Introduction
Quantum error correction is essential for protecting quantum information against noise and enabling reliable and scalable quantum computation. Of particular importance are transversal gates, which implement logical operations through physical gates that each act on at most one qubit per code block, preventing a single gate fault from spreading within a block [1 ] . However, the Eastin-Knill theorem rules out a universal transversal gate set on a fixed code that detects arbitrary single-qubit errors [2 ] . Nevertheless, transversal non-Clifford gates remain valuable for universal fault-tolerant computation through magic-state distillation [3 , 4 ] , code switching, and gauge fixing [5 , 6 ] . A further goal is to apply these logical gates selectively to individual logical qubits [7 ] . This property, known as addressability, allows a prescribed gate to act on any chosen logical qubit while leaving the others unchanged [8 , 9 ] . In addition, asymptotically good code families combine bounded data-qubit overhead per logical qubit with a minimum distance that grows linearly with the blocklength [10 , 11 , 12 ] . This motivates studying codes that support addressable transversal non-Clifford gates while maintaining a nonvanishing encoding rate and relative distance.
Algebraic-geometric (AG) codes already support combining positive asymptotic rate and relative distance with non-Clifford transversal operations. He et al. constructed asymptotically good codes with addressable transversal CCZ gates [13 ] . For single-qubit phase gates Z ( h ) = diag ( 1 , e 2 π i / 2 h ) Z(h)=\operatorname{diag}(1,e^{2\pi i/2^{h}}) Z ( h ) = diag ( 1 , e 2 π i / 2 h )
Table 1. Common asymptotic lower bounds on encoding rate and relative distance for codes supporting fully addressable transversal T T T T T T
José obtained asymptotically good CSS codes with addressable level-h h h h ≥ 3 h \ge3 h ≥ 3 ( h + 1 ) (h+1) ( h + 1 ) [14 ] , whereas Lin, Wu, and Cheng achieved fully addressable transversal T T T [9 ] . Meiburg gave both CSS constructions with fully addressable transversal T T T [15 ] . However, as shown in Table I, although these constructions give asymptotically good codes, the simultaneous rate and relative-distance guarantees remain unrealistically small, limiting their practical applicability. This motivates a natural question: Can these parameters be improved to practically meaningful values, such as a few percent, while supporting fully addressable transversal phase gates?
In this work, we construct quantum codes with fully addressable transversal phase gates and substantially improved rate and relative distance. Our approach uses an encoding over the cyclic ring R h = Z / 2 h Z \mathcal{R}_{h}=\mathbb{Z}/2^{h}\mathbb{Z} R h = Z / 2 h Z h h h
For the polynomial-time construction, we start with nested AG codes over F 256 \mathbb{F}_{256} F 256 [16 , 17 , 18 ] . A binary basis expansion gives blocks of eight bits. Completing a basis adapted to the nested codes and lifting it to R h \mathcal{R}_h R h
For the T T T h ≥ 3 h \ge3 h ≥ 3 1 / h 1/h 1/ h h h h [14 , 15 ] . We also derive a universal upper bound on the common asymptotic rate–distance coefficient for codes supporting fully addressable transversal Z ( h ) Z(h) Z ( h ) 1 / h 1/h 1/ h T T T
The remainder of the paper is organized as follows. Section II introduces the encoding and its classical ingredients. Section III proves the polynomial-time construction and the existence result with exact exponential-time generation. Section IV derives upper bounds on the simultaneously achievable rate and relative distance, and Sec. V illustrates the framework with two finite codes. Technical lemmas, construction algorithms, counting details, and exact certification criteria appear in SM.
Preliminaries and cyclic encoding
Here we introduce the code parameters and addressability condition, review the required properties of AG codes, and present the cyclic encoding together with its transversal logical phase action.
Code parameters and addressability
A binary quantum code ( ( N , 2 K , D ) ) 2 ((N,2^{K},D))_{2} (( N , 2 K , D ) ) 2 K K K N N N D D D E E E O O O E † O E = λ O I 2 K E^{\dagger}OE=\lambda_{O}I_{2^{K}} E † O E = λ O I 2 K λ O ∈ C \lambda_{O}\in\mathbb{C} λ O ∈ C [19 ] . The code’s rate and relative distance are R = K / N R=K/N R = K / N d = D / N d=D/N d = D / N N i → ∞ N_i\to\infty N i → ∞
R ∞ : = lim inf i K i / N i , d ∞ : = lim inf i D i / N i , R_{\infty}:=\liminf_{i} K_i/N_i,\qquad d_{\infty}:=\liminf_{i} D_i/N_i, R ∞ := i lim inf K i / N i , d ∞ := i lim inf D i / N i , and define η : = min { R ∞ , d ∞ } \eta:=\min\{R_{\infty},d_{\infty}\} η := min { R ∞ , d ∞ } h h h
For an integer h ≥ 1 h\geq1 h ≥ 1
ν : = 2 h , R h : = Z / ν Z , Z ( h ) : = diag ( 1 , e 2 π i / ν ) . \nu:=2^{h},\qquad\mathcal{R}_{h}:=\mathbb{Z}/\nu\mathbb{Z},\qquad Z(h):=\operatorname{diag}(1,e^{2\pi i/\nu}). ν := 2 h , R h := Z / ν Z , Z ( h ) := diag ( 1 , e 2 π i / ν ) . The levels h = 1 , 2 , 3 h=1,2,3 h = 1 , 2 , 3 Z Z Z S S S T T T E E E F ∈ R h K × N F\in\mathcal{R}_{h}^{K\times N} F ∈ R h K × N
( ⨂ j = 1 N Z ( h ) ( c F ) j ) E = E ( ⨂ ℓ = 1 K Z ( h ) c ℓ ) ( c ∈ R h K ) . (1) \left(\bigotimes_{j=1}^{N}Z(h)^{(cF)_{j}}\right)E
=
E\left(\bigotimes_{\ell=1}^{K}Z(h)^{c_{\ell}}\right)
\qquad(c\in\mathcal{R}_{h}^{K}). \tag*{(1)} ( j = 1 ⨂ N Z ( h ) ( c F ) j ) E = E ( ℓ = 1 ⨂ K Z ( h ) c ℓ ) ( c ∈ R h K ) . (1) Thus, the vector c c c h h h
Nested algebraic-geometric codes
We use nested one-point evaluation codes. Let X / F q X/\mathbb{F}_{q} X / F q g g g P ∞ P_{\infty} P ∞ D D D n n n L ( a P ∞ ) L(aP_{\infty}) L ( a P ∞ ) P ∞ P_{\infty} P ∞ a a a D D D C L ( D , a P ∞ ) C_{L}(D,aP_{\infty}) C L ( D , a P ∞ ) [20 ] .
Lemma 1 (Ref. [20 ] ). If 2 g − 2 < a < n 2g-2<a<n 2 g − 2 < a < n C = C L ( D , a P ∞ ) C=C_{L}(D,aP_{\infty}) C = C L ( D , a P ∞ ) dim C = a + 1 − g \dim C=a+1-g dim C = a + 1 − g C C C C ⊥ C^{\perp} C ⊥ n − a n-a n − a a + 2 − 2 g a+2-2g a + 2 − 2 g a 0 < a 1 a_{0}<a_{1} a 0 < a 1 C L ( D , a 0 P ∞ ) ⊆ C L ( D , a 1 P ∞ ) C_{L}(D,a_{0}P_{\infty})\subseteq C_{L}(D,a_{1}P_{\infty}) C L ( D , a 0 P ∞ ) ⊆ C L ( D , a 1 P ∞ )
The Garcia–Stichtenoth tower over F 256 \mathbb{F}_{256} F 256 n i → ∞ n_i\to\infty n i → ∞ g i / n i → 1 / 15 g_i/n_i\to1/15 g i / n i → 1/15 [16 , 17 ] . With suitable choices of divisor degree, Lemma 1 therefore yields AG code families with positive asymptotic rate and relative minimum distance. Its evaluation matrices can be constructed deterministically in polynomial time [18 ] . Thus, these AG code families are asymptotically good and can be constructed deterministically and efficiently.
Encoding and logical addressability
Each element y j ∈ R h y_j\in\mathcal{R}_h y j ∈ R h y j = ∑ b = 0 h − 1 2 b y j , b y_j=\sum_{b=0}^{h-1}2^{b}y_{j,b} y j = ∑ b = 0 h − 1 2 b y j , b dig ( y j ) = ( y j , 0 , … , y j , h − 1 ) \operatorname{dig}(y_j)=(y_{j,0},\ldots,y_{j,h-1}) dig ( y j ) = ( y j , 0 , … , y j , h − 1 ) h h h M M M N = h M N=hM N = h M 2 h 2^{h} 2 h
Choose an invertible M × M M\times M M × M B B B R h \mathcal{R}_{h} R h G G G r r r L L L K K K r + K ≤ M r+K\leq M r + K ≤ M R h r : = { ( v 1 , … , v r ) : v j ∈ R h for 1 ≤ j ≤ r } \mathcal{R}_{h}^{r}:=\{(v_{1},\ldots,v_{r}):v_{j}\in\mathcal{R}_{h}\text{ for }1\leq j\leq r\} R h r := {( v 1 , … , v r ) : v j ∈ R h for 1 ≤ j ≤ r } M × M M\times M M × M A A A E : C 2 K → C 2 N E:\mathbb{C}^{2^{K}}\to\mathbb{C}^{2^{N}} E : C 2 K → C 2 N
E ∣ x ⟩ = ν − r / 2 ∑ v ∈ R h r e π i A y T / ν ∣ dig ( y ) ⟩ , (2) E|x\rangle=\nu^{-r/2}\sum_{v\in\mathcal{R}_{h}^{r}}e^{\pi i A y^{T}/\nu}|\operatorname{dig}(y)\rangle, \tag*{(2)} E ∣ x ⟩ = ν − r /2 v ∈ R h r ∑ e π i A y T / ν ∣ dig ( y )⟩ , (2) for every x ∈ { 0 , 1 } K x\in\{0,1\}^{K} x ∈ { 0 , 1 } K y = v G + x L ∈ R h M y=vG+xL\in\mathcal{R}_{h}^{M} y = v G + xL ∈ R h M { 0 , … , ν − 1 } \{0,\ldots,\nu-1\} { 0 , … , ν − 1 } A A A y y y A = 0 A=0 A = 0 A A A B B B 1 ≤ ℓ ≤ K 1\leq\ell\leq K 1 ≤ ℓ ≤ K z ℓ T z_{\ell}^{T} z ℓ T r + ℓ r+\ell r + ℓ B − 1 B^{-1} B − 1
Lemma 2. Equation (2) defines an encoding isometry E E E E E E Z ( h ) Z(h) Z ( h ) F ∈ R h K × h M F\in\mathcal{R}_{h}^{K\times hM} F ∈ R h K × h M
F ℓ , ( j , b ) = 2 b z ℓ , j ( m o d ν ) , (3) F_{\ell,(j,b)}=2^{b}z_{\ell,j}\pmod{\nu}, \tag*{(3)} F ℓ , ( j , b ) = 2 b z ℓ , j ( mod ν ) , (3) where 1 ≤ j ≤ M 1\leq j\leq M 1 ≤ j ≤ M 0 ≤ b < h 0\leq b<h 0 ≤ b < h
Proof. Invertibility of B B B ( v , x ) ↦ v G + x L (v,x)\mapsto vG+xL ( v , x ) ↦ v G + xL (2) corresponding to distinct binary labels x x x ν r \nu^{r} ν r ν − r / 2 \nu^{-r/2} ν − r /2 E E E
Since y = ( v , x , 0 ) B y=(v,x,0)B y = ( v , x , 0 ) B z ℓ T z_{\ell}^{T} z ℓ T r + ℓ r+\ell r + ℓ B − 1 B^{-1} B − 1
∑ j = 1 M z ℓ , j y j = z ℓ T y ≡ x ℓ ( m o d ν ) . \sum_{j=1}^{M}z_{\ell,j}y_{j}=z_{\ell}^{T}y\equiv x_{\ell}\pmod{\nu}. j = 1 ∑ M z ℓ , j y j = z ℓ T y ≡ x ℓ ( mod ν ) . Substituting y j = ∑ b = 0 h − 1 2 b y j , b y_{j}=\sum_{b=0}^{h-1}2^{b}y_{j,b} y j = ∑ b = 0 h − 1 2 b y j , b
x ℓ ≡ ∑ j = 1 M ∑ b = 0 h − 1 ( 2 b z ℓ , j ) y j , b ( m o d ν ) . x_{\ell}\equiv\sum_{j=1}^{M}\sum_{b=0}^{h-1}(2^{b}z_{\ell,j})y_{j,b}\pmod{\nu}. x ℓ ≡ j = 1 ∑ M b = 0 ∑ h − 1 ( 2 b z ℓ , j ) y j , b ( mod ν ) . The coefficient of each physical bit therefore gives the addressing entry F ℓ , ( j , b ) = 2 b z ℓ , j ( m o d ν ) F_{\ell,(j,b)}=2^{b}z_{\ell,j}\pmod{\nu} F ℓ , ( j , b ) = 2 b z ℓ , j ( mod ν ) Z ( h ) F ℓ , ( j , b ) Z(h)^{F_{\ell,(j,b)}} Z ( h ) F ℓ , ( j , b ) ( j , b ) (j,b) ( j , b ) e 2 π i F ℓ , ( j , b ) y j , b / ν e^{2\pi i F_{\ell,(j,b)}y_{j,b}/\nu} e 2 π i F ℓ , ( j , b ) y j , b / ν e 2 π i x ℓ / ν e^{2\pi i x_\ell/\nu} e 2 π i x ℓ / ν E ∣ x ⟩ E|x\rangle E ∣ x ⟩ Z ( h ) Z(h) Z ( h ) ℓ \ell ℓ c ℓ c_\ell c ℓ (1) .
The vectors z ℓ z_\ell z ℓ F F F B − 1 B^{-1} B − 1 B B B F F F A A A A A A
Once B B B A A A h h h w ↦ w B w\mapsto wB w ↦ w B O h ( M 2 ) O_h(M^2) O h ( M 2 ) [21 Sec. IV] [22 ] . The uniform superposition requires h r = O h ( M ) hr=O_h(M) h r = O h ( M ) (2) into O h ( M 2 ) O_h(M^2) O h ( M 2 )
Asymptotically Good Code Constructions
This section gives two constructions of quantum codes with fully addressable Z ( h ) Z(h) Z ( h ) h ≥ 3 h\ge3 h ≥ 3 Theorem 2 ). Figure 1 shows the shared phase mechanism and two approaches to obtaining distance.
Figure 1. Overview of quantum code constructions with fully addressable transversal Z ( h ) Z(h) Z ( h ) B − 1 \mathrm{B}^{-1} B − 1 Z ( h ) Z(h) Z ( h ) B \mathrm{B} B A \mathrm{A} A F 256 \mathbb{F}_{256} F 256 A = 0 \mathrm{A}=0 A = 0 ( B , A ) (\mathrm{B},\mathrm{A}) ( B , A ) h h h
Polynomial-time algebraic-geometric construction
Our construction concatenates an outer quantum code with an inner code. The outer code is constructed from nested one-point AG codes over F 256 \mathbb{F}_{256} F 256
The inner encoding acts on each eight-register block by introducing a uniformly superposed common offset and two additional registers that store the offset and a modular linear check. The resulting ten-register code detects every error supported on a single output register and allows the outer logical phase operations to remain transversal. Each ring register is represented by h h h ∣ s ⟩ |s\rangle ∣ s ⟩ ∣ dig ( s ) ⟩ |\operatorname{dig}(s)\rangle ∣ dig ( s )⟩ s ∈ R h s\in\mathcal{R}_h s ∈ R h
We now define this inner encoding and prove these two properties. Its extension to arbitrary even block sizes and the full concatenation argument appear in § 3.2.2 of SM.
Set ϵ a = 1 \epsilon_a=1 ϵ a = 1 1 ≤ a ≤ 4 1\le a\le4 1 ≤ a ≤ 4 ϵ a = − 1 \epsilon_a=-1 ϵ a = − 1 5 ≤ a ≤ 8 5\le a\le8 5 ≤ a ≤ 8 ∑ a ϵ a = 0 \sum_a\epsilon_a=0 ∑ a ϵ a = 0 u ∈ R h 8 u\in\mathcal{R}_h^8 u ∈ R h 8
J 8 ∣ u ⟩ = ν − 1 / 2 ∑ t ∈ R h ∣ u 1 + t , … , u 8 + t ⟩ J_8|u\rangle=\nu^{-1/2}\sum_{t\in\mathcal{R}_h}|u_1+t,\ldots,u_8+t\rangle J 8 ∣ u ⟩ = ν − 1/2 t ∈ R h ∑ ∣ u 1 + t , … , u 8 + t ⟩ ⊗ ∣ t ⟩ ⊗ ∣ t − ∑ a = 1 8 ϵ a u a ∣ . (4) \otimes|t\rangle\otimes\left|t-\sum_{a=1}^{8}\epsilon_a u_a\right|. \tag*{(4)} ⊗ ∣ t ⟩ ⊗ t − a = 1 ∑ 8 ϵ a u a . (4) The inner code defined by J 8 J_8 J 8
Lemma 3. The map J 8 J_8 J 8
Proof. The ninth output register takes the value t t t t t t u u u ν \nu ν ν − 1 / 2 \nu^{-1/2} ν − 1/2 w w w
∑ a = 1 8 ϵ a w a − w 9 + w 10 = 0. \sum_{a=1}^{8}\epsilon_a w_a-w_9+w_{10}=0. a = 1 ∑ 8 ϵ a w a − w 9 + w 10 = 0. The single-register Weyl operators Q a , b Q_{a,b} Q a , b Q a , b ∣ s ⟩ = e 2 π i b s / ν ∣ s + a ⟩ Q_{a,b}|s\rangle=e^{2\pi i b s/\nu}|s+a\rangle Q a , b ∣ s ⟩ = e 2 π ib s / ν ∣ s + a ⟩ a , b ∈ R h a,b\in\mathcal{R}_h a , b ∈ R h a a a e 2 π i b s / ν e^{2\pi i b s/\nu} e 2 π ib s / ν [21 ] . If a ≠ 0 a\ne0 a = 0 a a a a a a ± a ≠ 0 \pm a\ne0 ± a = 0 a = 0 a=0 a = 0 b ≠ 0 b\ne0 b = 0 t + f ( u ) t+f(u) t + f ( u ) Q Q Q e 2 π i b ( t + f ( u ) ) / ν e^{2\pi i b(t+f(u))/\nu} e 2 π ib ( t + f ( u )) / ν
⟨ u ∣ J 8 † Q J 8 ∣ u ⟩ = ν − 1 e 2 π i b f ( u ) / ν ∑ t ∈ R h e 2 π i b t / ν = 0. \langle u|J_8^\dagger QJ_8|u\rangle
=\nu^{-1}e^{2\pi i b f(u)/\nu}\sum_{t\in\mathcal{R}_h}e^{2\pi i bt/\nu}
=0. ⟨ u ∣ J 8 † Q J 8 ∣ u ⟩ = ν − 1 e 2 π ib f ( u ) / ν t ∈ R h ∑ e 2 π ib t / ν = 0. Inner products with other encoded basis states also vanish, since phase operators preserve the disjoint supports of distinct inputs. Thus J 8 † Q J 8 = 0 J_8^\dagger QJ_8=0 J 8 † Q J 8 = 0 Q Q Q J 8 † O J 8 J_8^\dagger OJ_8 J 8 † O J 8 O O O
For a ring register, write Z ν ( b ) ∣ s ⟩ = e 2 π i b s / ν ∣ s ⟩ Z_\nu(b)|s\rangle=e^{2\pi i bs/\nu}|s\rangle Z ν ( b ) ∣ s ⟩ = e 2 π ib s / ν ∣ s ⟩ Z ν ( b ) = ⨂ j = 0 h − 1 Z ( j ) 2 j b Z_\nu(b)=\bigotimes_{j=0}^{h-1}Z^{(j)2^jb} Z ν ( b ) = ⨂ j = 0 h − 1 Z ( j ) 2 j b
Lemma 4. For z ∈ R h 8 z \in\mathbb{R}_h^8 z ∈ R h 8 z ~ = ( z 1 , … , z 8 , − ∑ a = 1 8 z a , 0 ) \tilde{z}=(z_1,\ldots,z_8,-\sum_{a=1}^8 z_a,0) z ~ = ( z 1 , … , z 8 , − ∑ a = 1 8 z a , 0 )
( ⨂ j = 1 10 Z ν ( z ~ j ) ) J 8 = J 8 ( ⨂ a = 1 8 Z ν ( z a ) ) . (5) \left(\bigotimes_{j=1}^{10} Z_\nu(\tilde{z}_j)\right)J_8=J_8\left(\bigotimes_{a=1}^{8} Z_\nu(z_a)\right). \tag*{(5)} ( j = 1 ⨂ 10 Z ν ( z ~ j ) ) J 8 = J 8 ( a = 1 ⨂ 8 Z ν ( z a ) ) . (5) Proof. For each summand of Eq. (4) , the phase exponent is
z ~ w T = ∑ a = 1 8 z a ( u a + t ) − ( ∑ a = 1 8 z a ) t = ∑ a = 1 8 z a u a ( m o d ν ) . \begin{aligned}
\tilde{z}w^{\mathsf{T}}=\sum_{a=1}^{8}z_a(u_a+t)-\left(\sum_{a=1}^{8}z_a\right)t \\
&=\sum_{a=1}^{8}z_au_a\pmod{\nu}.
\end{aligned} z ~ w T = a = 1 ∑ 8 z a ( u a + t ) − ( a = 1 ∑ 8 z a ) t = a = 1 ∑ 8 z a u a ( mod ν ) . It is independent of t t t
Combining these properties with the AG distance bounds gives the following family.
