + 1 G i , 2 k + 1 G j , 2 k − G i , 2 k G j , 2 k + 1 , (206) where we have used Wick’s theorem [144 ] to obtain the latter expression. We measure each stabilizer subsequently from the left end of the chain to the right end, updating the covariance matrix at each step.
Finally, we implement the majority vote X s X_s X s F s ∣ e \mathcal{F}_{s|e} F s ∣ e ∣ ψ g ( s ∣ e ) ⟩ ≡ P s ∣ e − 1 / 2 X s Π ^ s X e ∣ 0 g ⟩ \lvert\psi_g(s|e)\rangle\equiv P_{s|e}^{-1/2}X_s\widehat{\Pi}_sX_e\lvert0_g\rangle ∣ ψ g ( s ∣ e )⟩ ≡ P s ∣ e − 1/2 X s Π s X e ∣ 0 g ⟩ b 0 ∣ 0 0 ⟩ + b 1 ∣ 1 0 ⟩ b_0\lvert0_0\rangle+b_1\lvert1_0\rangle b 0 ∣ 0 0 ⟩ + b 1 ∣ 1 0 ⟩
F s ∣ e ( ∣ 0 g ⟩ ) = ∣ b 0 ∣ 2 = 1 2 [ 1 + ⟨ ψ g ( s ∣ e ) ∣ Z 1 ∣ ψ g ( s ∣ e ) ⟩ ] . (207) \mathcal{F}_{s|e}(\lvert0_g\rangle)=\lvert b_0\rvert^2=\frac{1}{2}[1+\langle\psi_g(s|e)\lvert Z_1\lvert\psi_g(s|e)\rangle]. \tag*{(207)} F s ∣ e (∣ 0 g ⟩) = ∣ b 0 ∣ 2 = 2 1 [ 1 + ⟨ ψ g ( s ∣ e ) ∣ Z 1 ∣ ψ g ( s ∣ e )⟩] . (207) In other words, F s ∣ e \mathcal{F}_{s|e} F s ∣ e Z 1 = i χ ℓ γ 1 Z_1=i\chi_{\ell}\gamma_1 Z 1 = i χ ℓ γ 1 Z 1 Z_1 Z 1 X s X_s X s Z 1 Z_1 Z 1 s j s_j s j
m 1 = ∑ j = 2 L Z 1 Z j = ∑ j = 1 L − 1 ( − 1 ) ∑ i ≤ j s i . (208) m_1=\sum_{j=2}^{L}Z_1Z_j=\sum_{j=1}^{L-1}(-1)^{\sum_{i\leq j}s_i}. \tag*{(208)} m 1 = j = 2 ∑ L Z 1 Z j = j = 1 ∑ L − 1 ( − 1 ) ∑ i ≤ j s i . (208) Qubit 1 belongs to the majority (minority) domain if m 1 > 0 m_1>0 m 1 > 0 m 1 < 0 m_1<0 m 1 < 0 Z 1 Z_1 Z 1 m 1 < 0 m_1<0 m 1 < 0 L L L m 1 m_1 m 1
Additional Numerical Plots
For completeness, here we provide a few supplementary numerical plots for the decoding protocol described above and in Sec. VID.
Figure 18 depicts the decoding fidelity as a function of the bit-flip probability p p p g g g g c = 1 g_c=1 g c = 1 L L L g < 1 g<1 g < 1 Figure 11 (a); in particular, for all p < 1 2 p<\frac{1}{2} p < 2 1 F ‾ \overline{\mathcal{F}} F g > 1 g>1 g > 1 p p p F ‾ \overline{\mathcal{F}} F L L L F ‾ \overline{\mathcal{F}} F p = 1 2 p=\frac{1}{2} p = 2 1 g > 1 g > 1 g > 1 F ‾ ( 1 − p ) = 1 − F ‾ ( p ) \overline{F}(1-p)=1-\overline{F}(p) F ( 1 − p ) = 1 − F ( p ) g g g F ‾ \overline{F} F L L L
Figure 19 depicts the decoding fidelity as a function of the transverse field g g g p p p p = 0 p=0 p = 0 [72 ] ; in particular, an excellent finite-size scaling collapse is achieved using the correlation length exponent ν = 1 \nu=1 ν = 1 p p p p = 0 p=0 p = 0 F ‾ \overline{F} F p p p p p p L L L g > 1 g>1 g > 1 g > 1 g>1 g > 1
Finally, Figure 20 depicts the decoding fidelity as a function of system size; Figure 20 (a) depicts several values of p p p g = 0.99 g=0.99 g = 0.99 Figure 20 (b) depicts several values of g g g p = 0.4 p=0.4 p = 0.4 g < 1 g<1 g < 1 p < 1 2 p<\frac{1}{2} p < 2 1
Figure 20. Deviation of the average decoding fidelity F ˉ \bar{F} F ˉ ( 1 + 1 ) d (1+1)d ( 1 + 1 ) d L L L g g g p p p 1 − F ˉ 1-\bar{F} 1 − F ˉ L L L
References 