Theorem 1. For each fixed h ≥ 3 h \ge3 h ≥ 3 Z ( h ) Z(h) Z ( h )
lim i K i N i = 52 675 h , lim inf i D i N i ≥ 52 675 h . (6) \lim_i\frac{K_i}{N_i}=\frac{52}{675h},\qquad\liminf_i\frac{D_i}{N_i}\ge\frac{52}{675h}. \tag*{(6)} i lim N i K i = 675 h 52 , i lim inf N i D i ≥ 675 h 52 . (6) Proof. Fix h ≥ 3 h \ge3 h ≥ 3 J 8 J_8 J 8 F 256 \mathbb{F}_{256} F 256 n n n g g g 2 g − 2 < a 0 < a 1 < n 2g-2<a_0<a_1<n 2 g − 2 < a 0 < a 1 < n A b = C L ( D , a b P ∞ ) A_b=C_L(D,a_bP_\infty) A b = C L ( D , a b P ∞ ) b = 0 , 1 b=0,1 b = 0 , 1
dim A b = a b + 1 − g , \dim A_b=a_b+1-g, dim A b = a b + 1 − g , d ( A 1 ) ≥ n − a 1 , d ( A 0 ⊥ ) ≥ a 0 + 2 − 2 g . d(A_1)\ge n-a_1,\qquad d(A_0^\perp)\ge a_0+2-2g. d ( A 1 ) ≥ n − a 1 , d ( A 0 ⊥ ) ≥ a 0 + 2 − 2 g . The dimension difference determines the number of logical qubits, and the two distance bounds control shifts and phases.
Expand each field coordinate in a fixed binary basis to obtain B 0 ⊂ B 1 ⊆ F 2 8 n B_0\subset B_1\subseteq\mathbb{F}_2^{8n} B 0 ⊂ B 1 ⊆ F 2 8 n B 0 B_0 B 0 B 1 B_1 B 1 F 2 8 n \mathbb{F}_2^{8n} F 2 8 n R h R_h R h B o u t B_{\mathrm{out}} B out G G G r o u t r_{\mathrm{out}} r out L L L K K K
r o u t = 8 ( a 0 + 1 − g ) , K = 8 ( a 1 − a 0 ) . r_{\mathrm{out}}=8(a_0+1-g),\qquad K=8(a_1-a_0). r out = 8 ( a 0 + 1 − g ) , K = 8 ( a 1 − a 0 ) . Setting A = 0 A=0 A = 0 (2) defines the outer encoding
E o u t ∣ x ⟩ = ν − r o u t / 2 ∑ v ∈ R h r o u t ∣ v G + x L ⟩ , x ∈ { 0 , 1 } K . (7) E_{\mathrm{out}}|x\rangle=\nu^{-r_{\mathrm{out}}/2}\sum_{v\in R_h^{r_{\mathrm{out}}}}|vG+xL\rangle,\qquad x\in\{0,1\}^{K}. \tag*{(7)} E out ∣ x ⟩ = ν − r out /2 v ∈ R h r out ∑ ∣ v G + xL ⟩ , x ∈ { 0 , 1 } K . (7) By Lemma 2, it is an isometry with fully addressable logical Z ( h ) Z(h) Z ( h )
Binary expansion preserves nonzero field-coordinate blocks, and expanding A 0 ⊥ A_0^\perp A 0 ⊥ [23 ] . Thus every nonzero word of B 1 B_1 B 1 n − a 1 n-a_1 n − a 1 B 0 ⊥ B_0^\perp B 0 ⊥ a 0 + 2 − 2 g a_0+2-2g a 0 + 2 − 2 g
For any nonzero ring codeword, write c = 2 k α ( G ) c=2^k\alpha^{(G)} c = 2 k α ( G ) 0 ≤ k < h 0\le k<h 0 ≤ k < h α m o d 2 ≠ 0 \alpha\bmod2\ne0 α mod 2 = 0 b = α ( G ) m o d 2 b=\alpha^{(G)}\bmod2 b = α ( G ) mod 2 B 1 B_1 B 1 b j = 1 b_j=1 b j = 1 c j c_j c j 2 k 2^k 2 k 2 h 2^h 2 h b b b c c c n − a 1 n-a_1 n − a 1
Similarly, for a nonzero ring vector z z z G z T = 0 Gz^{\mathsf{T}}=0 G z T = 0 z = 2 k z ′ z=2^kz' z = 2 k z ′ 0 ≤ k < h 0\le k<h 0 ≤ k < h z ′ z' z ′ G z ′ T ≡ 0 ( m o d 2 h − k ) Gz'^{\mathsf{T}}\equiv0\pmod{2^{h-k}} G z ′ T ≡ 0 ( mod 2 h − k ) z ′ m o d 2 z'\bmod2 z ′ mod 2 B 0 ⊥ B_0^\perp B 0 ⊥ z z z z z z a 0 + 2 − 2 g a_0+2-2g a 0 + 2 − 2 g
Set
d ∗ = min { n − a 1 , a 0 + 2 − 2 g } . d_*=\min\{n-a_1,a_0+2-2g\}. d ∗ = min { n − a 1 , a 0 + 2 − 2 g } . A nonzero shift on fewer than d ∗ d_* d ∗ Q β = ⨂ j = 1 8 n Z ν ( β j ) Q_\beta=\bigotimes_{j=1}^{8n}Z_\nu(\beta_j) Q β = ⨂ j = 1 8 n Z ν ( β j ) 0 ≠ β ∈ R h 8 n 0\ne\beta\in R_h^{8n} 0 = β ∈ R h 8 n e 2 π i ( v G + x L ) β T / ν e^{2\pi i(vG+xL)\beta^{\mathsf{T}}/\nu} e 2 π i ( v G + xL ) β T / ν G β T ≠ 0 G\beta^{\mathsf{T}}\ne0 G β T = 0 v v v R h r o u t R_h^{r_{\mathrm{out}}} R h r out E o u t † Q E o u t = 0 E_{\mathrm{out}}^\dagger QE_{\mathrm{out}}=0 E out † Q E out = 0 Q Q Q d ∗ d_* d ∗ Q Q Q E o u t † Q E o u t E_{\mathrm{out}}^\dagger QE_{\mathrm{out}} E out † Q E out d ∗ d_* d ∗
Next apply J 8 J_8 J 8 n n n h h h E E E N = 10 h n N=10hn N = 10 hn K K K P P P w < 2 d ∗ w<2d_* w < 2 d ∗ P = ⨂ p = 1 n P p P=\bigotimes_{p=1}^{n}P_p P = ⨂ p = 1 n P p P p P_p P p 10 h 10h 10 h p p p P p P_p P p J 8 † P p J 8 J_8^\dagger P_pJ_8 J 8 † P p J 8 ⌊ w / 2 ⌋ < d ∗ \lfloor w/2\rfloor<d_* ⌊ w /2 ⌋ < d ∗ E † P E E^\dagger PE E † P E 2 d ∗ 2d_* 2 d ∗
N = 10 h n , K = 8 ( a 1 − a 0 ) , N=10hn,\qquad K=8(a_1-a_0), N = 10 hn , K = 8 ( a 1 − a 0 ) , D ≥ 2 min { n − a 1 , a 0 + 2 − 2 g } . (8) D\ge2\min\{n-a_1,a_0+2-2g\}. \tag*{(8)} D ≥ 2 min { n − a 1 , a 0 + 2 − 2 g } . (8) To retain addressability after concatenation, let z ~ ℓ T \tilde{z}_{\ell}^{\mathsf{T}} z ~ ℓ T r o u t + ℓ r_{\mathrm{out}}+\ell r out + ℓ B o u t − 1 B_{\mathrm{out}}^{-1} B out − 1 z ℓ z_{\ell} z ℓ z ℓ , p z_{\ell,p} z ℓ , p z ~ ℓ , p \tilde{z}_{\ell,p} z ~ ℓ , p (5) , the corresponding phase operators implement the same outer logical action after concatenation. Expanding each output register into binary digits gives the physical addressing matrix F F F
F ℓ , ( p , j , b ) = 2 b z ~ ℓ , p , j ( m o d ν ) , 1 ≤ p ≤ n , 1 ≤ j ≤ 10 , 0 ≤ b < h . F_{\ell,(p,j,b)} = 2^{b}\tilde{z}_{\ell,p,j}\pmod{\nu},
\qquad1 \le p \le n,\quad1 \le j \le10,\quad0 \le b < h. F ℓ , ( p , j , b ) = 2 b z ~ ℓ , p , j ( mod ν ) , 1 ≤ p ≤ n , 1 ≤ j ≤ 10 , 0 ≤ b < h . For each logical qubit ℓ \ell ℓ Z ( h ) Z(h) Z ( h ) (1) for every logical exponent vector on this fixed encoding.
Finally, take the Garcia–Stichtenoth tower over F 256 \mathbb{F}_{256} F 256 n i → ∞ n_i \to\infty n i → ∞ g i / n i → 1 / 15 g_i/n_i \to1/15 g i / n i → 1/15 [16 , 17 ] . For degree fractions a b , i / n i → β b a_{b,i}/n_i \to\beta_b a b , i / n i → β b 2 / 15 < β 0 < β 1 < 1 2/15 < \beta_0 < \beta_1 < 1 2/15 < β 0 < β 1 < 1 (8) gives
R ∞ = 4 5 h ( β 1 − β 0 ) , d ∞ ≥ 1 5 h min { 1 − β 1 , β 0 − 2 / 15 } . (9) R_{\infty} = \frac{4}{5h}(\beta_1-\beta_0),
\qquad
d_{\infty} \ge\frac{1}{5h}\min\{1-\beta_1,\beta_0-2/15\}. \tag*{(9)} R ∞ = 5 h 4 ( β 1 − β 0 ) , d ∞ ≥ 5 h 1 min { 1 − β 1 , β 0 − 2/15 } . (9) Balancing the rate and the two distance terms gives
4 ( β 1 − β 0 ) = 1 − β 1 = β 0 − 2 / 15 , 4(\beta_1-\beta_0)=1-\beta_1=\beta_0-2/15, 4 ( β 1 − β 0 ) = 1 − β 1 = β 0 − 2/15 , so choose β 0 = 70 / 135 \beta_0=70/135 β 0 = 70/135 β 1 = 83 / 135 \beta_1=83/135 β 1 = 83/135
a 0 , i = ⌊ 70 n i / 135 ⌋ , a 1 , i = ⌊ 83 n i / 135 ⌋ . a_{0,i}=\lfloor70n_i/135\rfloor,\qquad a_{1,i}=\lfloor83n_i/135\rfloor. a 0 , i = ⌊ 70 n i /135 ⌋ , a 1 , i = ⌊ 83 n i /135 ⌋ . These degrees satisfy 2 g i − 2 < a 0 , i < a 1 , i < n i 2g_i-2<a_{0,i}<a_{1,i}<n_i 2 g i − 2 < a 0 , i < a 1 , i < n i i i i 1 / n i 1/n_i 1/ n i (9) yields Eq. (6) . □
For fixed h h h [18 ] . Binary expansion, basis completion, ring-matrix inversion and the inner encodings also take polynomial time in N i N_i N i § 3.2.3 and S2.4 of SM, respectively.
The same proof gives a rate–distance tradeoff. For 0 < τ d < 13 / 30 0<\tau_d<13/30 0 < τ d < 13/30 β 0 = 2 / 15 + τ d \beta_0=2/15+\tau_d β 0 = 2/15 + τ d β 1 = 1 − τ d \beta_1=1-\tau_d β 1 = 1 − τ d (9) . Then
R ∞ = 4 5 h ( 13 15 − 2 τ d ) , d ∞ ≥ τ d 5 h . (10) R_{\infty}=\frac{4}{5h}\left(\frac{13}{15}-2\tau_d\right),
\qquad
d_{\infty}\ge\frac{\tau_d}{5h}. \tag*{(10)} R ∞ = 5 h 4 ( 15 13 − 2 τ d ) , d ∞ ≥ 5 h τ d . (10) Increasing τ d \tau_d τ d τ d = 52 / 135 \tau_d=52/135 τ d = 52/135 T = Z ( 3 ) T=Z(3) T = Z ( 3 ) 52 / 2025 52/2025 52/2025 2025 / 52 ≃ 38.94 2025/52\simeq38.94 2025/52 ≃ 38.94
Existence with exact exponential-time generation
The second construction uses a random quadratic phase to obtain codes that detect low-weight errors. Fix h ≥ 3 h\ge3 h ≥ 3 ν = 2 h \nu=2^h ν = 2 h 1 ≤ K < M 1\le K<M 1 ≤ K < M 1 ≤ d ≤ M 1\le d\le M 1 ≤ d ≤ M r = M − K r=M-K r = M − K B B B M × M M\times M M × M R h \mathcal{R}_h R h A A A
B = ( G L ) , B=\begin{pmatrix}G\\L\end{pmatrix}, B = ( G L ) , with G G G r r r E E E (2) , with N = h M N=hM N = h M
For a , b ∈ R h M a,b\in\mathcal{R}_h^M a , b ∈ R h M W ( a , b ) = ⨂ j = 1 M X ν ( a j ) Z ν ( b j ) W(a,b)=\bigotimes_{j=1}^{M}X_{\nu}(a_j)Z_{\nu}(b_j) W ( a , b ) = ⨂ j = 1 M X ν ( a j ) Z ν ( b j ) [21 ] , where X ν ( a j ) ∣ y j ⟩ = ∣ y j + a j ⟩ X_{\nu}(a_j)|y_j\rangle=|y_j+a_j\rangle X ν ( a j ) ∣ y j ⟩ = ∣ y j + a j ⟩ Z ν ( b j ) ∣ y j ⟩ = e 2 π i b j y j / ν ∣ y j ⟩ Z_{\nu}(b_j)|y_j\rangle=e^{2\pi i b_jy_j/\nu}|y_j\rangle Z ν ( b j ) ∣ y j ⟩ = e 2 π i b j y j / ν ∣ y j ⟩ y j ∈ R h y_j\in\mathcal{R}_h y j ∈ R h P P P
P = 2 − N ∑ a , b Tr ( W ( a , b ) † P ) W ( a , b ) . P=2^{-N}\sum_{a,b}\operatorname{Tr}(W(a,b)^{\dagger}P)W(a,b). P = 2 − N a , b ∑ Tr ( W ( a , b ) † P ) W ( a , b ) . We first characterize when a Weyl operator has a nonzero matrix element between encoded basis states.
Lemma 5. For x , x ′ ∈ { 0 , 1 } K x,x'\in\{0,1\}^{K} x , x ′ ∈ { 0 , 1 } K ⟨ x ′ ∣ E † W ( a , b ) E ∣ x ⟩ \langle x'|E^{\dagger}W(a,b)E|x\rangle ⟨ x ′ ∣ E † W ( a , b ) E ∣ x ⟩
G ( b − a A ) T = 0 , ( a B − 1 ) r + 1 : M = x ′ − x (11) G(b-aA)^{\mathsf{T}}=0,\qquad(aB^{-1})_{r+1:M}=x'-x \tag*{(11)} G ( b − a A ) T = 0 , ( a B − 1 ) r + 1 : M = x ′ − x (11) over R h \mathcal{R}_h R h
The proof is given in § 4.1 of SM.
To identify the Weyl terms that must be tested, we assign each a cost that lower-bounds the weight of any qubit Pauli whose Weyl expansion contains that pair. Thus all terms arising from Pauli errors of weight below d d d d d d ( a , b ) (a,b) ( a , b ) 2 s 2^s 2 s 2 s ( a , b ) = 0 2^s(a,b)=0 2 s ( a , b ) = 0 2 s − 1 ( a , b ) ≠ 0 2^{s-1}(a,b)\ne0 2 s − 1 ( a , b ) = 0 R h \mathcal{R}_h R h
For exact order two, write ( a , b ) = 2 h − 1 ( α , β ) (a,b)=2^{h-1}(\alpha,\beta) ( a , b ) = 2 h − 1 ( α , β ) α , β ∈ { 0 , 1 } M \alpha,\beta\in\{0,1\}^{M} α , β ∈ { 0 , 1 } M wt ( α ) + wt ( β ) \operatorname{wt}(\alpha)+\operatorname{wt}(\beta) wt ( α ) + wt ( β )
For exact order four, each local pair satisfies 4 ( a j , b j ) = 0 4(a_j,b_j)=0 4 ( a j , b j ) = 0 ( 0 , 0 ) (0,0) ( 0 , 0 )
operator appears with a nonzero coefficient in the Weyl expansion of a single-qubit Pauli on that register. Assign local cost two to all remaining nonzero local pairs. The total cost is the sum of these local costs.
(3) For exact order 2 s 2^s 2 s 3 ≤ s ≤ h 3 \le s \le h 3 ≤ s ≤ h ( a j , b j ) ≠ ( 0 , 0 ) (a_j,b_j) \ne(0,0) ( a j , b j ) = ( 0 , 0 )
For a pair of exact order two, every entry of the logical shift ( a B − 1 ) r + 1 : M (a\mathbf{B}^{-1})_{r+1:M} ( a B − 1 ) r + 1 : M 2 h − 1 2^{h-1} 2 h − 1 h ≥ 3 h \ge3 h ≥ 3 x ′ − x x^\prime-x x ′ − x x ′ = x x^\prime=x x ′ = x
We now define a sufficient test for the distance bound D ≥ d D \ge d D ≥ d d d d G ( b − a A ) T = 0 G(b-aA)^{\mathsf T}=0 G ( b − a A ) T = 0 ( a B − 1 ) r + 1 : M = 0 (aB^{-1})_{r+1:M}=0 ( a B − 1 ) r + 1 : M = 0 G ( b − a A ) T = 0 G(b-aA)^{\mathsf T}=0 G ( b − a A ) T = 0 ( B , A ) (B,A) ( B , A )
Passing this test guarantees D ≥ d D \ge d D ≥ d Theorem 2 . To establish existence, we bound Pr ( r e j e c t ) \Pr(\mathrm{reject}) Pr ( reject )
B ( n , d ) = ∑ w = 1 d − 1 ( n w ) , B(n,d)=\sum_{w=1}^{d-1}\binom{n}{w}, B ( n , d ) = w = 1 ∑ d − 1 ( w n ) , W h ( M , d ) = ∑ w = 1 d − 1 z w [ 1 + c h z + ( 15 − c h ) z 2 ] M , (12) W_h(M,d)=\sum_{w=1}^{d-1}z^w\left[1+c_hz+(15-c_h)z^2\right]^M, \tag*{(12)} W h ( M , d ) = w = 1 ∑ d − 1 z w [ 1 + c h z + ( 15 − c h ) z 2 ] M , (12) V s ( M , d ) = ∑ w = 1 d − 1 ( 4 s − 1 ) w ( M w ) , V_s(M,d)=\sum_{w=1}^{d-1}(4^s-1)^w\binom{M}{w}, V s ( M , d ) = w = 1 ∑ d − 1 ( 4 s − 1 ) w ( w M ) , where c 3 = 10 c_3=10 c 3 = 10 c h = 6 c_h=6 c h = 6 h ≥ 4 h \ge4 h ≥ 4
Lemma 6 (Order-two errors). Among exact-order-two pairs of cost below d d d α ≠ 0 \alpha\ne0 α = 0 B ( 2 M , d ) B(2M,d) B ( 2 M , d ) 2 − M 2^{-M} 2 − M α = 0 \alpha=0 α = 0 B ( M , d ) B(M,d) B ( M , d ) 2 − r 2^{-r} 2 − r d d d d d d
The proof is given in Sec. S3.2 of SM.
Lemma 7 (Order-four errors). Among local pairs satisfying 4 ( a j , b j ) = 0 4(a_j,b_j)=0 4 ( a j , b j ) = 0 c h c_h c h W h ( M , d ) W_h(M,d) W h ( M , d ) d d d 2 − 2 r 2^{-2r} 2 − 2 r d d d d d d
The proof is given in Sec. S3.3 of SM. The local costs lower-bound the Pauli weight needed to produce a Weyl term.
Lemma 8 (Higher-order errors). For 3 ≤ s ≤ h 3 \le s \le h 3 ≤ s ≤ h V s ( M , d ) V_s(M,d) V s ( M , d ) 2 s 2^s 2 s d d d 2 − s r 2^{-sr} 2 − sr 2 s 2^s 2 s d d d d d d
The proof is given in Sec. S3.4 of SM. Combining these bounds by the union bound gives Pr ( r e j e c t ) ≤ Φ h ( M , K , d ) \Pr(\mathrm{reject}) \le\Phi_h(M,K,d) Pr ( reject ) ≤ Φ h ( M , K , d )
Φ h ( M , K , d ) = 2 − M B ( 2 M , d ) + 2 − r B ( M , d ) + 2 − 2 r W h ( M , d ) + ∑ s = 3 h 2 − s r V s ( M , d ) . (13) \Phi_h(M,K,d)=2^{-M}B(2M,d)
+2^{-r}B(M,d)+2^{-2r}W_h(M,d)
+\sum_{s=3}^{h}2^{-sr}V_s(M,d). \tag*{(13)} Φ h ( M , K , d ) = 2 − M B ( 2 M , d ) + 2 − r B ( M , d ) + 2 − 2 r W h ( M , d ) + s = 3 ∑ h 2 − sr V s ( M , d ) . (13) The entropy estimate below uses the threshold t ⋆ = 2 H 2 − 1 ( 1 / 2 ) t_\star=2H_2^{-1}(1/2) t ⋆ = 2 H 2 − 1 ( 1/2 )
H 2 ( t ) = − t log 2 t − ( 1 − t ) log 2 ( 1 − t ) H_2(t)=-t\log_2 t-(1-t)\log_2(1-t) H 2 ( t ) = − t log 2 t − ( 1 − t ) log 2 ( 1 − t ) is the binary entropy and the inverse is taken on [ 0 , 1 / 2 ] [0,1/2] [ 0 , 1/2 ] [24 ] yields the following lemma.
Lemma 9. Fix h ≥ 3 h \ge3 h ≥ 3 0 < t < t ⋆ 0<t<t_\star 0 < t < t ⋆ c > 0 c>0 c > 0 M 0 M_0 M 0 h , t h,t h , t M ≥ M 0 M \ge M_0 M ≥ M 0 K = d = ⌈ t M ⌉ K=d=\lceil tM\rceil K = d = ⌈ tM ⌉
Pr ( r e j e c t ) ≤ Φ h ( M , K , d ) ≤ 2 − c M . (14) \Pr(\mathrm{reject})\le\Phi_h(M,K,d)\le2^{-cM}. \tag*{(14)} Pr ( reject ) ≤ Φ h ( M , K , d ) ≤ 2 − c M . (14) The proof is given in Sec. S3.5 of SM. The preceding bounds yield the following computable code family.
Theorem 2. For each fixed h ≥ 3 h \ge3 h ≥ 3 Z ( h ) \mathbb{Z}(h) Z ( h )
lim i K i N i = t ⋆ h , lim inf i D i N i ≥ t ⋆ h . (15) \lim_i\frac{K_i}{N_i}=\frac{t_\star}{h},\qquad
\liminf_i\frac{D_i}{N_i}\ge\frac{t_\star}{h}. \tag*{(15)} i lim N i K i = h t ⋆ , i lim inf N i D i ≥ h t ⋆ . (15) The defining matrices B i \mathbf{B}_i B i A i \mathbf{A}_i A i 2 O h ( N i 2 ) 2^{O^h(N_i^2)} 2 O h ( N i 2 ) 2 O h ( N i ) 2^{O^h(N_i)} 2 O h ( N i )
Proof. Fix h ≥ 3 h \ge3 h ≥ 3 N = h M N=hM N = h M r = M − K r=M-K r = M − K ( B , A ) (B,A) ( B , A )
Let P P P d d d 2 − N Tr P = 0 2^{-N}\operatorname{Tr}P=0 2 − N Tr P = 0 d d d Lemma 5 gives E W ( a , b ) E = 0 \mathbb{E}^W(a,b)E=0 E W ( a , b ) E = 0 E † P E = 0 E^\dagger P E=0 E † P E = 0 d d d D ≥ d D \ge d D ≥ d Pr ( r e j e c t ) ≤ Φ h ( M , K , d ) \Pr(\mathrm{reject})\le\Phi_h(M,K,d) Pr ( reject ) ≤ Φ h ( M , K , d )
For i ≥ 3 i \ge3 i ≥ 3 t i t_i t i 2 − i 2^{-i} 2 − i ( 0 , 1 / 2 ) (0,1/2) ( 0 , 1/2 ) 2 H 2 ( t i / 2 ) < 1 2H_2(t_i/2)<1 2 H 2 ( t i /2 ) < 1 ( t i ) (t_i) ( t i ) t ⋆ t_\star t ⋆ 0 < t i < t ⋆ 0 < t_i < t_\star 0 < t i < t ⋆ i i i M i ≥ 2 i M_i \ge2^i M i ≥ 2 i
K i = d i = ⌈ t i M i ⌉ , Φ h ( M i , K i , d i ) < 1 2 . K_i=d_i=\lceil t_iM_i\rceil,\qquad\Phi_h(M_i,K_i,d_i)<\frac{1}{2}. K i = d i = ⌈ t i M i ⌉ , Φ h ( M i , K i , d i ) < 2 1 . Lemma 9 guarantees that such an integer exists and can be found by testing successive lengths. At every selected length, some matrix pair triggers no rejection and therefore defines a code with D i ≥ d i D_i\ge d_i D i ≥ d i N i = h M i N_i=hM_i N i = h M i
K i N i = d i N i = t i h + O ( M i − 1 ) , \frac{K_i}{N_i}=\frac{d_i}{N_i}=\frac{t_i}{h}+O(M_i^{-1}), N i K i = N i d i = h t i + O ( M i − 1 ) , which implies Eq. (15) .
These parameter choices are effective. For rational t i t_i t i Φ h \Phi_h Φ h 2 i ≤ M i 2^i\le M_i 2 i ≤ M i N i N_i N i
For each candidate matrix pair, enumerate the 2 2 h M i 2^{2hM_i} 2 2 h M i d i d_i d i 2 O h ( N i ) 2^{O_h(N_i)} 2 O h ( N i ) 2 O h ( M i 2 ) 2^{O_h(M_i^2)} 2 O h ( M i 2 ) 2 O h ( N i 2 ) 2^{O_h(N_i^2)} 2 O h ( N i 2 )
Uniform admissible matrix pairs can be sampled in polynomial expected time: a uniformly sampled ring matrix is invertible with probability greater than 1 / 4 1/4 1/4 Pr ( r e j e c t ) ≤ Φ h ( M i , K i , d i ) < 1 / 2 \Pr(\mathrm{reject})\le\Phi_h(M_i,K_i,d_i)<1/2 Pr ( reject ) ≤ Φ h ( M i , K i , d i ) < 1/2 1 / 2 1/2 1/2 2 O h ( N i ) 2^{O_h(N_i)} 2 O h ( N i )
For T = Z ( 3 ) T=Z(3) T = Z ( 3 ) 0.0733519 … 0.0733519\ldots 0.0733519 …
The AG construction also applies to h = 1 , 2 h=1,2 h = 1 , 2 S S S H 2 − 1 ( 1 / 2 ) H_2^{-1}(1/2) H 2 − 1 ( 1/2 ) Z Z Z β ⋆ \beta_\star β ⋆ β + H 2 ( β ) + β log 2 3 = 1 \beta+H_2(\beta)+\beta\log_2 3=1 β + H 2 ( β ) + β log 2 3 = 1 § 4.1.5 of SM).
Upper bounds from addressability
The addressing matrix F F F (1) imposes a classical distance constraint. Set B F = F m o d 2 B_F=F\bmod2 B F = F mod 2 s F s_F s F B F B_F B F
Lemma 10. Let an ( ( N , 2 K , D ) ) 2 ((N,2^K,D))_2 (( N , 2 K , D ) ) 2 (1) with h ≥ 1 h\ge1 h ≥ 1 K ≥ 1 K\ge1 K ≥ 1 B F B_F B F [ s F , K , ≥ D ] [s_F,K,\ge D] [ s F , K , ≥ D ]
Proof. Substitute c = 2 h − 1 v c=2^{h-1}v c = 2 h − 1 v v ∈ F 2 K v\in\mathbb{F}_2^K v ∈ F 2 K (1) . Since Z ( h ) 2 h − 1 = Z Z(h)^{2^{h-1}}=Z Z ( h ) 2 h − 1 = Z
Z v B F E = E Z v . Z^{vB_F}E=EZ^v. Z v B F E = E Z v . For v ≠ 0 v\ne0 v = 0 Z v Z^v Z v v B F ≠ 0 vB_F\ne0 v B F = 0 K K K wt ( v B F ) ≥ D \operatorname{wt}(vB_F)\ge D wt ( v B F ) ≥ D
Therefore, the parameters of the quantum code must satisfy the following limits,
2 K ≤ A 2 ( s F , D ) , K + D − 1 ≤ s F , (16) 2^K\le A_2(s_F,D),\qquad K+D-1\le s_F, \tag*{(16)} 2 K ≤ A 2 ( s F , D ) , K + D − 1 ≤ s F , (16) where A 2 ( s F , d ) A_2(s_F,d) A 2 ( s F , d ) s F s_F s F d d d [24 ] , obtained by puncturing D − 1 D-1 D − 1
Theorem 3. Consider a family of binary quantum codes with parameters ( ( N i , 2 K i , D i ) ) 2 ((N_i,2^{K_i},D_i))_2 (( N i , 2 K i , D i ) ) 2 N i → ∞ N_i\to\infty N i → ∞ h ≥ 1 h\ge1 h ≥ 1 Z ( h ) = diag ( 1 , e i 2 π / 2 h ) Z(h)=\operatorname{diag}(1,e^{i2\pi/2^h}) Z ( h ) = diag ( 1 , e i 2 π / 2 h ) F i F_i F i η \eta η η ≤ c u p \eta\le c_{\mathrm{up}} η ≤ c up c u p = 282652727463 / 10 12 c_{\mathrm{up}}=282652727463/10^{12} c up = 282652727463/1 0 12 F i F_i F i η ≤ c u p / h \eta\le c_{\mathrm{up}}/h η ≤ c up / h
Proof. For binary codes of relative distance at least t ∈ ( 0 , 1 / 2 ) t\in(0,1/2) t ∈ ( 0 , 1/2 ) U ( t ) U(t) U ( t ) [25 Corollary 23] . The function U U U U ( c u p ) < c u p U(c_{\mathrm{up}})<c_{\mathrm{up}} U ( c up ) < c up N N N 0 < t < c u p 0<t<c_{\mathrm{up}} 0 < t < c up U ( t ) < t U(t)<t U ( t ) < t η > c u p \eta>c_{\mathrm{up}} η > c up t t t t < U ( t ) < t t<U(t)<t t < U ( t ) < t η ≤ c u p \eta\le c_{\mathrm{up}} η ≤ c up
Under the additional condition on the addressing matrices, the factor 2 b 2^b 2 b b ≥ 1 b\ge1 b ≥ 1 b = 0 b=0 b = 0 M M M s F ≤ M s_F\le M s F ≤ M M = N / h M=N/h M = N / h K / M = h K / N K/M=hK/N K / M = h K / N D / M = h D / N D/M=hD/N D / M = h D / N h η ≤ c u p h\eta\le c_{\mathrm{up}} h η ≤ c up η ≤ c u p / h \eta\le c_{\mathrm{up}}/h η ≤ c up / h
Every code family admitting fully addressable logical Z ( h ) Z(h) Z ( h ) η ≤ c u p \eta\le c_{\mathrm{up}} η ≤ c up (3) , so they obey the stronger bound η ≤ c u p / h \eta\le c_{\mathrm{up}}/h η ≤ c up / h Table II compares the achievable lower bounds on η \eta η (3) , including their values for $T=Z(3)`.
Table II. Values of η \eta η Z ( h ) Z(h) Z ( h ) h ≥ 3 h\ge3 h ≥ 3 η = min { R ∞ , d ∞ } \eta=\min\{R_{\infty},d_{\infty}\} η = min { R ∞ , d ∞ }
Finite codes and encoding circuits
We obtain two finite quantum codes with fully addressable transversal logical T T T Q 48 = ( ( 48 , 16 , 5 ) ) 2 Q_{48} = ((48, 16, 5))_2 Q 48 = (( 48 , 16 , 5 ) ) 2 Q 90 = ( ( 90 , 256 , 7 ) ) 2 Q_{90} = ((90, 256, 7))_2 Q 90 = (( 90 , 256 , 7 ) ) 2 § 2 B at h = 3 h = 3 h = 3 D D D t = ⌊ ( D − 1 ) / 2 ⌋ t = \lfloor(D - 1)/2\rfloor t = ⌊( D − 1 ) /2 ⌋ [19 ] . Thus, the two codes correct arbitrary errors on up to 2 and 3 qubits, respectively. The rejection tests and logical witnesses are given in § 4.4.2 of SM; complete matrices and machine-readable certificates are available from the authors upon reasonable request.
By Eq. (1) with h = 3 h = 3 h = 3 T T T Q 48 Q_{48} Q 48 T T T
An ancilla-free encoding circuit for Q 48 Q_{48} Q 48 x ↦ u x m o d 8 x \mapsto ux \bmod8 x ↦ ux mod 8 u u u ( x , y ) ↦ ( x , y + c x m o d 8 ) (x, y) \mapsto(x, y + cx \bmod8) ( x , y ) ↦ ( x , y + c x mod 8 ) § 4.4.3 of SM.
Discussion
Independent phase control can coexist with a nonvanishing density of protected logical qubits. The cyclic encoding makes the mechanism explicit: inverse-matrix columns select logical phases, while classical code distances or quadratic phase counting provide error detection. This separation produces quantitative guarantees for general quantum subspaces and connects their addressability to classical coding bounds.
The two constructions offer different parameter and construction-cost tradeoffs. Algebraic-geometric evaluation matrices give a uniform polynomial-time family. Random quadratic phases, coupled with exact verification, give a larger common rate–distance coefficient. The parity-code converse places both within a common framework and identifies how the physical digit structure limits the simultaneous rate and distance.
A natural next step is to combine these selectors with efficient decoding, low-weight error checks, and scalable syndrome extraction. For physical implementations, the simple transversal phase layer should be assessed together with encoding, recovery, connectivity, and a specified noise model. The explicit finite encoders provide concrete starting points for such studies. On the coding side, tighter error counting, improved inner encodings, and other physical addressing structures offer routes to narrowing the remaining gap between construction guarantees and converse bounds.
Acknowledgments
OpenAI’s GPT models were used to assist in developing proofs and searching for optimal code parameters. The authors take full responsibility for the results and proofs presented in this work. B.W. and T.L. acknowledge support from the National Natural Science Foundation of China (Grant No. 12405014).
Data availability
The data supporting the findings of this study are available from the authors upon reasonable request.
References
[22 ] The subscript h h h h h h
Supplemental Material
Supplemental Material for:
Substantially Improved Rate and Distance for Addressable Transversal Phase Gates
This Supplemental Material provides the proofs and construction details supporting the main text. Section S1 establishes addressability and derives error-compression identities for the encoding. Sections S2 and S3 develop the polynomial-time AG construction and the counting proof with exact exponential-time generation, respectively. Section S4 derives universal rate–distance upper bounds from addressability and stronger bounds for cyclic-digit encodings. Section S5 compares the parameter guarantees of prior constructions. Section S6 verifies the addressable logical gates and exact distances of the finite examples and describes an ancilla-free encoder for the 48-qubit code.
Encoding and logical phase gates
A binary quantum code ( ( N , 2 K , D ) ) 2 ((N,2^K,D))_2 (( N , 2 K , D ) ) 2 K K K N N N D D D R = K / N R=K/N R = K / N d = D / N d=D/N d = D / N
A code projector P P P O O O P O P POP P O P P P P E E E P = E E † P=EE^\dagger P = E E † E † O E E^\dagger O E E † O E [19 ] . The distance is the minimum Pauli weight for which this fails. For N i → ∞ N_i\to\infty N i → ∞
R ∞ = lim inf i K i / N i , d ∞ = lim inf i D i / N i , η = min { R ∞ , d ∞ } . R_\infty=\liminf_i K_i/N_i,\qquad d_\infty=\liminf_i D_i/N_i,\qquad\eta=\min\{R_\infty,d_\infty\}. R ∞ = i lim inf K i / N i , d ∞ = i lim inf D i / N i , η = min { R ∞ , d ∞ } . Lower asymptotic targets R 0 , d 0 R_0,d_0 R 0 , d 0 R ∞ ≥ R 0 R_\infty\ge R_0 R ∞ ≥ R 0 d ∞ ≥ d 0 d_\infty\ge d_0 d ∞ ≥ d 0
For an integer h ≥ 1 h\ge1 h ≥ 1
Z ( h ) = diag ( 1 , e 2 π i / 2 h ) . Z(h)=\operatorname{diag}(1,e^{2\pi i/2^h}). Z ( h ) = diag ( 1 , e 2 π i / 2 h ) . Set
ν = 2 h , R h = Z / ν Z . \nu=2^h,\qquad\mathcal{R}_h=\mathbb{Z}/\nu\mathbb{Z}. ν = 2 h , R h = Z / ν Z . The phase level h h h log \log log ln \ln ln [9 Definition 1.1] .
Definition S1 (Full addressability). An encoding isometry E : ( C 2 ) ⊗ K → ( C 2 ) ⊗ N E:(\mathbb{C}^2)^{\otimes K}\to(\mathbb{C}^2)^{\otimes N} E : ( C 2 ) ⊗ K → ( C 2 ) ⊗ N h h h F ∈ R h K × N F\in\mathcal{R}_h^{K\times N} F ∈ R h K × N
( ⨂ j = 1 N Z ( h ) ( c F ) j ) E = E ( ⨂ ℓ = 1 K Z ( h ) c ℓ ) for every c ∈ R h K . (17) \left(\bigotimes_{j=1}^{N} Z(h)^{(cF)_j}\right)E
=
E\left(\bigotimes_{\ell=1}^{K} Z(h)^{c_\ell}\right)
\quad\text{for every }c\in\mathcal{R}_h^K. \tag*{(17)} ( j = 1 ⨂ N Z ( h ) ( c F ) j ) E = E ( ℓ = 1 ⨂ K Z ( h ) c ℓ ) for every c ∈ R h K . (17) Physical powers, including inverse powers, are allowed. The encoding is fixed for every logical exponent vector c c c
For y ∈ R h y\in\mathcal{R}_h y ∈ R h dig ( y ) = ( y 0 , … , y h − 1 ) \operatorname{dig}(y)=(y_0,\ldots,y_{h-1}) dig ( y ) = ( y 0 , … , y h − 1 ) y = ∑ t = 0 h − 1 2 t y t y=\sum_{t=0}^{h-1}2^t y_t y = ∑ t = 0 h − 1 2 t y t y t ∈ { 0 , 1 } y_t\in\{0,1\} y t ∈ { 0 , 1 } h h h dig \operatorname{dig} dig ∣ y ⟩ ↦ ∣ dig ( y ) ⟩ = ∣ y 0 , … , y h − 1 ⟩ |y\rangle\mapsto|\operatorname{dig}(y)\rangle=|y_0,\ldots,y_{h-1}\rangle ∣ y ⟩ ↦ ∣ dig ( y )⟩ = ∣ y 0 , … , y h − 1 ⟩ h h h
X ν ( a ) ∣ y ⟩ = ∣ y + a ⟩ , Z ν ( b ) ∣ y ⟩ = e 2 π i b y / ν ∣ y ⟩ . X_\nu(a)|y\rangle=|y+a\rangle,\qquad Z_\nu(b)|y\rangle=e^{2\pi iby/\nu}|y\rangle. X ν ( a ) ∣ y ⟩ = ∣ y + a ⟩ , Z ν ( b ) ∣ y ⟩ = e 2 π ib y / ν ∣ y ⟩ . For M M M a , b ∈ R h M a,b\in\mathcal{R}_h^M a , b ∈ R h M W ( a , b ) = ⨂ j = 1 M X ν ( a j ) Z ν ( b j ) W(a,b)=\bigotimes_{j=1}^{M}X_\nu(a_j)Z_\nu(b_j) W ( a , b ) = ⨂ j = 1 M X ν ( a j ) Z ν ( b j ) ν \nu ν Tr ( W ( a , b ) † W ( a ′ , b ′ ) ) = ν M 1 ( a , b ) = ( a ′ , b ′ ) \operatorname{Tr}(W(a,b)^\dagger W(a',b'))=\nu^M\mathbf{1}_{(a,b)=(a',b')} Tr ( W ( a , b ) † W ( a ′ , b ′ )) = ν M 1 ( a , b ) = ( a ′ , b ′ ) a = a ′ a=a' a = a ′ O O O
O = ν − M ∑ a , b Tr ( W ( a , b ) † O ) W ( a , b ) . (18) O=\nu^{-M}\sum_{a,b}\operatorname{Tr}(W(a,b)^\dagger O)W(a,b). \tag*{(18)} O = ν − M a , b ∑ Tr ( W ( a , b ) † O ) W ( a , b ) . (18) The number of registers on which W ( a , b ) W(a,b) W ( a , b ) wts ( a , b ) = ∣ supp ( a ) ∪ supp ( b ) ∣ \operatorname{wts}(a,b)=|\operatorname{supp}(a)\cup\operatorname{supp}(b)| wts ( a , b ) = ∣ supp ( a ) ∪ supp ( b ) ∣
The following encoding will be used in both constructions. Choose B ∈ G L M ( R h ) B \in\mathrm{GL}_{M}(\mathcal{R}_{h}) B ∈ GL M ( R h ) r r r K K K r + K ≤ M r+K \le M r + K ≤ M A A A R h \mathcal{R}_{h} R h G G G L L L r r r K K K B B B τ = e π i / ν \tau= e^{\pi i/\nu} τ = e π i / ν
E B , A ∣ x ⟩ = ν − r / 2 ∑ v ∈ R h r τ y A y T ∣ dig ( y ) ⟩ , y = v G + x L , x ∈ { 0 , 1 } K , (19) E_{B,A}|x\rangle= \nu^{-r/2} \sum_{v\in\mathcal{R}_{h}^{r}} \tau^{yAy^{\mathsf{T}}}|\operatorname{dig}(y)\rangle,\qquad y=vG+xL,\quad x\in\{0,1\}^{K}, \tag*{(19)} E B , A ∣ x ⟩ = ν − r /2 v ∈ R h r ∑ τ y A y T ∣ dig ( y )⟩ , y = v G + xL , x ∈ { 0 , 1 } K , (19) where in the quadratic phase, we use integer representatives in { 0 , … , ν − 1 } \{0,\ldots,\nu-1\} { 0 , … , ν − 1 } A A A y y y N = h M N=hM N = h M
Lemma S1. Eq. (19) defines an isometry, and its addressing matrix is
F ℓ , ( j , t ) = 2 t ( B − 1 ) j , r + ℓ ( m o d ν ) , 1 ≤ ℓ ≤ K , 1 ≤ j ≤ M , 0 ≤ t < h . (20) F_{\ell,(j,t)} = 2^{t}(B^{-1})_{j,r+\ell}\pmod{\nu},\qquad1\le\ell\le K,\quad1\le j\le M,\quad0\le t<h. \tag*{(20)} F ℓ , ( j , t ) = 2 t ( B − 1 ) j , r + ℓ ( mod ν ) , 1 ≤ ℓ ≤ K , 1 ≤ j ≤ M , 0 ≤ t < h . (20) Let C 1 C_{1} C 1 r + K r+K r + K B B B R h \mathcal{R}_{h} R h a , b ∈ R h M a,b\in\mathcal{R}_{h}^{M} a , b ∈ R h M x , x ′ ∈ { 0 , 1 } K x,x'\in\{0,1\}^{K} x , x ′ ∈ { 0 , 1 } K ⟨ x ′ ∣ E B , A † W ( a , b ) E B , A ∣ x ⟩ \langle x'|E_{B,A}^{\dagger}W(a,b)E_{B,A}|x\rangle ⟨ x ′ ∣ E B , A † W ( a , b ) E B , A ∣ x ⟩
a ∈ C 1 , G ( b − a A ) T = 0 , x ′ − x = ( a B − 1 ) r + 1 : r + K over R h . (21) a\in C_{1},\qquad G(b-aA)^{\mathsf{T}}=0,\qquad x'-x=(aB^{-1})_{r+1:r+K}\quad\text{over }\mathcal{R}_{h}. \tag*{(21)} a ∈ C 1 , G ( b − a A ) T = 0 , x ′ − x = ( a B − 1 ) r + 1 : r + K over R h . (21) Proof. The first r + K r+K r + K v ↦ v G + x L v\mapsto vG+xL v ↦ v G + xL ν r \nu^{r} ν r ν − r / 2 \nu^{-r/2} ν − r /2 Q A ∣ y ⟩ = τ y A y T ∣ y ⟩ Q_{A}|y\rangle=\tau^{yAy^{\mathsf{T}}}|y\rangle Q A ∣ y ⟩ = τ y A y T ∣ y ⟩
Q A † W ( a , b ) Q A = τ − a A a T W ( a , b − a A ) . (22) Q_{A}^{\dagger}W(a,b)Q_{A}=\tau^{-aAa^{\mathsf{T}}}W(a,b-aA). \tag*{(22)} Q A † W ( a , b ) Q A = τ − a A a T W ( a , b − a A ) . (22) Write γ = b − a A \gamma=b-aA γ = b − a A q = a B − 1 q=aB^{-1} q = a B − 1 q r + K + 1 : M = 0 q_{r+K+1:M}=0 q r + K + 1 : M = 0 q r + 1 : r + K = x ′ − x q_{r+1:r+K}=x'-x q r + 1 : r + K = x ′ − x
⟨ x ′ ∣ E B , A † W ( a , b ) E B , A ∣ x ⟩ = τ − a A a T e 2 π i γ ( x L ) T / ν ν − r ∑ v ∈ R h r e 2 π i γ ( v G ) T / ν . (23) \langle x'|E_{B,A}^{\dagger}W(a,b)E_{B,A}|x\rangle
=\tau^{-aAa^{\mathsf{T}}}e^{2\pi i\gamma(xL)^{\mathsf{T}}/\nu}
\nu^{-r}\sum_{v\in\mathcal{R}_{h}^{r}}e^{2\pi i\gamma(vG)^{\mathsf{T}}/\nu}. \tag*{(23)} ⟨ x ′ ∣ E B , A † W ( a , b ) E B , A ∣ x ⟩ = τ − a A a T e 2 π iγ ( xL ) T / ν ν − r v ∈ R h r ∑ e 2 π iγ ( v G ) T / ν . (23) The normalized character sum equals one if G γ T = 0 G\gamma^{\mathsf{T}}=0 G γ T = 0 (21) over the cyclic ring.
Let z ℓ z_{\ell} z ℓ r + ℓ r+\ell r + ℓ B − T B^{-\mathsf{T}} B − T G z ℓ T = 0 Gz_{\ell}^{\mathsf{T}}=0 G z ℓ T = 0 L z ℓ T = e ℓ T Lz_{\ell}^{\mathsf{T}}=e_{\ell}^{\mathsf{T}} L z ℓ T = e ℓ T ⨂ j Z ν ( z ℓ , j ) \bigotimes_{j}Z_{\nu}(z_{\ell,j}) ⨂ j Z ν ( z ℓ , j ) e 2 π i x ℓ / ν e^{2\pi ix_{\ell}/\nu} e 2 π i x ℓ / ν
Z ν ( a ) = ⨂ t = 0 h − 1 Z ( h ) 2 t a (24) Z_{\nu}(a)=\bigotimes_{t=0}^{h-1}Z(h)^{2^{t}a} \tag*{(24)} Z ν ( a ) = t = 0 ⨂ h − 1 Z ( h ) 2 t a (24) under the digit representation. This proves Eq. (20) and Eq. (17) ; the quadratic phase commutes with these gates.
The cyclic-digit class consists of the encodings in Eq. (19) with the addressing form in Eq. (20) . The upper bound for this class will use only that form, and therefore also applies to other encodings admitting such a matrix. The matrices compactly specify the encoding and its gates. For fixed h h h y ↦ y B y\mapsto yB y ↦ y B O ( M 2 ) O(M^{2}) O ( M 2 ) [21 Secs. III–IV] . The diagonal Q A Q_{A} Q A O ( M 2 ) O(M^{2}) O ( M 2 ) h h h
Polynomial-time algebraic-geometric construction
We give the ring-lifting and outer-code arguments used in § 2 A of the main text, then extend its inner encoding from eight registers to any positive even number m m m
Ring lifting and the outer code
For a fixed partition of coordinates into blocks, block weight counts the blocks containing a nonzero coordinate. Write d b l ( C ) d_{\mathrm{bl}}(C) d bl ( C ) C C C
Lemma S2. Let B 2 ∈ G L M ( F 2 ) \mathbf{B}_2 \in\mathrm{GL}_M(\mathbb{F}_2) B 2 ∈ GL M ( F 2 ) B ∈ G L M ( R h ) \mathbf{B} \in\mathrm{GL}_M(\mathcal{R}_h) B ∈ GL M ( R h ) 1 ≤ s < M 1 \leq s < M 1 ≤ s < M B s B_s B s C s C_s C s s s s d b l ( C s ) = d b l ( B s ) d_{\mathrm{bl}}(C_s)=d_{\mathrm{bl}}(B_s) d bl ( C s ) = d bl ( B s ) d b l ( C s ⊥ ) = d b l ( B s ⊥ ) d_{\mathrm{bl}}(C_s^\perp)=d_{\mathrm{bl}}(B_s^\perp) d bl ( C s ⊥ ) = d bl ( B s ⊥ ) C s ⊥ = { z ∈ R h M : c z T = 0 for every c ∈ C s } C_s^\perp=\{z\in\mathcal{R}_h^M:cz^{\mathsf T}=0\text{ for every }c\in C_s\} C s ⊥ = { z ∈ R h M : c z T = 0 for every c ∈ C s } 2 h 2^h 2 h
Proof. Since B m o d 2 \mathbf{B}\bmod2 B mod 2 det B \det\mathbf{B} det B R h \mathcal{R}_h R h B \mathbf{B} B R h \mathcal{R}_h R h
Let G G G s s s B \mathbf{B} B G 2 = G m o d 2 G_2=G\bmod2 G 2 = G mod 2 C s C_s C s B s B_s B s
Take any nonzero codeword c ∈ C s c\in C_s c ∈ C s c = u G c=uG c = u G u = 2 k u ′ u=2^k u' u = 2 k u ′ 0 ≤ k < h 0\leq k<h 0 ≤ k < h u ′ u' u ′
w = u ′ G , b = w m o d 2 = ( u ′ m o d 2 ) G 2 . w=u'G,\qquad b=w\bmod2=(u'\bmod2)G_2. w = u ′ G , b = w mod 2 = ( u ′ mod 2 ) G 2 . The rows of G 2 G_2 G 2 b b b B s B_s B s b j = 1 b_j=1 b j = 1 w j w_j w j c j = 2 k w j c_j=2^k w_j c j = 2 k w j 2 h 2^h 2 h b b b c c c
wt b l ( c ) ≥ wt b l ( b ) ≥ d b l ( B s ) . \operatorname{wt}_{\mathrm{bl}}(c)\geq\operatorname{wt}_{\mathrm{bl}}(b)\geq d_{\mathrm{bl}}(B_s). wt bl ( c ) ≥ wt bl ( b ) ≥ d bl ( B s ) . Since this holds for every nonzero c ∈ C s c\in C_s c ∈ C s d b l ( C s ) ≥ d b l ( B s ) d_{\mathrm{bl}}(C_s)\geq d_{\mathrm{bl}}(B_s) d bl ( C s ) ≥ d bl ( B s )
For the reverse inequality, choose a nonzero codeword b = α G 2 b=\alpha G_2 b = α G 2 B s B_s B s α \alpha α R h \mathcal{R}_h R h
c ∗ = 2 h − 1 α G ∈ C s . c_* = 2^{h-1}\alpha G\in C_s. c ∗ = 2 h − 1 α G ∈ C s . Because α G m o d 2 = b \alpha G\bmod2=b α G mod 2 = b 2 h − 1 2^{h-1} 2 h − 1
( c ∗ ) j = { 0 , b j = 0 , 2 h − 1 , b j = 1. (c_*)_j=
\begin{cases}
0, & b_j=0,\\
2^{h-1}, & b_j=1.
\end{cases} ( c ∗ ) j = { 0 , 2 h − 1 , b j = 0 , b j = 1. Hence c ∗ c_* c ∗ b b b
d b l ( C s ) ≤ wt b l ( c ∗ ) = d b l ( B s ) , d_{\mathrm{bl}}(C_s)\leq\operatorname{wt}_{\mathrm{bl}}(c_*)=d_{\mathrm{bl}}(B_s), d bl ( C s ) ≤ wt bl ( c ∗ ) = d bl ( B s ) , which proves the first equality.
For the dual codes, let H H H M − s M-s M − s B − T \mathbf{B}^{-\mathsf T} B − T B ( B − T ) T = I \mathbf{B}(\mathbf{B}^{-\mathsf T})^{\mathsf T}=I B ( B − T ) T = I C s ⊥ C_s^\perp C s ⊥
B − T m o d 2 = ( B m o d 2 ) − T , \mathbf{B}^{-\mathsf T}\bmod2=(\mathbf{B}\bmod2)^{-\mathsf T}, B − T mod 2 = ( B mod 2 ) − T , so the rows of H m o d 2 H\bmod2 H mod 2 B s ⊥ B_s^\perp B s ⊥ H H H d b l ( C s ⊥ ) = d b l ( B s ⊥ ) d_{\mathrm{bl}}(C_s^\perp)=d_{\mathrm{bl}}(B_s^\perp) d bl ( C s ⊥ ) = d bl ( B s ⊥ )
Let X / F 2 m X/\mathbb{F}_{2^m} X / F 2 m g g g P ∞ P_{\infty} P ∞ n n n D \mathcal{D} D 2 g − 2 < a 0 < a 1 < n 2g-2<a_0<a_1<n 2 g − 2 < a 0 < a 1 < n
A b = C L ( D , a b P ∞ ) , b = 0 , 1. \mathcal{A}_b=\mathcal{C}_L(\mathcal{D},a_bP_{\infty}),\qquad b=0,1. A b = C L ( D , a b P ∞ ) , b = 0 , 1. Expand them in a fixed binary basis to obtain B 0 ⊂ B 1 B_0\subset B_1 B 0 ⊂ B 1 m m m B 0 ⊂ B 1 B_0\subset B_1 B 0 ⊂ B 1 B o u t ∈ GL m n ( R h ) B_{\mathrm{out}}\in\operatorname{GL}_{mn}(\mathcal{R}_h) B out ∈ GL mn ( R h ) r o u t = m ( a 0 + 1 − g ) r_{\mathrm{out}}=m(a_0+1-g) r out = m ( a 0 + 1 − g ) G G G K = m ( a 1 − a 0 ) K=m(a_1-a_0) K = m ( a 1 − a 0 ) L L L [20 ] .
Lemma S3. For G G G L L L
E o u t ∣ x ⟩ = ν − r o u t / 2 ∑ v ∈ R h r o u t ∣ v G + x L ⟩ , x ∈ { 0 , 1 } K , (25) E_{\mathrm{out}}|x\rangle=\nu^{-r_{\mathrm{out}}/2}\sum_{v\in\mathcal{R}_h^{r_{\mathrm{out}}}}|vG+xL\rangle,\qquad x\in\{0,1\}^{K}, \tag*{(25)} E out ∣ x ⟩ = ν − r out /2 v ∈ R h r out ∑ ∣ v G + xL ⟩ , x ∈ { 0 , 1 } K , (25) detects every operator acting on fewer than
d ∗ = min { n − a 1 , a 0 + 2 − 2 g } (26) d_*=\min\{n-a_1,a_0+2-2g\} \tag*{(26)} d ∗ = min { n − a 1 , a 0 + 2 − 2 g } (26) blocks, where each block consists of the m m m
Proof. The rows of G G G L L L R h m n \mathcal{R}_h^{mn} R h mn ( v , x ) (v,x) ( v , x ) (25) therefore makes E o u t E_{\mathrm{out}} E out d ∗ = 1 d_*=1 d ∗ = 1 d ∗ ≥ 2 d_*\ge2 d ∗ ≥ 2
We first obtain the two classical distance bounds needed for the proof. Let θ 1 , … , θ m \theta_1,\ldots,\theta_m θ 1 , … , θ m θ 1 ∗ , … , θ m ∗ \theta_1^*,\ldots,\theta_m^* θ 1 ∗ , … , θ m ∗ [23 Chap. 2] : Tr ( θ j θ ℓ ∗ ) = δ j ℓ \operatorname{Tr}(\theta_j\theta_\ell^*)=\delta_{j\ell} Tr ( θ j θ ℓ ∗ ) = δ j ℓ Tr \operatorname{Tr} Tr F 2 \mathbb{F}_2 F 2 ξ = ∑ j ξ j θ j \xi=\sum_j\xi_j\theta_j ξ = ∑ j ξ j θ j ζ = ∑ j ζ j θ j ∗ \zeta=\sum_j\zeta_j\theta_j^* ζ = ∑ j ζ j θ j ∗
Tr ( ξ ζ ) = ∑ j ξ j ζ j . \operatorname{Tr}(\xi\zeta)=\sum_j\xi_j\zeta_j. Tr ( ξ ζ ) = j ∑ ξ j ζ j . Applying this identity coordinatewise shows that expanding A 0 ⊥ \mathcal{A}_0^\perp A 0 ⊥ B 0 B_0 B 0 B 0 ⊥ B_0^\perp B 0 ⊥ m n − r o u t mn-r_{\mathrm{out}} mn − r out [20 Chap. 2] thus imply
d b l ( B 1 ) ≥ n − a 1 , d b l ( B 0 ⊥ ) ≥ a 0 + 2 − 2 g . d_{\mathrm{bl}}(B_1)\ge n-a_1,\qquad d_{\mathrm{bl}}(B_0^\perp)\ge a_0+2-2g. d bl ( B 1 ) ≥ n − a 1 , d bl ( B 0 ⊥ ) ≥ a 0 + 2 − 2 g . By Lemma S2, every nonzero ring linear combination of the rows of G G G L L L n − a 1 n-a_1 n − a 1 b b b G b T = 0 Gb^{\mathsf{T}}=0 G b T = 0 a 0 + 2 − 2 g a_0+2-2g a 0 + 2 − 2 g
Now consider a nonidentity Weyl operator W ( a , b ) W(a,b) W ( a , b ) d ∗ d_* d ∗ a a a b b b d ∗ d_* d ∗
If a ≠ 0 a\ne0 a = 0
v G + x L + a = v ′ G + x ′ L vG+xL+a=v'G+x'L v G + xL + a = v ′ G + x ′ L for some v , v ′ , x , x ′ v,v',x,x' v , v ′ , x , x ′
a = ( v ′ − v ) G + ( x ′ − x ) L . a=(v'-v)G+(x'-x)L. a = ( v ′ − v ) G + ( x ′ − x ) L . This would make a a a G G G L L L n − a 1 n-a_1 n − a 1
E o u t † W ( a , b ) E o u t = 0. E_{\mathrm{out}}^{\dagger}W(a,b)E_{\mathrm{out}}=0. E out † W ( a , b ) E out = 0. If a = 0 a=0 a = 0 b ≠ 0 b\ne0 b = 0 G b T ≠ 0 G b^{\mathsf T}\ne0 G b T = 0 W ( 0 , b ) W(0,b) W ( 0 , b ) (23) with A = 0 A=0 A = 0 a = 0 a=0 a = 0
⟨ x ∣ E o u t † W ( 0 , b ) E o u t ∣ x ⟩ = e 2 π i ( x L ) b T / ν ν r o u t ∑ v ∈ R h r o u t e 2 π i v G b T / ν . \langle x|E_{\mathrm{out}}^\dagger W(0,b)E_{\mathrm{out}}|x\rangle=\frac{e^{2\pi i(xL)b^{\mathsf T}/\nu}}{\nu^{r_{\mathrm{out}}}}\sum_{v\in\mathbb{R}_{h}^{r_{\mathrm{out}}}}e^{2\pi i vG b^{\mathsf T}/\nu}. ⟨ x ∣ E out † W ( 0 , b ) E out ∣ x ⟩ = ν r out e 2 π i ( xL ) b T / ν v ∈ R h r out ∑ e 2 π i v G b T / ν . Since G b T G b^{\mathsf T} G b T v v v E o u t † W ( 0 , b ) E o u t = 0 E_{\mathrm{out}}^\dagger W(0,b)E_{\mathrm{out}}=0 E out † W ( 0 , b ) E out = 0
By Eq. (18) , every operator O O O d ∗ d_* d ∗ E o u t † E o u t = I E_{\mathrm{out}}^\dagger E_{\mathrm{out}}=I E out † E out = I
E o u t † O E o u t = λ O I , E_{\mathrm{out}}^\dagger O E_{\mathrm{out}}=\lambda_O I, E out † O E out = λ O I , which is precisely the error-detection condition. □ \square □
Inner encoding and concatenation
The outer bound of Lemma S3 counts affected field-coordinate blocks. We now encode each block to detect single-register errors while preserving the phase operations used for addressability.
For positive even m m m ϵ a = 1 \epsilon_a=1 ϵ a = 1 a ≤ m / 2 a\le m/2 a ≤ m /2 ϵ a = − 1 \epsilon_a=-1 ϵ a = − 1 ∑ a ϵ a = 0 \sum_a\epsilon_a=0 ∑ a ϵ a = 0 u ∈ R h m u\in\mathbb{R}_h^m u ∈ R h m
J m ∣ u ⟩ = ν − 1 / 2 ∑ t ∈ R h ∣ u 1 + t , … , u m + t , t , t − ∑ a = 1 m ϵ a u a ⟩ . (27) J_m|u\rangle=\nu^{-1/2}\sum_{t\in\mathbb{R}_h}|u_1+t,\ldots,u_m+t,t,t-\sum_{a=1}^{m}\epsilon_a u_a\rangle. \tag*{(27)} J m ∣ u ⟩ = ν − 1/2 t ∈ R h ∑ ∣ u 1 + t , … , u m + t , t , t − a = 1 ∑ m ϵ a u a ⟩ . (27) All arithmetic is modulo ν \nu ν m m m m + 2 m+2 m + 2 § 2 A of the main text at m = 8 m=8 m = 8
Lemma S4. The map J m J_m J m For z ∈ R h m z\in\mathbb{R}_h^m z ∈ R h m z ~ = ( z 1 , … , z m , − ∑ a z a , 0 ) \widetilde{z}=(z_1,\ldots,z_m,-\sum_a z_a,0) z = ( z 1 , … , z m , − ∑ a z a , 0 )
( ⨂ j = 1 m + 2 Z ν ( z ~ j ) ) J m = J m ( ⨂ a = 1 m Z ν ( z a ) ) . (28) \left(\bigotimes_{j=1}^{m+2}Z_\nu(\widetilde{z}_j)\right)J_m=J_m\left(\bigotimes_{a=1}^{m}Z_\nu(z_a)\right). \tag*{(28)} ( j = 1 ⨂ m + 2 Z ν ( z j ) ) J m = J m ( a = 1 ⨂ m Z ν ( z a ) ) . (28) Proof. For fixed u u u t t t (27) as
w = w ( u , t ) = ( w 1 , … , w m + 2 ) = ( u 1 + t , … , u m + t , t , t − ∑ a = 1 m ϵ a u a ) . w=w(u,t)=(w_1,\ldots,w_{m+2})=\left(u_1+t,\ldots,u_m+t,t,t-\sum_{a=1}^{m}\epsilon_a u_a\right). w = w ( u , t ) = ( w 1 , … , w m + 2 ) = ( u 1 + t , … , u m + t , t , t − a = 1 ∑ m ϵ a u a ) . Its ( m + 1 ) (m+1) ( m + 1 ) t t t u a = w a − t u_a=w_a-t u a = w a − t ( u , t ) (u,t) ( u , t ) ν \nu ν ν − 1 / 2 \nu^{-1/2} ν − 1/2 J m J_m J m
We next prove detection of every single-register error. Since ∑ a ϵ a = 0 \sum_a\epsilon_a=0 ∑ a ϵ a = 0
∑ a = 1 m ϵ a w a − w m + 1 + w m + 2 = 0 ( m o d ν ) . \sum_{a=1}^{m}\epsilon_a w_a-w_{m+1}+w_{m+2}=0\pmod{\nu}. a = 1 ∑ m ϵ a w a − w m + 1 + w m + 2 = 0 ( mod ν ) . Consider a nonidentity Weyl operator Q Q Q a a a b b b
If a ≠ 0 a \ne0 a = 0 w w w − 1 -1 − 1 a a a − a -a − a ν \nu ν
J m † Q J m = 0. J_m^\dagger Q J_m = 0. J m † Q J m = 0. If a = 0 a = 0 a = 0 b ≠ 0 b \ne0 b = 0 Q Q Q j j j w j ( u , t ) = w j ( u , 0 ) + t w_j(u,t) = w_j(u,0) + t w j ( u , t ) = w j ( u , 0 ) + t
⟨ u ∣ J m † Q J m ∣ u ⟩ = e 2 π i b w j ( u , 0 ) / ν ν ∑ t = 0 ν − 1 e 2 π i b t / ν = 0. \langle u|J_m^\dagger Q J_m|u\rangle= \frac{e^{2\pi i b w_j(u,0)/\nu}}{\nu}\sum_{t=0}^{\nu-1}e^{2\pi i b t/\nu}=0. ⟨ u ∣ J m † Q J m ∣ u ⟩ = ν e 2 π ib w j ( u , 0 ) / ν t = 0 ∑ ν − 1 e 2 π ib t / ν = 0. The last equality follows because b ≠ 0 b \ne0 b = 0 ν \nu ν J m † Q J m = 0 J_m^\dagger Q J_m = 0 J m † Q J m = 0 O O O
J m † O J m = λ O I . J_m^\dagger O J_m = \lambda_O I. J m † O J m = λ O I . Thus the register distance is at least two.
To prove Eq. (28) , evaluate the physical phase exponent on each basis label:
z ~ w ( u , t ) T = ∑ a = 1 m z a ( u a + t ) − ( ∑ a = 1 m z a ) t = ∑ a = 1 m z a u a . \begin{aligned}
\widetilde{z}w(u,t)^{\mathsf T}
= \sum_{a=1}^{m} z_a(u_a+t)-\left(\sum_{a=1}^{m}z_a\right)t \\
&= \sum_{a=1}^{m}z_a u_a.
\end{aligned} z w ( u , t ) T = a = 1 ∑ m z a ( u a + t ) − ( a = 1 ∑ m z a ) t = a = 1 ∑ m z a u a . This exponent is independent of t t t J m ∣ u ⟩ J_m|u\rangle J m ∣ u ⟩ ⨂ a = 1 m Z ν ( z a ) \bigotimes_{a=1}^{m} Z_\nu(z_a) ⨂ a = 1 m Z ν ( z a ) ∣ u ⟩ |u\rangle ∣ u ⟩
Finally, choose z = ( 1 , 0 , … , 0 ) z=(1,0,\ldots,0) z = ( 1 , 0 , … , 0 ) Z ν ( 1 ) Z_\nu(1) Z ν ( 1 ) Z ν ( − 1 ) Z_\nu(-1) Z ν ( − 1 ) m + 1 m+1 m + 1 (28) , its action on the encoded inputs is Z ν ( 1 ) Z_\nu(1) Z ν ( 1 )
Apply J m J_m J m dig \operatorname{dig} dig (28) preserves the logical phase operations.
Lemma S5. For even m m m E = dig ∘ J m ⊗ n ∘ E o u t E=\operatorname{dig}\circ J_m^{\otimes n}\circ E_{\mathrm{out}} E = dig ∘ J m ⊗ n ∘ E out
N = h ( m + 2 ) n , K = m ( a 1 − a 0 ) , D ≥ 2 min { n − a 1 , a 0 + 2 − 2 g } . (29) N=h(m+2)n,\qquad K=m(a_1-a_0),\qquad D\ge2\min\{n-a_1,a_0+2-2g\}. \tag*{(29)} N = h ( m + 2 ) n , K = m ( a 1 − a 0 ) , D ≥ 2 min { n − a 1 , a 0 + 2 − 2 g } . (29) Proof. Each stage of the encoding is an isometry. The inner encoding replaces each block of m m m m + 2 m+2 m + 2 h h h N = h ( m + 2 ) n N=h(m+2)n N = h ( m + 2 ) n K = m ( a 1 − a 0 ) K=m(a_1-a_0) K = m ( a 1 − a 0 )
Let d ∗ d_{*} d ∗ (26) , and consider a physical Pauli operator Q Q Q w < 2 d ∗ w < 2d_{*} w < 2 d ∗ J m † ( ⋅ ) J m J_m^{\dagger}(\cdot)J_m J m † ( ⋅ ) J m ⌊ w / 2 ⌋ < d ∗ \lfloor w/2 \rfloor< d_{*} ⌊ w /2 ⌋ < d ∗
E † Q E = λ Q I . E^{\dagger}QE = \lambda_Q I. E † QE = λ Q I . Thus every physical Pauli operator of weight below 2 d ∗ 2d_{*} 2 d ∗ D ≥ 2 d ∗ D \ge2d_{*} D ≥ 2 d ∗
For addressability, let z ℓ T z_{\ell}^{\mathsf{T}} z ℓ T r o u t + ℓ r_{\mathrm{out}}+\ell r out + ℓ B o u t − 1 B_{\mathrm{out}}^{-1} B out − 1
( v G + x L ) z ℓ T = x ℓ , (vG+xL)z_{\ell}^{\mathsf{T}}=x_{\ell}, ( v G + xL ) z ℓ T = x ℓ , so the register phases with exponents z ℓ z_{\ell} z ℓ Z ( h ) Z(h) Z ( h ) ℓ \ell ℓ
Partition z ℓ z_{\ell} z ℓ z ℓ , p ∈ R h m z_{\ell,p}\in\mathcal{R}_{h}^{m} z ℓ , p ∈ R h m
z ~ ℓ , p = ( z ℓ , p , − ∑ j = 1 m z ℓ , p , j , 0 ) . \widetilde{z}_{\ell,p}=\left(z_{\ell,p},-\sum_{j=1}^{m}z_{\ell,p,j},0\right). z ℓ , p = ( z ℓ , p , − j = 1 ∑ m z ℓ , p , j , 0 ) . Eq. (28) preserves this logical action through the inner encoding. The digit-phase identity in Eq. (24) then gives the physical addressing matrix
F ℓ , ( p , j , b ) = 2 b z ~ ℓ , p , j ( m o d ν ) , (30) F_{\ell,(p,j,b)}=2^{b}\widetilde{z}_{\ell,p,j}\pmod{\nu}, \tag*{(30)} F ℓ , ( p , j , b ) = 2 b z ℓ , p , j ( mod ν ) , (30) where 1 ≤ ℓ ≤ K 1\le\ell\le K 1 ≤ ℓ ≤ K 1 ≤ p ≤ n 1\le p\le n 1 ≤ p ≤ n 1 ≤ j ≤ m + 2 1\le j\le m+2 1 ≤ j ≤ m + 2 0 ≤ b < h 0\le b<h 0 ≤ b < h c ∈ R h K c\in\mathcal{R}_{h}^{K} c ∈ R h K c ℓ c_{\ell} c ℓ c F cF c F c c c (17) .
Finally, introduce an auxiliary input v ∈ R h v\in\mathcal{R}_{h} v ∈ R h
( u 1 , … , u m , t , v ) ⟼ ( u 1 + t , … , u m + t , t , t − ∑ a = 1 m ϵ a u a + v ) . (31) (u_{1},\ldots,u_{m},t,v)\longmapsto\left(u_{1}+t,\ldots,u_{m}+t,t,t-\sum_{a=1}^{m}\epsilon_{a}u_{a}+v\right). \tag*{(31)} ( u 1 , … , u m , t , v ) ⟼ ( u 1 + t , … , u m + t , t , t − a = 1 ∑ m ϵ a u a + v ) . (31) Output coordinate m + 1 m+1 m + 1 t t t u u u v v v (31) defines a unitary permutation of the computational basis. Preparing the two auxiliary registers in a uniform superposition and ∣ 0 ⟩ \lvert0\rangle ∣ 0 ⟩
∣ u ⟩ ⊗ ( ν − 1 / 2 ∑ t ∈ R h ∣ t ⟩ ) ⊗ ∣ 0 ⟩ \lvert u\rangle\otimes\left(\nu^{-1/2}\sum_{t\in\mathcal{R}_{h}}\lvert t\rangle\right)\otimes\lvert0\rangle ∣ u ⟩ ⊗ ( ν − 1/2 t ∈ R h ∑ ∣ t ⟩ ) ⊗ ∣ 0 ⟩ ⟼ ν − 1 / 2 ∑ t ∈ R h ∣ u 1 + t , … , u m + t , t , t − ∑ a = 1 m ϵ a u a ⟩ = J m ∣ u ⟩ , \begin{aligned}
\longmapsto\nu^{-1/2}\sum_{t\in\mathcal{R}_{h}}\left\lvert u_{1}+t,\ldots,u_{m}+t,t,t-\sum_{a=1}^{m}\epsilon_{a}u_{a}\right\rangle\\
&=J_m\lvert u\rangle,
\end{aligned} ⟼ ν − 1/2 t ∈ R h ∑ u 1 + t , … , u m + t , t , t − a = 1 ∑ m ϵ a u a ⟩ = J m ∣ u ⟩ , where the last equality is the definition in Eq. (27) . By linearity, the same implementation applies to any input state. Composing these maps with B o u t B_{\mathrm{out}} B out (19) with
A = 0 , M = ( m + 2 ) n , r = r o u t + n , A=0,\qquad M=(m+2)n,\qquad r=r_{\mathrm{out}}+n, A = 0 , M = ( m + 2 ) n , r = r out + n , and K K K
Tower parameters and rate–distance bounds
Fix h ≥ 3 h \ge3 h ≥ 3 (29) to choose the AG degrees and derive the asymptotic guarantees. For a fixed positive even m m m F 2 m \mathbb{F}_{2^m} F 2 m n i → ∞ n_i \to\infty n i → ∞ g i / n i → κ g_i/n_i \to\kappa g i / n i → κ 2 κ < β 0 < β 1 < 1 2\kappa< \beta_0 < \beta_1 < 1 2 κ < β 0 < β 1 < 1 a b , i = ⌊ β b n i ⌋ a_{b,i} = \lfloor\beta_b n_i \rfloor a b , i = ⌊ β b n i ⌋ b = 0 , 1 b = 0,1 b = 0 , 1 2 g i − 2 < a 0 , i < a 1 , i < n i 2g_i - 2 < a_{0,i} < a_{1,i} < n_i 2 g i − 2 < a 0 , i < a 1 , i < n i i i i 1 / n i 1/n_i 1/ n i (29) gives
lim i K i N i = m ( β 1 − β 0 ) h ( m + 2 ) , lim inf i D i N i ≥ 2 min { 1 − β 1 , β 0 − 2 κ } h ( m + 2 ) . (32) \lim_{i} \frac{K_i}{N_i} = \frac{m(\beta_1-\beta_0)}{h(m+2)},
\qquad
\liminf_{i} \frac{D_i}{N_i} \ge\frac{2\min\{1-\beta_1,\beta_0-2\kappa\}}{h(m+2)}. \tag*{(32)} i lim N i K i = h ( m + 2 ) m ( β 1 − β 0 ) , i lim inf N i D i ≥ h ( m + 2 ) 2 min { 1 − β 1 , β 0 − 2 κ } . (32) For the family constructed in § 2 A of the main text, take m = 8 m=8 m = 8 F 256 \mathbb{F}_{256} F 256 [16 , 17 ] :
F 1 = F 256 ( x 1 ) , F i + 1 = F i ( x i + 1 ) , x i + 1 16 + x i + 1 = x i 16 x i 15 + 1 . F_1=\mathbb{F}_{256}(x_1), \qquad F_{i+1}=F_i(x_{i+1}), \qquad x_{i+1}^{16}+x_{i+1}=\frac{x_i^{16}}{x_i^{15}+1}. F 1 = F 256 ( x 1 ) , F i + 1 = F i ( x i + 1 ) , x i + 1 16 + x i + 1 = x i 15 + 1 x i 16 . At level i ≥ 1 i \ge1 i ≥ 1 x 1 = α ∈ F 256 ∖ F 16 x_1=\alpha\in\mathbb{F}_{256}\setminus\mathbb{F}_{16} x 1 = α ∈ F 256 ∖ F 16 n i = 240 ⋅ 16 i − 1 n_i=240\cdot16^{i-1} n i = 240 ⋅ 1 6 i − 1 P ∞ , i P_{\infty,i} P ∞ , i x 1 x_1 x 1 [16 , 17 ]
g i = { ( 16 i / 2 − 1 ) 2 , i even , ( 16 ( i + 1 ) / 2 − 1 ) ( 16 ( i − 1 ) / 2 − 1 ) , i odd . g_i =
\begin{cases}
(16^{i/2}-1)^2, & i\ \text{even},\\
(16^{(i+1)/2}-1)(16^{(i-1)/2}-1), & i\ \text{odd}.
\end{cases} g i = { ( 1 6 i /2 − 1 ) 2 , ( 1 6 ( i + 1 ) /2 − 1 ) ( 1 6 ( i − 1 ) /2 − 1 ) , i even , i odd . Hence g i ≤ 16 i = n i / 15 g_i \le16^i=n_i/15 g i ≤ 1 6 i = n i /15 g i / n i → 1 / 15 g_i/n_i \to1/15 g i / n i → 1/15
At m = 8 m=8 m = 8 κ = 1 / 15 \kappa=1/15 κ = 1/15 (32) requires
4 ( β 1 − β 0 ) = 1 − β 1 = β 0 − 2 / 15. 4(\beta_1-\beta_0)=1-\beta_1=\beta_0-2/15. 4 ( β 1 − β 0 ) = 1 − β 1 = β 0 − 2/15. Solving gives β 0 = 70 / 135 \beta_0=70/135 β 0 = 70/135 β 1 = 83 / 135 \beta_1=83/135 β 1 = 83/135
a 0 , i = ⌊ 70 n i 135 ⌋ , a 1 , i = ⌊ 83 n i 135 ⌋ . (33) a_{0,i}=\left\lfloor\frac{70n_i}{135}\right\rfloor,\qquad
a_{1,i}=\left\lfloor\frac{83n_i}{135}\right\rfloor. \tag*{(33)} a 0 , i = ⌊ 135 70 n i ⌋ , a 1 , i = ⌊ 135 83 n i ⌋ . (33) The genus bound and rounding give
a 1 , i − a 0 , i ≥ 13 n i 135 − 1 , a_{1,i}-a_{0,i}\ge\frac{13n_i}{135}-1, a 1 , i − a 0 , i ≥ 135 13 n i − 1 , n i − a 1 , i ≥ 52 n i 135 , n_i-a_{1,i}\ge\frac{52n_i}{135}, n i − a 1 , i ≥ 135 52 n i , a 0 , i + 2 − 2 g i ≥ 52 n i 135 + 1. a_{0,i}+2-2g_i\ge\frac{52n_i}{135}+1. a 0 , i + 2 − 2 g i ≥ 135 52 n i + 1. Since n i ≥ 240 n_i\ge240 n i ≥ 240 2 g i − 2 < a 0 , i < a 1 , i < n i 2g_i-2<a_{0,i}<a_{1,i}<n_i 2 g i − 2 < a 0 , i < a 1 , i < n i i ≥ 1 i\ge1 i ≥ 1 (29) , with N i = 10 h n i N_i=10hn_i N i = 10 h n i
K i N i ≥ 52 675 h − 4 5 h n i , D i N i ≥ 52 675 h . \frac{K_i}{N_i}\ge\frac{52}{675h}-\frac{4}{5hn_i},
\qquad
\frac{D_i}{N_i}\ge\frac{52}{675h}. N i K i ≥ 675 h 52 − 5 h n i 4 , N i D i ≥ 675 h 52 . Moreover, ( a 1 , i − a 0 , i ) / n i → 13 / 135 (a_{1,i}-a_{0,i})/n_i\to13/135 ( a 1 , i − a 0 , i ) / n i → 13/135
lim i K i N i = 52 675 h , lim inf i D i N i ≥ 52 675 h . \lim_i\frac{K_i}{N_i}=\frac{52}{675h},
\qquad
\liminf_i\frac{D_i}{N_i}\ge\frac{52}{675h}. i lim N i K i = 675 h 52 , i lim inf N i D i ≥ 675 h 52 . as stated in § 2 A of the main text. The encoding and addressing matrices are generated by the polynomial-time procedure in Sec. S2.4.
To obtain the rate–distance tradeoff, keep the same tower and choose
β 0 = 2 15 + τ d , β 1 = 1 − τ d , 0 < τ d < 13 30 . \beta_0 = \frac{2}{15}+\tau_d,\qquad\beta_1 = 1-\tau_d,\qquad0<\tau_d<\frac{13}{30}. β 0 = 15 2 + τ d , β 1 = 1 − τ d , 0 < τ d < 30 13 . The two distance terms are then equal, and Eq. (32) becomes
R ∞ = 4 5 h ( 13 15 − 2 τ d ) , d ∞ ≥ τ d 5 h . R_\infty=\frac{4}{5h}\left(\frac{13}{15}-2\tau_d\right),\qquad d_\infty\geq\frac{\tau_d}{5h}. R ∞ = 5 h 4 ( 15 13 − 2 τ d ) , d ∞ ≥ 5 h τ d . Thus a positive target pair ( R 0 , d 0 ) (R_0,d_0) ( R 0 , d 0 )
5 4 h R 0 + 10 h d 0 < 13 15 . (34) \frac{5}{4}hR_0+10hd_0<\frac{13}{15}. \tag*{(34)} 4 5 h R 0 + 10 h d 0 < 15 13 . (34) Indeed, this condition makes the interval
5 h d 0 < τ d < 13 30 − 5 h 8 R 0 5hd_0<\tau_d<\frac{13}{30}-\frac{5h}{8}R_0 5 h d 0 < τ d < 30 13 − 8 5 h R 0 nonempty. Choosing a rational τ d \tau_d τ d τ d = 52 / 135 \tau_d=52/135 τ d = 52/135 h = 3 h=3 h = 3 52 / 2025 52/2025 52/2025
Finally, we compare the common guarantees supplied by Eq. (32) for fixed square binary fields. For a fixed genus ratio κ \kappa κ η > 0 \eta>0 η > 0
β 1 − β 0 ≥ h ( m + 2 ) m η , 1 − β 1 ≥ h ( m + 2 ) 2 η , β 0 − 2 κ ≥ h ( m + 2 ) 2 η . \beta_1-\beta_0\geq\frac{h(m+2)}{m}\eta,\qquad1-\beta_1\geq\frac{h(m+2)}{2}\eta,\qquad\beta_0-2\kappa\geq\frac{h(m+2)}{2}\eta. β 1 − β 0 ≥ m h ( m + 2 ) η , 1 − β 1 ≥ 2 h ( m + 2 ) η , β 0 − 2 κ ≥ 2 h ( m + 2 ) η . The left sides sum to 1 − 2 κ 1-2\kappa 1 − 2 κ
η A G , m = m ( 1 − 2 κ ) h ( m + 1 ) ( m + 2 ) , 2 κ < 1. (35) \eta_{\mathrm{AG},m}=\frac{m(1-2\kappa)}{h(m+1)(m+2)},\qquad2\kappa<1. \tag*{(35)} η AG , m = h ( m + 1 ) ( m + 2 ) m ( 1 − 2 κ ) , 2 κ < 1. (35) It is attained by taking β 0 = 2 κ + τ d \beta_0=2\kappa+\tau_d β 0 = 2 κ + τ d β 1 = 1 − τ d \beta_1=1-\tau_d β 1 = 1 − τ d τ d = m ( 1 − 2 κ ) / ( 2 ( m + 1 ) ) \tau_d=m(1-2\kappa)/(2(m+1)) τ d = m ( 1 − 2 κ ) / ( 2 ( m + 1 ))
The Drinfeld–Vlăduţ bound [26 ] gives κ ≥ ( 2 m / 2 − 1 ) − 1 \kappa\geq(2^{m/2}-1)^{-1} κ ≥ ( 2 m /2 − 1 ) − 1 m = 4 , 6 , 8 m=4,6,8 m = 4 , 6 , 8 h η A G , m h\eta_{\mathrm{AG},m} h η AG , m 2 / 45 2/45 2/45 15 / 196 15/196 15/196 52 / 675 52/675 52/675 m = 2 m=2 m = 2 2 κ < 1 2\kappa<1 2 κ < 1 m ≥ 10 m\geq10 m ≥ 10
h η A G , m < m ( m + 1 ) ( m + 2 ) ≤ 5 66 < 52 675 . h\eta_{\mathrm{AG},m}<\frac{m}{(m+1)(m+2)}\leq\frac{5}{66}<\frac{52}{675}. h η AG , m < ( m + 1 ) ( m + 2 ) m ≤ 66 5 < 675 52 . The F 256 \mathbb{F}_{256} F 256 κ = 1 / 15 \kappa=1/15 κ = 1/15 (32) with the inner encodings J m J_m J m
Polynomial-time generation
For the balanced family, the input is a fixed phase level h h h i i i F F F (29) at m = 8 m=8 m = 8
Construct the F 256 \mathbb{F}_{256} F 256 n i n_i n i P ∞ , i P_{\infty,i} P ∞ , i (33) .
Lift the basis to B o u t \mathrm{B}_{\mathrm{out}} B out R h \mathcal{R}_h R h r o u t = 8 ( a 0 , i + 1 − g i ) r_{\mathrm{out}} = 8(a_{0,i} + 1 - g_i) r out = 8 ( a 0 , i + 1 − g i ) K i = 8 ( a 1 , i − a 0 , i ) K_i = 8(a_{1,i} - a_{0,i}) K i = 8 ( a 1 , i − a 0 , i ) G G G L L L B o u t − 1 \mathrm{B}_{\mathrm{out}}^{-1} B out − 1
For each block, use the invertible ring-linear transformation defined in Eq. (31) with m = 8 m = 8 m = 8 B o u t \mathrm{B}_{\mathrm{out}} B out F F F (30) .
The resulting cyclic encoding has A = 0 A = 0 A = 0 M i = 10 n i M_i = 10n_i M i = 10 n i r i = r o u t + n i r_i = r_{\mathrm{out}} + n_i r i = r out + n i N i = 10 h n i N_i = 10hn_i N i = 10 h n i [18 ] computes the evaluation matrices in O ( n i 3 ( log n i ) 3 ) O(n_i^3(\log n_i)^3) O ( n i 3 ( log n i ) 3 ) n i n_i n i h h h N i N_i N i ∣ 0 ⟩ \lvert0 \rangle ∣ 0 ⟩ h h h t t t (31) with m = 8 m = 8 m = 8 O h ( N i 2 ) O_h(N_i^2) O h ( N i 2 )
Counting proof and certified exponential construction
We prove existence by showing that a random encoding has positive probability of satisfying a sufficient condition for distance at least d d d
Throughout this section, 1 ≤ K < M 1 \le K < M 1 ≤ K < M 1 ≤ d ≤ M 1 \le d \le M 1 ≤ d ≤ M r = M − K r = M - K r = M − K ν = 2 h \nu= 2^h ν = 2 h B \mathrm{B} B G L M ( R h ) \mathrm{GL}_M(\mathcal{R}_h) GL M ( R h ) A A A (19) . The first r r r K K K B \mathrm{B} B G G G L L L ( a , b ) (a,b) ( a , b ) 2 s 2^s 2 s 2 s ( a , b ) = 0 2^s(a,b) = 0 2 s ( a , b ) = 0 2 s − 1 ( a , b ) ≠ 0 2^{s-1}(a,b) \ne0 2 s − 1 ( a , b ) = 0 2 h − s 2^{h-s} 2 h − s
B ( n , d ) = ∑ w = 1 d − 1 ( n w ) , B(n,d) = \sum_{w=1}^{d-1} \binom{n}{w}, B ( n , d ) = w = 1 ∑ d − 1 ( w n ) , V s ( M , d ) = ∑ w = 1 d − 1 ( 4 s − 1 ) w ( M w ) , V_s(M,d) = \sum_{w=1}^{d-1} (4^s - 1)^w \binom{M}{w}, V s ( M , d ) = w = 1 ∑ d − 1 ( 4 s − 1 ) w ( w M ) , W h ( M , d ) = ∑ w = 1 d − 1 [ z w ] ( 1 + c h z + ( 15 − c h ) z 2 ) M , c h = { 10 , h = 2 , 3 , 6 , h ≥ 4. (36) W_h(M,d) = \sum_{w=1}^{d-1} [z^w](1 + c_h z + (15-c_h)z^2)^M,\qquad c_h =
\begin{cases}
10, & h = 2,3,\\
6, & h \ge4.
\end{cases} \tag*{(36)} W h ( M , d ) = w = 1 ∑ d − 1 [ z w ] ( 1 + c h z + ( 15 − c h ) z 2 ) M , c h = { 10 , 6 , h = 2 , 3 , h ≥ 4. (36) The last function is used for h ≥ 2 h \ge2 h ≥ 2
Compression and a common probability bound
Here we derive the rejection test and a probability bound for each fixed Weyl pair. We start by applying Lemma S1 with r + K = M r + K = M r + K = M
Lemma S6. For x , x ′ ∈ { 0 , 1 } K x,x' \in\{0,1\}^{K} x , x ′ ∈ { 0 , 1 } K ⟨ x ′ ∣ E † W ( a , b ) E ∣ x ⟩ \langle x'|E^{\dagger}W(a,b)E|x\rangle ⟨ x ′ ∣ E † W ( a , b ) E ∣ x ⟩
G ( b − a A ) T = 0 , ( a B − 1 ) r + 1 : M = x ′ − x . G(b-aA)^{\mathsf{T}}=0,\qquad(aB^{-1})_{r+1:M}=x'-x. G ( b − a A ) T = 0 , ( a B − 1 ) r + 1 : M = x ′ − x . Proof. Apply Lemma S1 with r + K = M r+K=M r + K = M R h M \mathbb{R}^{M}_{h} R h M C 1 = R h M C_{1}=\mathbb{R}^{M}_{h} C 1 = R h M a ∈ C 1 a\in C_{1} a ∈ C 1
For h ≥ 2 h\geq2 h ≥ 2 d d d G ( b − a A ) T = 0 G(b-aA)^{\mathsf{T}}=0 G ( b − a A ) T = 0 ( a B − 1 ) r + 1 : M = 0 (aB^{-1})_{r+1:M}=0 ( a B − 1 ) r + 1 : M = 0 Table S1 , together with the counting and probability bounds proved below. Let reject denote the event that at least one such pair exists. At exact order two, the logical shift has entries in { 0 , 2 h − 1 } \{0,2^{h-1}\} { 0 , 2 h − 1 } D ≥ d D\geq d D ≥ d E † W ( a , b ) E = 0 E^{\dagger}W(a,b)E=0 E † W ( a , b ) E = 0
Table S1. Costs and bounds for nonzero Weyl pairs of cost below d d d h ≥ 2 h\geq2 h ≥ 2
At exact order two, the cost equals the physical Pauli weight of W ( a , b ) W(a,b) W ( a , b )
For order two, ( a , b ) = 2 h − 1 ( α , β ) (a,b)=2^{h-1}(\alpha,\beta) ( a , b ) = 2 h − 1 ( α , β ) α , β ∈ F 2 M \alpha,\beta\in\mathbb{F}_{2}^{M} α , β ∈ F 2 M ( a j , b j ) = 2 h − 2 ( α j , β j ) (a_j,b_j)=2^{h-2}(\alpha_j,\beta_j) ( a j , b j ) = 2 h − 2 ( α j , β j ) α j , β j ∈ Z / 4 Z \alpha_j,\beta_j\in\mathbb{Z}/4\mathbb{Z} α j , β j ∈ Z /4 Z ( 0 , 0 ) (0,0) ( 0 , 0 ) (37) , and two otherwise.
The following estimate supplies the probability bound for each exact order.
Lemma S7. For a fixed Weyl pair ( a , b ) (a,b) ( a , b ) 2 s 2^s 2 s 1 ≤ s ≤ h 1\leq s\leq h 1 ≤ s ≤ h
Pr [ G ( b − a A ) T = 0 ] ≤ 2 − s r . \Pr\left[G(b-aA)^{\mathsf{T}}=0\right]\leq2^{-sr}. Pr [ G ( b − a A ) T = 0 ] ≤ 2 − sr . Proof. Divide ( a , b ) (a,b) ( a , b ) 2 h − s 2^{h-s} 2 h − s 2 s 2^s 2 s ( a ′ , b ′ ) (a',b') ( a ′ , b ′ )
Suppose a j ′ a'_j a j ′ 2 s 2^s 2 s w w w ( A k ℓ ) k , ℓ ≠ j (A_{k\ell})_{k,\ell\ne j} ( A k ℓ ) k , ℓ = j ℓ ≠ j \ell\ne j ℓ = j
a j ′ A j ℓ = w ℓ − ∑ k ≠ j a k ′ A k ℓ a'_jA_{j\ell}=w_\ell-\sum_{k\ne j}a'_kA_{k\ell} a j ′ A j ℓ = w ℓ − k = j ∑ a k ′ A k ℓ uniquely determines A j ℓ = A ℓ j A_{j\ell}=A_{\ell j} A j ℓ = A ℓ j ( a ′ A ) j = w j (a'A)_j=w_j ( a ′ A ) j = w j A j j A_{jj} A j j 2 s M ( M − 1 ) / 2 2^{sM(M-1)/2} 2 s M ( M − 1 ) /2 a ′ A = w a'A=w a ′ A = w w w w a ′ A a'A a ′ A γ = b ′ − a ′ A \gamma=b'-a'A γ = b ′ − a ′ A
Invertibility of B implies that B γ T B\gamma^{\mathsf{T}} B γ T G γ T = 0 G\gamma^{\mathsf{T}}=0 G γ T = 0 r r r K K K
Pr [ G γ T = 0 ∣ B ] = 2 s K 2 s M = 2 − s r . \Pr\left[G\gamma^{\mathsf{T}}=0\mid B\right]=\frac{2^{sK}}{2^{sM}}=2^{-sr}. Pr [ G γ T = 0 ∣ B ] = 2 s M 2 sK = 2 − sr . Otherwise, all coordinates of a ′ a' a ′ b ′ b' b ′ A A A γ = b ′ − a ′ A \gamma= b' - a'A γ = b ′ − a ′ A
For uniformly chosen invertible B B B B γ T B\gamma^{\mathsf{T}} B γ T B B B
There are 2 s M − 2 ( s − 1 ) M 2^{sM} - 2^{(s-1)M} 2 s M − 2 ( s − 1 ) M G γ T = 0 G\gamma^{\mathsf{T}} = 0 G γ T = 0 r r r K K K
Pr [ G γ T = 0 ∣ A ] = 2 s K − 2 ( s − 1 ) K 2 s M − 2 ( s − 1 ) M = 2 − s r 1 − 2 − K 1 − 2 − M < 2 − s r . \Pr[G\gamma^{\mathsf{T}} = 0 \mid A] = \frac{2^{sK} - 2^{(s-1)K}}{2^{sM} - 2^{(s-1)M}} = 2^{-sr}\frac{1 - 2^{-K}}{1 - 2^{-M}} < 2^{-sr}. Pr [ G γ T = 0 ∣ A ] = 2 s M − 2 ( s − 1 ) M 2 sK − 2 ( s − 1 ) K = 2 − sr 1 − 2 − M 1 − 2 − K < 2 − sr . where the last inequality uses K < M K < M K < M
This proves the bound in both cases.
Order-two errors
We now count Weyl errors of exact order two with weight below d d d d d d α ≠ 0 \alpha\ne0 α = 0 α = 0 \alpha= 0 α = 0 a = 2 h − 1 α a = 2^{h-1}\alpha a = 2 h − 1 α α ∈ { 0 , 1 } M \alpha\in\{0,1\}^{M} α ∈ { 0 , 1 } M
Lemma S8. For h ≥ 2 h \ge2 h ≥ 2 ( a , b ) = 2 h − 1 ( α , β ) (a,b) = 2^{h-1}(\alpha,\beta) ( a , b ) = 2 h − 1 ( α , β ) α , β ∈ { 0 , 1 } M \alpha,\beta\in\{0,1\}^{M} α , β ∈ { 0 , 1 } M wt ( α ) + wt ( β ) < d \operatorname{wt}(\alpha) + \operatorname{wt}(\beta) < d wt ( α ) + wt ( β ) < d α ≠ 0 \alpha\ne0 α = 0 α = 0 \alpha= 0 α = 0 B ( 2 M , d ) B(2M,d) B ( 2 M , d ) B ( M , d ) B(M,d) B ( M , d ) 2 − M 2^{-M} 2 − M 2 − r 2^{-r} 2 − r
Proof. The operators X ν ( 2 h − 1 ) X_\nu(2^{h-1}) X ν ( 2 h − 1 ) Z ν ( 2 h − 1 ) Z_\nu(2^{h-1}) Z ν ( 2 h − 1 ) X X X Z Z Z h ≥ 2 h \ge2 h ≥ 2 W ( a , b ) W(a,b) W ( a , b ) wt ( α ) + wt ( β ) \operatorname{wt}(\alpha) + \operatorname{wt}(\beta) wt ( α ) + wt ( β ) P P P wt ( P ) \operatorname{wt}(P) wt ( P ) wt ( P ) < d \operatorname{wt}(P) < d wt ( P ) < d d d d 2 M 2M 2 M B ( 2 M , d ) B(2M,d) B ( 2 M , d ) α = 0 \alpha= 0 α = 0 M M M B ( M , d ) B(M,d) B ( M , d )
For α ≠ 0 \alpha\ne0 α = 0 B ‾ = B m o d 2 \overline{B} = B \bmod2 B = B mod 2 q = α B ‾ − 1 q = \alpha\overline{B}^{-1} q = α B − 1 ( a B ‾ − 1 ) r + 1 : M = 0 (a\overline{B}^{-1})_{r+1:M} = 0 ( a B − 1 ) r + 1 : M = 0 a = 2 h − 1 α a = 2^{h-1}\alpha a = 2 h − 1 α q r + 1 : M = 0 q_{r+1:M} = 0 q r + 1 : M = 0 B ‾ \overline{B} B q q q F 2 M ∖ { 0 } \mathbb{F}_2^M \setminus\{0\} F 2 M ∖ { 0 } 2 r − 1 2^r - 1 2 r − 1 K K K
Pr [ ( a B ‾ − 1 ) r + 1 : M = 0 ] = 2 r − 1 2 M − 1 < 2 − K . \Pr[(a\overline{B}^{-1})_{r+1:M} = 0] = \frac{2^r - 1}{2^M - 1} < 2^{-K}. Pr [( a B − 1 ) r + 1 : M = 0 ] = 2 M − 1 2 r − 1 < 2 − K . For every fixed B B B s = 1 s = 1 s = 1 α ≠ 0 \alpha\ne0 α = 0
Pr [ G ( b − a A ) T = 0 ∣ B ] = 2 − r . \Pr[G(b-aA)^{\mathsf{T}} = 0 \mid B] = 2^{-r}. Pr [ G ( b − a A ) T = 0 ∣ B ] = 2 − r . Since the shift condition depends only on $B,
Pr [ G ( b − a A ) T = 0 , ( a B ‾ − 1 ) r + 1 : M = 0 ] = 2 − r 2 r − 1 2 M − 1 < 2 − M . \begin{aligned}
\Pr[G(b-aA)^{\mathsf{T}} = 0,\ (a\overline{B}^{-1})_{r+1:M} = 0] \\
&= 2^{-r}\frac{2^r - 1}{2^M - 1} < 2^{-M}.
\end{aligned} Pr [ G ( b − a A ) T = 0 , ( a B − 1 ) r + 1 : M = 0 ] = 2 − r 2 M − 1 2 r − 1 < 2 − M . For α = 0 \alpha= 0 α = 0 a = 0 a = 0 a = 0 G b T = 0 Gb^{\mathsf{T}} = 0 G b T = 0 2 − r 2^{-r} 2 − r
Order-four errors
We next bound the number of order-four Weyl terms that can appear in qubit Pauli errors of weight below d d d
Lemma S9. For h ≥ 2 h \ge2 h ≥ 2 W h ( M , d ) W_h(M,d) W h ( M , d ) d d d 2 − 2 r 2^{-2r} 2 − 2 r
Proof. Write a local Weyl pair as ( a , b ) = 2 h − 2 ( α , β ) (a,b)=2^{h-2}(\alpha,\beta) ( a , b ) = 2 h − 2 ( α , β ) α , β ∈ Z 4 \alpha,\beta\in\mathbb{Z}_4 α , β ∈ Z 4 A h A_h A h ( α , β ) (\alpha,\beta) ( α , β ) P j P_j P j j j j 0 ≤ j < h 0 \le j < h 0 ≤ j < h (18) with M = 1 M=1 M = 1
1 ν Tr ( W ( a , b ) † P j ) = 1 ν ∑ y = 0 ν − 1 e − 2 π i β y / 4 ⟨ y + a ∣ P j ∣ y ⟩ . \frac{1}{\nu}\operatorname{Tr}\left(W(a,b)^\dagger P_j\right)=\frac{1}{\nu}\sum_{y=0}^{\nu-1}e^{-2\pi i\beta y/4}\langle y+a|P_j|y\rangle. ν 1 Tr ( W ( a , b ) † P j ) = ν 1 y = 0 ∑ ν − 1 e − 2 π i β y /4 ⟨ y + a ∣ P j ∣ y ⟩ . We evaluate it using y = ∑ ℓ = 0 h − 1 2 ℓ y ℓ y=\sum_{\ell=0}^{h-1}2^\ell y_\ell y = ∑ ℓ = 0 h − 1 2 ℓ y ℓ y ℓ ∈ { 0 , 1 } y_\ell\in\{0,1\} y ℓ ∈ { 0 , 1 }
For Z j Z_j Z j α = 0 \alpha=0 α = 0 Z j ∣ y ⟩ = ( − 1 ) y j ∣ y ⟩ Z_j|y\rangle=(-1)^{y_j}|y\rangle Z j ∣ y ⟩ = ( − 1 ) y j ∣ y ⟩
1 ν ( 1 − e − 2 π i β 2 j / 4 ) ∏ 0 ≤ ℓ < h ℓ ≠ j ( 1 + e − 2 π i β 2 ℓ / 4 ) . \frac{1}{\nu}\left(1-e^{-2\pi i\beta2^j/4}\right)\prod_{\substack{0\le\ell<h\\\ell\ne j}}\left(1+e^{-2\pi i\beta2^\ell/4}\right). ν 1 ( 1 − e − 2 π i β 2 j /4 ) 0 ≤ ℓ < h ℓ = j ∏ ( 1 + e − 2 π i β 2 ℓ /4 ) . It is nonzero exactly for ( j , β ) = ( 0 , 2 ) , ( 1 , 1 ) , ( 1 , 3 ) (j,\beta)=(0,2),(1,1),(1,3) ( j , β ) = ( 0 , 2 ) , ( 1 , 1 ) , ( 1 , 3 ) j ≥ 2 j\ge2 j ≥ 2 j = 0 , 1 j=0,1 j = 0 , 1 β ∈ Z 4 \beta\in\mathbb{Z}_4 β ∈ Z 4 Z j Z_j Z j α = 0 \alpha=0 α = 0 ( α , β ) = ( 0 , 2 ) (\alpha,\beta)=(0,2) ( α , β ) = ( 0 , 2 ) Z 0 Z_0 Z 0 ( α , β ) = ( 0 , 1 ) , ( 0 , 3 ) (\alpha,\beta)=(0,1),(0,3) ( α , β ) = ( 0 , 1 ) , ( 0 , 3 ) Z 1 Z_1 Z 1
For X j X_j X j Y j Y_j Y j j < h − 1 j<h-1 j < h − 1 a = ± 2 j a=\pm2^j a = ± 2 j a ≡ 0 ( m o d 2 h − 2 ) a\equiv0\pmod{2^{h-2}} a ≡ 0 ( mod 2 h − 2 ) j = h − 2 j=h-2 j = h − 2 α = 1 \alpha=1 α = 1 3 3 3 y j y_j y j ν − 1 \nu^{-1} ν − 1
∏ 0 ≤ ℓ < h ℓ ≠ h − 2 ( 1 + e − 2 π i β 2 ℓ / 4 ) . \prod_{\substack{0\le\ell<h\\\ell\ne h-2}}\left(1+e^{-2\pi i\beta2^\ell/4}\right). 0 ≤ ℓ < h ℓ = h − 2 ∏ ( 1 + e − 2 π i β 2 ℓ /4 ) . The X j X_j X j Y j Y_j Y j h = 2 h=2 h = 2 1 + ( − 1 ) β 1+(-1)^\beta 1 + ( − 1 ) β h = 3 h=3 h = 3 2 ( 1 + e − π i β / 2 ) 2(1+e^{-\pi i\beta/2}) 2 ( 1 + e − π i β /2 ) h ≥ 4 h\ge4 h ≥ 4 2 h − 3 ( 1 + e − π i β / 2 ) ( 1 + ( − 1 ) β ) 2^{h-3}(1+e^{-\pi i\beta/2})(1+(-1)^\beta) 2 h − 3 ( 1 + e − π i β /2 ) ( 1 + ( − 1 ) β )
α ∈ { 1 , 3 } , β ∈ { { 0 , 2 } , h = 2 , { 0 , 1 , 3 } , h = 3 , { 0 } , h ≥ 4. \alpha\in\{1,3\},\qquad
\beta\in
\begin{cases}
\{0,2\}, & h=2,\\
\{0,1,3\}, & h=3,\\
\{0\}, & h\ge4.
\end{cases} α ∈ { 1 , 3 } , β ∈ ⎩ ⎨ ⎧ { 0 , 2 } , { 0 , 1 , 3 } , { 0 } , h = 2 , h = 3 , h ≥ 4. At the highest digit, X h − 1 = X ν ( 2 h − 1 ) X_{h-1}=X_\nu(2^{h-1}) X h − 1 = X ν ( 2 h − 1 ) ( α , β ) = ( 2 , 0 ) (\alpha,\beta)=(2,0) ( α , β ) = ( 2 , 0 ) Y h − 1 = i X h − 1 Z h − 1 Y_{h-1}=iX_{h-1}Z_{h-1} Y h − 1 = i X h − 1 Z h − 1 Z h − 1 Z_{h-1} Z h − 1 h = 2 h=2 h = 2 β ∈ { 1 , 3 } \beta\in\{1,3\} β ∈ { 1 , 3 } ( 2 , 1 ) , ( 2 , 3 ) (2,1),(2,3) ( 2 , 1 ) , ( 2 , 3 )
Combining the three cases gives
A h = { { ( 0 , 1 ) , ( 0 , 2 ) , ( 0 , 3 ) , ( 2 , 0 ) , ( 2 , 1 ) , ( 2 , 3 ) } ∪ { 1 , 3 } × { 0 , 2 } , h = 2 , { ( 0 , 1 ) , ( 0 , 2 ) , ( 0 , 3 ) , ( 2 , 0 ) } ∪ { 1 , 3 } × { 0 , 1 , 3 } , h = 3 , { 0 } × { 1 , 2 , 3 } ∪ { 1 , 2 , 3 } × { 0 } , h ≥ 4. (37) A_h=
\begin{cases}
\{(0,1),(0,2),(0,3),(2,0),(2,1),(2,3)\}\cup\{1,3\}\times\{0,2\}, & h=2,\\
\{(0,1),(0,2),(0,3),(2,0)\}\cup\{1,3\}\times\{0,1,3\}, & h=3,\\
\{0\}\times\{1,2,3\}\cup\{1,2,3\}\times\{0\}, & h\ge4.
\end{cases} \tag*{(37)} A h = ⎩ ⎨ ⎧ {( 0 , 1 ) , ( 0 , 2 ) , ( 0 , 3 ) , ( 2 , 0 ) , ( 2 , 1 ) , ( 2 , 3 )} ∪ { 1 , 3 } × { 0 , 2 } , {( 0 , 1 ) , ( 0 , 2 ) , ( 0 , 3 ) , ( 2 , 0 )} ∪ { 1 , 3 } × { 0 , 1 , 3 } , { 0 } × { 1 , 2 , 3 } ∪ { 1 , 2 , 3 } × { 0 } , h = 2 , h = 3 , h ≥ 4. (37) The groups with α = 0 \alpha= 0 α = 0 α = 2 \alpha= 2 α = 2 α ∈ { 1 , 3 } \alpha\in\{1,3\} α ∈ { 1 , 3 }
c h = ∣ A h ∣ = { 3 + 3 + 4 = 10 , h = 2 , 3 + 1 + 6 = 10 , h = 3 , 3 + 1 + 2 = 6 , h ≥ 4. c_h = |\mathcal{A}_h| =
\begin{cases}
3+3+4=10, & h=2,\\
3+1+6=10, & h=3,\\
3+1+2=6, & h\geq4.
\end{cases} c h = ∣ A h ∣ = ⎩ ⎨ ⎧ 3 + 3 + 4 = 10 , 3 + 1 + 6 = 10 , 3 + 1 + 2 = 6 , h = 2 , h = 3 , h ≥ 4. For counting, assign local weight zero to the identity, one to pairs in A h \mathcal{A}_h A h A h \mathcal{A}_h A h d d d d d d
Each register has one pair of weight zero, c h c_h c h 15 − c h 15-c_h 15 − c h d d d
∑ w = 1 d − 1 [ z w ] ( 1 + c h z + ( 15 − c h ) z 2 ) M = W h ( M , d ) . \sum_{w=1}^{d-1}[z^w]\left(1+c_hz+(15-c_h)z^2\right)^M = W_h(M,d). w = 1 ∑ d − 1 [ z w ] ( 1 + c h z + ( 15 − c h ) z 2 ) M = W h ( M , d ) . This count may also include pairs of order two, which only increases the upper bound. For every pair of exact order four, Lemma S7 with s = 2 s=2 s = 2 2 − 2 r 2^{-2r} 2 − 2 r
Higher-order errors
For 3 ≤ s ≤ h 3\leq s\leq h 3 ≤ s ≤ h 2 s 2^s 2 s d d d
Lemma S10. For $3\leq s\leq h$, give a Weyl pair cost equal to its number of nonzero registers. At most V s ( M , d ) V_s(M,d) V s ( M , d ) 2 s 2^s 2 s d d d 2 − s r 2^{-sr} 2 − sr 2 s 2^s 2 s d d d
Proof. A qubit Pauli of weight below d d d d d d d d d w w w ( M w ) \binom{M}{w} ( w M ) ( 4 s − 1 ) w (4^s-1)^w ( 4 s − 1 ) w 1 ≤ w < d 1\leq w<d 1 ≤ w < d V s ( M , d ) V_s(M,d) V s ( M , d ) 2 s 2^s 2 s d d d 2 − s r 2^{-sr} 2 − sr
The total rejection bound
Combining the three error classes gives the finite rejection bound.
Lemma S11. For h ≥ 2 h\geq2 h ≥ 2
Φ h ( M , K , d ) : = 2 − M B ( 2 M , d ) + 2 − r B ( M , d ) + 2 − 2 r W h ( M , d ) + ∑ s = 3 h 2 − s r V s ( M , d ) . (38) \Phi_h(M,K,d) := 2^{-M}B(2M,d)+2^{-r}B(M,d)+2^{-2r}W_h(M,d)+\sum_{s=3}^{h}2^{-sr}V_s(M,d). \tag*{(38)} Φ h ( M , K , d ) := 2 − M B ( 2 M , d ) + 2 − r B ( M , d ) + 2 − 2 r W h ( M , d ) + s = 3 ∑ h 2 − sr V s ( M , d ) . (38) For h = 1 h=1 h = 1
Φ 1 ( M , K , d ) : = 2 − r ∑ w = 1 d − 1 3 w ( M w ) . (39) \Phi_1(M,K,d) := 2^{-r}\sum_{w=1}^{d-1}3^w\binom{M}{w}. \tag*{(39)} Φ 1 ( M , K , d ) := 2 − r w = 1 ∑ d − 1 3 w ( w M ) . (39) In either case, if Φ h < 1 \Phi_h < 1 Φ h < 1 N = h M N = hM N = h M D ≥ d D \ge d D ≥ d
Proof. For h ≥ 2 h \ge2 h ≥ 2 Pr ( r e j e c t ) ≤ Φ h \Pr(\mathrm{reject}) \le\Phi_h Pr ( reject ) ≤ Φ h Φ h < 1 \Phi_h < 1 Φ h < 1
Let P P P d d d ν − M Tr P = 0 \nu^{-M}\operatorname{Tr} P = 0 ν − M Tr P = 0 (18) . Lemmas S8–S10 ensure that every remaining Weyl term with nonzero coefficient satisfies the weight restriction for its order. Since no rejection occurs, Lemma S6 gives E † W ( a , b ) E = 0 E^\dagger W(a,b)E = 0 E † W ( a , b ) E = 0 E † P E = 0 E^\dagger P E = 0 E † P E = 0 D ≥ d D \ge d D ≥ d N = h M N = hM N = h M
For h = 1 h = 1 h = 1 ( a , b ) (a,b) ( a , b ) ∣ supp ( a ) ∪ supp ( b ) ∣ |\operatorname{supp}(a) \cup\operatorname{supp}(b)| ∣ supp ( a ) ∪ supp ( b ) ∣ d d d G ( b − a A ) T = 0 G(b-aA)^{\mathsf T} = 0 G ( b − a A ) T = 0 3 w ( M w ) 3^w\binom{M}{w} 3 w ( w M ) w w w s = 1 s = 1 s = 1 Pr ( r e j e c t ) ≤ Φ 1 \Pr(\mathrm{reject}) \le\Phi_1 Pr ( reject ) ≤ Φ 1 Φ 1 < 1 \Phi_1 < 1 Φ 1 < 1 d d d G ( b − a A ) T ≠ 0 G(b-aA)^{\mathsf T} \ne0 G ( b − a A ) T = 0 E † W ( a , b ) E = 0 E^\dagger W(a,b)E = 0 E † W ( a , b ) E = 0 D ≥ d D \ge d D ≥ d N = M N = M N = M
We next show that the rejection bound decays exponentially when K K K d d d M M M
H 2 ( t ) = − t log 2 t − ( 1 − t ) log 2 ( 1 − t ) , t ⋆ = 2 H 2 − 1 ( 1 / 2 ) . H_2(t) = -t\log_2 t - (1-t)\log_2(1-t), \qquad t_\star= 2H_2^{-1}(1/2). H 2 ( t ) = − t log 2 t − ( 1 − t ) log 2 ( 1 − t ) , t ⋆ = 2 H 2 − 1 ( 1/2 ) . where the inverse is taken on [ 0 , 1 / 2 ] [0,1/2] [ 0 , 1/2 ]
Lemma S12. Fix h ≥ 2 h \ge2 h ≥ 2 0 < t < t ⋆ 0 < t < t_\star 0 < t < t ⋆ c > 0 c > 0 c > 0 M 0 M_0 M 0 h , t h,t h , t M ≥ M 0 M \ge M_0 M ≥ M 0 K = d = ⌈ t M ⌉ K = d = \lceil tM\rceil K = d = ⌈ tM ⌉
Φ h ( M , K , d ) ≤ 2 − c M . \Phi_h(M,K,d) \le2^{-cM}. Φ h ( M , K , d ) ≤ 2 − c M . Proof. Put K = d = ⌈ t M ⌉ K = d = \lceil tM\rceil K = d = ⌈ tM ⌉ r = M − K r = M-K r = M − K d − 1 < t M d-1 < tM d − 1 < tM r > ( 1 − t ) M − 1 r > (1-t)M-1 r > ( 1 − t ) M − 1 (38) separately and show that each decays exponentially in M M M
The standard binary Hamming-ball estimate [24 ] gives
B ( 2 M , d ) ≤ 2 2 M H 2 ( t / 2 ) , B ( M , d ) ≤ 2 M H 2 ( t ) . B(2M,d) \le2^{2M H_2(t/2)}, \qquad B(M,d) \le2^{M H_2(t)}. B ( 2 M , d ) ≤ 2 2 M H 2 ( t /2 ) , B ( M , d ) ≤ 2 M H 2 ( t ) . For the first term, corresponding to order two with α ≠ 0 \alpha\ne0 α = 0
2 − M B ( 2 M , d ) ≤ 2 − M [ 1 − 2 H 2 ( t / 2 ) ] . 2^{-M}B(2M,d) \le2^{-M[1-2H_2(t/2)]}. 2 − M B ( 2 M , d ) ≤ 2 − M [ 1 − 2 H 2 ( t /2 )] . The decay rate 1 − 2 H 2 ( t / 2 ) 1-2H_2(t/2) 1 − 2 H 2 ( t /2 ) t < t ⋆ t < t_\star t < t ⋆
For the second term, corresponding to order two with α = 0 \alpha= 0 α = 0 r > ( 1 − t ) M − 1 r > (1-t)M-1 r > ( 1 − t ) M − 1
2 − r B ( M , d ) ≤ 2 ⋅ 2 − M [ 1 − t − H 2 ( t ) ] . 2^{-r}B(M,d) \le2 \cdot2^{-M[1-t-H_2(t)]}. 2 − r B ( M , d ) ≤ 2 ⋅ 2 − M [ 1 − t − H 2 ( t )] . To verify positivity of this decay rate, put t ˉ = p / q = 221 / 1000 \bar{t} = p/q = 221/1000 t ˉ = p / q = 221/1000
( 2 q ) 2 q > 2 4 q p p ( 2 q − p ) 2 q − p , q q < 2 q − p p p ( q − p ) q − p (2q)^{2q} > 2^{4q}p^p(2q-p)^{2q-p}, \qquad q^q < 2^{q-p}p^p(q-p)^{q-p} ( 2 q ) 2 q > 2 4 q p p ( 2 q − p ) 2 q − p , q q < 2 q − p p p ( q − p ) q − p give 2 H 2 ( t ˉ / 2 ) > 1 2H_2(\bar{t}/2) > 1 2 H 2 ( t ˉ /2 ) > 1 t ˉ + H 2 ( t ˉ ) < 1 \bar{t}+H_2(\bar{t}) < 1 t ˉ + H 2 ( t ˉ ) < 1 t < t ⋆ < t ˉ t < t_\star< \bar{t} t < t ⋆ < t ˉ t + H 2 ( t ) t+H_2(t) t + H 2 ( t ) ( 0 , 1 / 2 ) (0,1/2) ( 0 , 1/2 ) 1 − t − H 2 ( t ) > 0 1-t-H_2(t) > 0 1 − t − H 2 ( t ) > 0 P h ( z ) = 1 + c h z + ( 15 − c h ) z 2 P_h(z)=1+c_hz+(15-c_h)z^2 P h ( z ) = 1 + c h z + ( 15 − c h ) z 2 W h W_h W h (36) . For 0 < x < 1 0<x<1 0 < x < 1 w < d w<d w < d x w ≥ x t M x^w\ge x^{tM} x w ≥ x tM P h ( z ) M P_h(z)^M P h ( z ) M
W h ( M , d ) = ∑ w = 1 d − 1 [ z w ] P h ( z ) M ≤ x − t M P h ( x ) M . W_h(M,d)=\sum_{w=1}^{d-1}[z^w]P_h(z)^M\le x^{-tM}P_h(x)^M. W h ( M , d ) = w = 1 ∑ d − 1 [ z w ] P h ( z ) M ≤ x − tM P h ( x ) M . Choose x = 1 / 36 x=1/36 x = 1/36 c h = 6 , 10 c_h=6,10 c h = 6 , 10
P h ( 1 / 36 ) ≤ 1 + 10 36 + 5 36 2 = 1661 1296 . P_h(1/36)\le1+\frac{10}{36}+\frac{5}{36^2}=\frac{1661}{1296}. P h ( 1/36 ) ≤ 1 + 36 10 + 3 6 2 5 = 1296 1661 . Consequently,
2 − 2 r W h ( M , d ) ≤ 4 2 − M [ 2 − 2 t − log 2 ( 1661 / 1296 ) − t log 2 36 ] . 2^{-2r}W_h(M,d)\le4\,2^{-M[2-2t-\log_2(1661/1296)-t\log_2 36]}. 2 − 2 r W h ( M , d ) ≤ 4 2 − M [ 2 − 2 t − l o g 2 ( 1661/1296 ) − t l o g 2 36 ] . The exact comparison
1661 q 36 p < 2 2 ( q − p ) 1296 q 1661^q36^p<2^{2(q-p)}1296^q 166 1 q 3 6 p < 2 2 ( q − p ) 129 6 q shows that 2 t ˉ + log 2 ( 1661 / 1296 ) + t ˉ log 2 36 < 2 2\bar{t}+\log_2(1661/1296)+\bar{t}\log_2 36<2 2 t ˉ + log 2 ( 1661/1296 ) + t ˉ log 2 36 < 2 t t t t < t ⋆ t<t_\star t < t ⋆
For the fourth contribution, fix 3 ≤ s ≤ h 3\le s\le h 3 ≤ s ≤ h w ≤ d − 1 < t M w\le d-1<tM w ≤ d − 1 < tM
V s ( M , d ) ≤ ( 4 s − 1 ) t M B ( M , d ) ≤ 2 M [ H 2 ( t ) + t log 2 ( 4 s − 1 ) ] . V_s(M,d)\le(4^s-1)^{tM}B(M,d)\le2^{M[H_2(t)+t\log_2(4^s-1)]}. V s ( M , d ) ≤ ( 4 s − 1 ) tM B ( M , d ) ≤ 2 M [ H 2 ( t ) + t l o g 2 ( 4 s − 1 )] . It follows that
2 − s r V s ( M , d ) ≤ 2 s 2 − M [ s − s t − H 2 ( t ) − t log 2 ( 4 s − 1 ) ] . 2^{-sr}V_s(M,d)\le2^s2^{-M[s-st-H_2(t)-t\log_2(4^s-1)]}. 2 − sr V s ( M , d ) ≤ 2 s 2 − M [ s − s t − H 2 ( t ) − t l o g 2 ( 4 s − 1 )] . Using H 2 ( t ) ≤ 1 H_2(t)\le1 H 2 ( t ) ≤ 1 log 2 ( 4 s − 1 ) < 2 s \log_2(4^s-1)<2s log 2 ( 4 s − 1 ) < 2 s t ≤ t ˉ < 2 / 9 t\le\bar{t}<2/9 t ≤ t ˉ < 2/9
s t + H 2 ( t ) + t log 2 ( 4 s − 1 ) < 3 s t + 1 < 2 s 3 + 1 ≤ s . st+H_2(t)+t\log_2(4^s-1)<3st+1<\frac{2s}{3}+1\le s. s t + H 2 ( t ) + t log 2 ( 4 s − 1 ) < 3 s t + 1 < 3 2 s + 1 ≤ s . Thus every higher-order term also has a positive decay rate. This contribution is absent when h = 2 h=2 h = 2
Finally, let δ > 0 \delta>0 δ > 0 h , t h,t h , t 1 + 2 + 4 + ∑ s = 3 h 2 s = 2 h + 1 − 1 1+2+4+\sum_{s=3}^h2^s=2^{h+1}-1 1 + 2 + 4 + ∑ s = 3 h 2 s = 2 h + 1 − 1
Φ h ( M , K , d ) ≤ ( 2 h + 1 − 1 ) 2 − δ M . \Phi_h(M,K,d)\le(2^{h+1}-1)2^{-\delta M}. Φ h ( M , K , d ) ≤ ( 2 h + 1 − 1 ) 2 − δ M . Set c = δ / 2 c=\delta/2 c = δ /2 M 0 M_0 M 0 K < M K<M K < M 2 h + 1 − 1 ≤ 2 c M 2^{h+1}-1\le2^{cM} 2 h + 1 − 1 ≤ 2 c M M ≥ M 0 M\ge M_0 M ≥ M 0 Φ h ( M , K , d ) ≤ 2 − c M \Phi_h(M,K,d)\le2^{-cM} Φ h ( M , K , d ) ≤ 2 − c M
Exponential-time construction algorithm
For fixed h ≥ 2 h\ge2 h ≥ 2 i ≥ 3 i\ge3 i ≥ 3 t i t_i t i 2 − i 2^{-i} 2 − i 0 < t i < 1 / 2 0<t_i<1/2 0 < t i < 1/2 2 H 2 ( t i / 2 ) < 1 2H_2(t_i/2)<1 2 H 2 ( t i /2 ) < 1 0 < t i < t ⋆ 0<t_i<t_\star 0 < t i < t ⋆ t ⋆ t_\star t ⋆ t i t_i t i t i = μ i / q i t_i=\mu_i/q_i t i = μ i / q i q i = 2 i q_i=2^i q i = 2 i μ i ∈ { 1 , … , q i / 2 − 1 } \mu_i\in\{1,\ldots,q_i/2-1\} μ i ∈ { 1 , … , q i /2 − 1 } 2 H 2 ( t i / 2 ) < 1 2H_2(t_i/2)<1 2 H 2 ( t i /2 ) < 1
( 2 q i ) 2 q i < 2 q i μ i 2 q i ( 2 q i − μ i ) 2 q i − μ i . (2q_i)^{2q_i}<2^{q_i}\mu_i^{2q_i}(2q_i-\mu_i)^{2q_i-\mu_i}. ( 2 q i ) 2 q i < 2 q i μ i 2 q i ( 2 q i − μ i ) 2 q i − μ i . With this t i t_i t i M = 2 i , 2 i + 1 , … M = 2^i, 2^i + 1, \ldots M = 2 i , 2 i + 1 , … K = d = ⌈ t i M ⌉ K = d = \lceil t_i M\rceil K = d = ⌈ t i M ⌉ Φ h ( M , K , d ) < 1 / 2 \Phi_h(M,K,d) < 1/2 Φ h ( M , K , d ) < 1/2 ( M i , K i , d i ) (M_i,K_i,d_i) ( M i , K i , d i ) i i i M i M_i M i M i ≥ 2 i M_i \ge2^i M i ≥ 2 i N i = h M i N_i = hM_i N i = h M i i → ∞ i \to\infty i → ∞
By Lemma S11, these parameters admit codes with D i ≥ d i D_i \ge d_i D i ≥ d i K i = d i = ⌈ t i M i ⌉ K_i = d_i = \lceil t_iM_i\rceil K i = d i = ⌈ t i M i ⌉ t i → t ⋆ t_i \to t_\star t i → t ⋆
K i N i = d i N i = t i h + O ( M i − 1 ) ⟶ t ⋆ h . (40) \frac{K_i}{N_i} = \frac{d_i}{N_i} = \frac{t_i}{h} + O(M_i^{-1}) \longrightarrow\frac{t_\star}{h}. \tag*{(40)} N i K i = N i d i = h t i + O ( M i − 1 ) ⟶ h t ⋆ . (40) Thus,
R ∞ = t ⋆ h , d ∞ ≥ t ⋆ h . (41) R_\infty= \frac{t_\star}{h}, \qquad d_\infty\ge\frac{t_\star}{h}. \tag*{(41)} R ∞ = h t ⋆ , d ∞ ≥ h t ⋆ . (41) Since H 2 ( t i / 2 ) H_2(t_i/2) H 2 ( t i /2 ) t i t_i t i μ i \mu_i μ i O ( 2 i ) O(2^i) O ( 2 i ) M i M_i M i Φ h \Phi_h Φ h O h ( M i ) O_h(M_i) O h ( M i ) M i M_i M i N i N_i N i
For S S S h = 2 h = 2 h = 2 H 2 − 1 ( 1 / 2 ) H_2^{-1}(1/2) H 2 − 1 ( 1/2 )
For Z Z Z h = 1 h = 1 h = 1 (39) instead gives R 0 + H 2 ( d 0 ) + d 0 log 2 3 < 1 R_0 + H_2(d_0) + d_0\log_2 3 < 1 R 0 + H 2 ( d 0 ) + d 0 log 2 3 < 1 β ⋆ ∈ ( 0 , 1 / 2 ) \beta_\star\in(0,1/2) β ⋆ ∈ ( 0 , 1/2 ) β ⋆ + H 2 ( β ⋆ ) + β ⋆ log 2 3 = 1 \beta_\star+ H_2(\beta_\star) + \beta_\star\log_2 3 = 1 β ⋆ + H 2 ( β ⋆ ) + β ⋆ log 2 3 = 1 t i = μ i / q i ∈ ( 0 , 1 / 2 ) t_i = \mu_i/q_i \in(0,1/2) t i = μ i / q i ∈ ( 0 , 1/2 )
q i q i 3 μ i < 2 q i − μ i μ i μ i ( q i − μ i ) q i − μ i , q_i^{q_i}3^{\mu_i} < 2^{q_i-\mu_i}\mu_i^{\mu_i}(q_i-\mu_i)^{q_i-\mu_i}, q i q i 3 μ i < 2 q i − μ i μ i μ i ( q i − μ i ) q i − μ i , which is exactly the strict balanced inequality. Then, t i → β ⋆ t_i \to\beta_\star t i → β ⋆ R ∞ = β ⋆ R_\infty= \beta_\star R ∞ = β ⋆ d ∞ ≥ β ⋆ d_\infty\ge\beta_\star d ∞ ≥ β ⋆
Given h h h ( M i , K i , d i ) (M_i,K_i,d_i) ( M i , K i , d i ) B B B A A A F F F N i = h M i N_i = hM_i N i = h M i K i K_i K i D i ≥ d i D_i \ge d_i D i ≥ d i
Lemma S11 proves termination and correctness. There are at most 2 h [ M 2 + M ( M + 1 ) / 2 ] 2^{h[M^2+M(M+1)/2]} 2 h [ M 2 + M ( M + 1 ) /2 ] 2 2 h M 2^{2hM} 2 2 h M 2 O h ( M ) 2^{O_h(M)} 2 O h ( M ) 2 O h ( N 2 ) 2^{O_h(N^2)} 2 O h ( N 2 ) 1 − Φ h 1-\Phi_h 1 − Φ h Φ h < 1 / 2 \Phi_h < 1/2 Φ h < 1/2 2 O h ( N ) 2^{O_h(N)} 2 O h ( N ) B B B F 2 \mathbb{F}_2 F 2 ∏ j = 1 M ( 1 − 2 − j ) > 1 / 4 \prod_{j=1}^{M}(1-2^{-j}) > 1/4 ∏ j = 1 M ( 1 − 2 − j ) > 1/4 N N N
Algorithm S1. Exact code construction in exponential time
Input: fixed h h h M , K , d M,K,d M , K , d Φ h ( M , K , d ) < 1 \Phi_h(M,K,d)<1 Φ h ( M , K , d ) < 1
Output: matrices B , A , F \mathrm{B},\mathrm{A},\mathrm{F} B , A , F ( ( h M , 2 K , ≥ d ) 2 ) ((hM,2^K,\ge d)_2) (( h M , 2 K , ≥ d ) 2 )
Set r = M − K r=M-K r = M − K B \mathrm{B} B A \mathrm{A} A R h \mathcal{R}_h R h B \mathrm{B} B B m o d 2 \mathrm{B}\bmod2 B mod 2 G = B 1 : r , : G=\mathrm{B}_{1:r,:} G = B 1 : r , : B − 1 \mathrm{B}^{-1} B − 1
Enumerate nonzero Weyl pairs ( a , b ) (a,b) ( a , b ) 2 s 2^s 2 s d d d h ≥ 2 h\ge2 h ≥ 2 s = 1 s=1 s = 1 wt ( α ) + wt ( β ) \operatorname{wt}(\alpha)+\operatorname{wt}(\beta) wt ( α ) + wt ( β ) ( a , b ) = 2 h − 1 ( α , β ) (a,b)=2^{h-1}(\alpha,\beta) ( a , b ) = 2 h − 1 ( α , β ) s = 2 s=2 s = 2 s ≥ 3 s\ge3 s ≥ 3 wt s ( a , b ) \operatorname{wt}_s(a,b) wt s ( a , b ) h h h h = 1 h=1 h = 1
For h ≥ 2 h\ge2 h ≥ 2 s = 1 s=1 s = 1 ( a B − 1 ) r + 1 : M ≠ 0 (a\mathrm{B}^{-1})_{r+1:M}\ne0 ( a B − 1 ) r + 1 : M = 0 G ( b − a A ) T = 0 ( m o d 2 h ) G(b-a\mathrm{A})^{\mathsf T}=0\pmod{2^h} G ( b − a A ) T = 0 ( mod 2 h )
Return the first matrix pair for which no Weyl pair triggers rejection, using Eq. (19) and the addressing matrix in Eq. (20) .
Rate-distance upper bounds from addressability
Let A 2 ( n , d ) A_2(n,d) A 2 ( n , d ) n n n d d d 2 k 2^k 2 k k / n k/n k / n n − 1 log 2 A 2 ( n , d ) n^{-1}\log_2 A_2(n,d) n − 1 log 2 A 2 ( n , d ) t t t
R 2 ( t ) = lim sup n → ∞ 1 n log 2 A 2 ( n , ⌈ t n ⌉ ) , 0 ≤ t ≤ 1 2 . (42) R_2(t)=\limsup_{n\to\infty}\frac{1}{n}\log_2 A_2(n,\lceil tn\rceil),\qquad0\le t\le\frac{1}{2}. \tag*{(42)} R 2 ( t ) = n → ∞ lim sup n 1 log 2 A 2 ( n , ⌈ t n ⌉) , 0 ≤ t ≤ 2 1 . (42) Define
H ( x ) = H 2 ( 1 − 1 − x 2 ) , \mathcal{H}(x)=H_2\left(\frac{1-\sqrt{1-x}}{2}\right), H ( x ) = H 2 ( 2 1 − 1 − x ) , U ( t ) = min 1 − 2 t ≤ u ≤ 1 { H ( u 2 ) − H ( u ( u − 1 + 2 t ) ) } , 0 < t < 1 2 . (43) \mathcal{U}(t)=\min_{1-2t\le u\le1}\left\{\mathcal{H}(u^2)-\mathcal{H}\left(u(u-1+2t)\right)\right\},\qquad0<t<\frac{1}{2}. \tag*{(43)} U ( t ) = 1 − 2 t ≤ u ≤ 1 min { H ( u 2 ) − H ( u ( u − 1 + 2 t ) ) } , 0 < t < 2 1 . (43) where H 2 ( p ) = − p log 2 p − ( 1 − p ) log 2 ( 1 − p ) H_2(p)=-p\log_2p-(1-p)\log_2(1-p) H 2 ( p ) = − p log 2 p − ( 1 − p ) log 2 ( 1 − p ) [25 ] gives the asymptotic upper bound R 2 ( t ) ≤ U ( t ) R_2(t)\le\mathcal{U}(t) R 2 ( t ) ≤ U ( t ) 0 < t < 1 / 2 0<t<1/2 0 < t < 1/2 u u u (43) gives U ( t ) \mathcal{U}(t) U ( t ) u = 1 − 2 t + 2 t s u=1-2t+2ts u = 1 − 2 t + 2 t s s ∈ [ 0 , 1 ] s\in[0,1] s ∈ [ 0 , 1 ] U \mathcal{U} U [ 0 , 1 / 2 ] [0,1/2] [ 0 , 1/2 ] U ( 0 ) = 1 \mathcal{U}(0)=1 U ( 0 ) = 1 U ( 1 / 2 ) = 0 \mathcal{U}(1/2)=0 U ( 1/2 ) = 0
This bound also applies to arbitrary code families whose relative distances are specified by a lower limit. Consider binary classical codes with lengths n i → ∞ n_i\to\infty n i → ∞ 2 k i 2^{k_i} 2 k i Δ i \Delta_i Δ i lim inf i Δ i / n i ≥ t \liminf_i\Delta_i/n_i\ge t lim inf i Δ i / n i ≥ t 0 < t ≤ 1 / 2 0<t\le1/2 0 < t ≤ 1/2 0 < ϵ < t 0<\epsilon<t 0 < ϵ < t Δ i / n i ≥ t − ϵ \Delta_i/n_i\ge t-\epsilon Δ i / n i ≥ t − ϵ i i i 2 k i ≤ A 2 ( n i , ⌈ ( t − ϵ ) n i ⌉ ) 2^{k_i}\le A_2(n_i,\lceil(t-\epsilon)n_i\rceil) 2 k i ≤ A 2 ( n i , ⌈( t − ϵ ) n i ⌉) lim sup i k i / n i ≤ R 2 ( t − ϵ ) ≤ U ( t − ϵ ) \limsup_i k_i/n_i\le R_2(t-\epsilon)\le\mathcal{U}(t-\epsilon) lim sup i k i / n i ≤ R 2 ( t − ϵ ) ≤ U ( t − ϵ ) ϵ ↓ 0 \epsilon\downarrow0 ϵ ↓ 0
lim sup i k i n i ≤ U ( t ) if lim inf i Δ i n i ≥ t , 0 ≤ t ≤ 1 2 . (44) \limsup_i\frac{k_i}{n_i}\le\mathcal{U}(t)\quad\text{if}\quad\liminf_i\frac{\Delta_i}{n_i}\ge t,\qquad0\le t\le\frac{1}{2}. \tag*{(44)} i lim sup n i k i ≤ U ( t ) if i lim inf n i Δ i ≥ t , 0 ≤ t ≤ 2 1 . (44) For t = 0 t=0 t = 0 2 k i ≤ 2 n i 2^{k_i}\le2^{n_i} 2 k i ≤ 2 n i U ( 0 ) = 1 \mathcal{U}(0)=1 U ( 0 ) = 1
To apply this bound to fully addressable quantum codes, we extract a classical binary code from the addressing matrix. The following theorem shows that this code has dimension K K K D D D
Theorem S1. Let E E E ( ( N , 2 K , D ) ) 2 ((N,2^{K},D))_{2} (( N , 2 K , D ) ) 2 K ≥ 1 K \ge1 K ≥ 1 E E E h ≥ 1 h \ge1 h ≥ 1 F F F s F s_{F} s F F F F 2 2 2 F F F 2 2 2 [ s F , K , ≥ D ] [s_{F},K,\ge D] [ s F , K , ≥ D ]
2 K ≤ A 2 ( s F , D ) , K + D − 1 ≤ s F . (45) 2^{K} \le A_{2}(s_{F},D), \qquad K+D-1 \le s_{F}. \tag*{(45)} 2 K ≤ A 2 ( s F , D ) , K + D − 1 ≤ s F . (45) Proof. Substitute c = 2 h − 1 v c=2^{h-1}v c = 2 h − 1 v v ∈ F 2 K v \in\mathbb{F}_{2}^{K} v ∈ F 2 K (17) to obtain Z v ( F m o d 2 ) E = E Z v Z^{v(F\ {\rm mod}\ 2)}E=EZ^{v} Z v ( F mod 2 ) E = E Z v v ≠ 0 v \ne0 v = 0 Z v ≠ I Z^{v} \ne I Z v = I v ( F m o d 2 ) ≠ 0 v(F\ {\rm mod}\ 2) \ne0 v ( F mod 2 ) = 0 F F F 2 2 2 K K K D D D P P P D D D E † P E = λ P I E^{\dagger}PE=\lambda_{P}I E † P E = λ P I λ P \lambda_{P} λ P Z v Z^{v} Z v + 1 +1 + 1 − 1 -1 − 1 v ≠ 0 v \ne0 v = 0 E † Z v ( F m o d 2 ) E = Z v E^{\dagger}Z^{v(F\ {\rm mod}\ 2)}E=Z^{v} E † Z v ( F mod 2 ) E = Z v wt ( v ( F m o d 2 ) ) ≥ D \operatorname{wt}(v(F\ {\rm mod}\ 2)) \ge D wt ( v ( F mod 2 )) ≥ D
Removing zero columns preserves these weights and the dimension, yielding a binary linear [ s F , K , ≥ D ] [s_{F},K,\ge D] [ s F , K , ≥ D ] 2 K ≤ A 2 ( s F , D ) 2^{K} \le A_{2}(s_{F},D) 2 K ≤ A 2 ( s F , D ) A 2 A_{2} A 2 [24 ] gives K + D − 1 ≤ s F K+D-1 \le s_{F} K + D − 1 ≤ s F
Consider a fully addressable code family with N i → ∞ N_{i} \to\infty N i → ∞ R ∞ > 0 R_{\infty} > 0 R ∞ > 0 N i N_{i} N i [ N i , K i , ≥ D i ] [N_{i},K_{i},\ge D_{i}] [ N i , K i , ≥ D i ] [25 Corollary 19] imply d ∞ < 1 / 2 d_{\infty}<1/2 d ∞ < 1/2 (44) with t = d ∞ t=d_{\infty} t = d ∞
R ∞ ≤ lim sup i K i N i ≤ U ( d ∞ ) . (46) R_{\infty} \le\limsup_{i} \frac{K_{i}}{N_{i}} \le U(d_{\infty}). \tag*{(46)} R ∞ ≤ i lim sup N i K i ≤ U ( d ∞ ) . (46) The classical rate bound yields the following explicit upper bound on the common coefficient η \eta η
Lemma S13. For each fixed h ≥ 1 h \ge1 h ≥ 1 N i → ∞ N_{i} \to\infty N i → ∞ h h h η = min { R ∞ , d ∞ } \eta=\min\{R_{\infty},d_{\infty}\} η = min { R ∞ , d ∞ }
η ≤ c u p = 0.282652727463. (47) \eta\le c_{\mathrm{up}}=0.282652727463. \tag*{(47)} η ≤ c up = 0.282652727463. (47) Proof. The case η = 0 \eta=0 η = 0 η > 0 \eta>0 η > 0 t 0 = c u p t_{0}=c_{\mathrm{up}} t 0 = c up u 0 = 0.454058689 u_{0}=0.454058689 u 0 = 0.454058689 1 − 2 t 0 < u 0 < 1 1-2t_{0}<u_{0}<1 1 − 2 t 0 < u 0 < 1 U ( t 0 ) < t 0 U(t_{0})<t_{0} U ( t 0 ) < t 0 10 − 24 10^{-24} 1 0 − 24 [ 1 , 2 ] [1,2] [ 1 , 2 ] z = ( x − 1 ) / ( x + 1 ) z=(x-1)/(x+1) z = ( x − 1 ) / ( x + 1 )
0 ≤ ln x − 2 ∑ j = 0 39 z 2 j + 1 2 j + 1 ≤ 2 z 81 81 ( 1 − z 2 ) . 0 \le\ln x-2\sum_{j=0}^{39}\frac{z^{2j+1}}{2j+1} \le\frac{2z^{81}}{81(1-z^{2})}. 0 ≤ ln x − 2 j = 0 ∑ 39 2 j + 1 z 2 j + 1 ≤ 81 ( 1 − z 2 ) 2 z 81 . The same enclosure at x = 2 x=2 x = 2 H 2 H_{2} H 2 [ 0 , 1 / 2 ] [0,1/2] [ 0 , 1/2 ]
0.2826527274543 < H ( u 0 2 ) − H ( u 0 ( u 0 − 1 + 2 t 0 ) ) < 0.2826527274544 < t 0 . (48) \begin{aligned}
0.2826527274543 < \mathcal{H}(u_{0}^{2})-\mathcal{H}\bigl(u_{0}(u_{0}-1+2t_{0})\bigr) \\
&< 0.2826527274544 < t_{0}. \tag*{(48)}
\end{aligned} 0.2826527274543 < H ( u 0 2 ) − H ( u 0 ( u 0 − 1 + 2 t 0 ) ) < 0.2826527274544 < t 0 . (48) Since u 0 u_{0} u 0 (43) , the witness in Eq. (48) implies U ( t 0 ) < t 0 U(t_{0})<t_{0} U ( t 0 ) < t 0 U U U t t t (43) , while the objective at each previously admissible u u u H \mathcal{H} H η ≥ t 0 \eta\ge t_0 η ≥ t 0 R ∞ R_{\infty} R ∞ d ∞ d_{\infty} d ∞ t 0 t_0 t 0 (46) would imply
t 0 ≤ R ∞ ≤ U ( d ∞ ) ≤ U ( t 0 ) < t 0 , t_0 \le R_{\infty} \le\mathcal{U}(d_{\infty}) \le\mathcal{U}(t_0) < t_0, t 0 ≤ R ∞ ≤ U ( d ∞ ) ≤ U ( t 0 ) < t 0 , a contradiction. Hence η ≤ c u p \eta\le c_{\mathrm{up}} η ≤ c up
The addressing form in Eq. (20) strengthens this bound by shortening the associated classical code.
Corollary S1. For a family as in Lemma S13, suppose N i = h M i N_i = hM_i N i = h M i ( F i ) ℓ , ( j , b ) = 2 b z ℓ , j ( i ) ( m o d 2 h ) (F_i)_{\ell,(j,b)} = 2^b z_{\ell,j}^{(i)} \pmod{2^h} ( F i ) ℓ , ( j , b ) = 2 b z ℓ , j ( i ) ( mod 2 h ) 0 ≤ b < h 0 \le b < h 0 ≤ b < h
η ≤ c u p h . (49) \eta\le\frac{c_{\mathrm{up}}}{h}. \tag*{(49)} η ≤ h c up . (49) Proof. Again, assume η > 0 \eta> 0 η > 0 b ≥ 1 b \ge1 b ≥ 1 s F i ≤ M i s_{F_i} \le M_i s F i ≤ M i [ M i , K i , ≥ D i ] [M_i, K_i, \ge D_i] [ M i , K i , ≥ D i ] h R ∞ hR_{\infty} h R ∞ h d ∞ hd_{\infty} h d ∞ N i = h M i N_i = hM_i N i = h M i [25 Corollary 19] ensure h d ∞ < 1 / 2 hd_{\infty} < 1/2 h d ∞ < 1/2 A 2 A_2 A 2 (44) , applied with n i = M i n_i = M_i n i = M i t = h d ∞ t = hd_{\infty} t = h d ∞
2 K i ≤ A 2 ( M i , D i ) , h R ∞ ≤ U ( h d ∞ ) . (50) 2^{K_i} \le A_2(M_i,D_i), \qquad hR_{\infty} \le\mathcal{U}(hd_{\infty}). \tag*{(50)} 2 K i ≤ A 2 ( M i , D i ) , h R ∞ ≤ U ( h d ∞ ) . (50) The proof of Lemma S13 establishes that U \mathcal{U} U U ( c u p ) < c u p \mathcal{U}(c_{\mathrm{up}}) < c_{\mathrm{up}} U ( c up ) < c up h η ≥ c u p h\eta\ge c_{\mathrm{up}} h η ≥ c up (50) would therefore give
c u p ≤ h R ∞ ≤ U ( h d ∞ ) ≤ U ( c u p ) < c u p , c_{\mathrm{up}} \le hR_{\infty} \le\mathcal{U}(hd_{\infty}) \le\mathcal{U}(c_{\mathrm{up}}) < c_{\mathrm{up}}, c up ≤ h R ∞ ≤ U ( h d ∞ ) ≤ U ( c up ) < c up , a contradiction. Hence η ≤ c u p / h \eta\le c_{\mathrm{up}}/h η ≤ c up / h
Quantitative comparison with prior constructions
We compare fully addressable transversal T T T T T T R ∞ R_{\infty} R ∞ d ∞ d_{\infty} d ∞
San-José constructs binary CSS codes by converting higher-order Schur-product conditions on AG codes into weighted binary divisibility [14 Proposition 4.5 and Theorem 6.4] . For addressable T T T w p w_p w p g a g_a g a X X X ϵ a ∈ { 1 , − 1 } \epsilon_a \in\{1,-1\} ϵ a ∈ { 1 , − 1 } F a , p = ϵ a w p g a , p ( m o d 8 ) F_{a,p} = \epsilon_a w_p g_{a,p} \pmod8 F a , p = ϵ a w p g a , p ( mod 8 ) m = 18 m = 18 m = 18 τ = 128 \tau= 128 τ = 128 [14 ] , applied with j = 8 j = 8 j = 8 ∑ a = 1 8 ( 18 a ) = 106761 \sum_{a=1}^{8} \binom{18}{a} = 106761 ∑ a = 1 8 ( a 18 ) = 106761 λ = 257 / 523264 \lambda= 257/523264 λ = 257/523264
R ∞ ≥ 257 55836750327 , d ∞ ≥ 257 55672124865 . (51) R_{\infty} \ge\frac{257}{55836750327}, \qquad d_{\infty} \ge\frac{257}{55672124865}. \tag*{(51)} R ∞ ≥ 55836750327 257 , d ∞ ≥ 55672124865 257 . (51) This explicit substitution therefore guarantees both quantities to be at least 4.6027033897 … × 10 − 9 4.6027033897\ldots\times10^{-9} 4.6027033897 … × 1 0 − 9 T T T [9 Theorem 4.5] . Their explicit construction uses a binary [ 244 , 10 , 100 ] [244, 10, 100] [ 244 , 10 , 100 ] K i = ⌊ n i / 992 ⌋ K_i = \lfloor n_i/992\rfloor K i = ⌊ n i /992 ⌋ N i = 244 ( n i − K i ) N_i = 244(n_i-K_i) N i = 244 ( n i − K i ) n i n_i n i [9 ] gives
R ∞ = 1 241804 , d ∞ ≥ 1 241804 . (52) R_\infty= \frac{1}{241804}, \qquad d_\infty\ge\frac{1}{241804}. \tag*{(52)} R ∞ = 241804 1 , d ∞ ≥ 241804 1 . (52) The common guarantee is ≃ 4.1355808837 × 10 − 6 \simeq4.1355808837 \times10^{-6} ≃ 4.1355808837 × 1 0 − 6
Meiburg obtains fully addressable transversal T T T [15 Secs. 3.3–3.5] . Before the coordinate-repetition step, the four-bit packing in Ref. [15 Sec. 3.5] gives K i = 4 t i K_i = 4t_i K i = 4 t i N i = 385 ( n i − t i ) N_i = 385(n_i-t_i) N i = 385 ( n i − t i ) t i / n i → 5 / 186 t_i/n_i \to5/186 t i / n i → 5/186 n i n_i n i t i t_i t i T T T T T T
R ∞ = 4 13937 , d ∞ ≥ 1 13937 . (53) R_\infty= \frac{4}{13937}, \qquad d_\infty\ge\frac{1}{13937}. \tag*{(53)} R ∞ = 13937 4 , d ∞ ≥ 13937 1 . (53) Thus the common guarantee is 1 / 13937 ≃ 7.1751 × 10 − 5 1/13937 \simeq7.1751 \times10^{-5} 1/13937 ≃ 7.1751 × 1 0 − 5
Meiburg’s modular-clock construction also applies to T T T [15 Sec. 3.6] . We specialize it to a Garcia–Stichtenoth tower over F 1024 \mathbb{F}_{1024} F 1024 g i / n i → 1 / 31 g_i/n_i \to1/31 g i / n i → 1/31 [16 ] , using Meiburg’s stated divisor choices a 0 , i / n i → 1 / 3 a_{0,i}/n_i \to1/3 a 0 , i / n i → 1/3 a 1 , i / n i → 2 / 3 a_{1,i}/n_i \to2/3 a 1 , i / n i → 2/3 10 n i 10n_i 10 n i 10 ( a 1 , i − a 0 , i ) 10(a_{1,i}-a_{0,i}) 10 ( a 1 , i − a 0 , i ) Z / 8 Z \mathbb{Z}/8\mathbb{Z} Z /8 Z min { n i − a 1 , i , a 0 , i − 2 g i + 2 } \min\{n_i-a_{1,i},a_{0,i}-2g_i+2\} min { n i − a 1 , i , a 0 , i − 2 g i + 2 } [15 Eq. (18)] uses ( ⌈ log 2 8 ⌉ + 1 ) ( 8 − 1 ) = 28 (\lceil\log_2 8\rceil+ 1)(8-1) = 28 (⌈ log 2 8 ⌉ + 1 ) ( 8 − 1 ) = 28 N i = 280 n i N_i = 280n_i N i = 280 n i K i = 10 ( a 1 , i − a 0 , i ) K_i = 10(a_{1,i}-a_{0,i}) K i = 10 ( a 1 , i − a 0 , i )
R ∞ = 1 84 , d ∞ ≥ 1 / 3 − 2 / 31 280 = 5 5208 . (54) R_\infty= \frac{1}{84}, \qquad d_\infty\ge\frac{1/3-2/31}{280} = \frac{5}{5208}. \tag*{(54)} R ∞ = 84 1 , d ∞ ≥ 280 1/3 − 2/31 = 5208 5 . (54) The common guarantee is 5 / 5208 ≃ 9.6006 × 10 − 4 5/5208 \simeq9.6006 \times10^{-4} 5/5208 ≃ 9.6006 × 1 0 − 4
All four prior constructions have positive asymptotic rate and relative distance. The prior-work entries in the comparison table (§ 1 of the main text) use the smaller of the two guarantees for each stated parameter choice. The prior constructions and our constructions meet the same logical and physical T T T § 3.2.3 and § 4.1.5 guarantee both quantities at least 52 / 2025 52/2025 52/2025 t ⋆ / 3 t_\star/3 t ⋆ /3
Finite examples and their exact certificates
The codes Q 48 = ( ( 48 , 16 , 5 ) ) 2 \mathcal{Q}_{48} = ((48,16,5))_2 Q 48 = (( 48 , 16 , 5 ) ) 2 Q 90 = ( ( 90 , 256 , 7 ) ) 2 \mathcal{Q}_{90} = ((90,256,7))_2 Q 90 = (( 90 , 256 , 7 ) ) 2 (19) with h = 3 h=3 h = 3 ( M , r , K ) = ( 16 , 12 , 4 ) (M,r,K) = (16,12,4) ( M , r , K ) = ( 16 , 12 , 4 ) ( 30 , 22 , 8 ) (30,22,8) ( 30 , 22 , 8 ) Z / 8 Z \mathbb{Z}/8\mathbb{Z} Z /8 Z Φ 3 < 1 \Phi_3 < 1 Φ 3 < 1 Φ 3 \Phi_3 Φ 3 (38) .
Addressable transversal T T T
The j j j ( 3 j − 2 , 3 j − 1 , 3 j ) (3j-2,3j-1,3j) ( 3 j − 2 , 3 j − 1 , 3 j ) G , L G,L G , L r r r K K K
Z a d d r = ( B − T ) r + 1 : M , : , G Z a d d r T = 0 , L Z a d d r T = I K ( m o d 8 ) . (55) Z_{\mathrm{addr}} = (\mathrm{B}^{-\mathsf{T}})_{r+1:M,:}, \qquad GZ_{\mathrm{addr}}^{\mathsf{T}} = 0,\qquad LZ_{\mathrm{addr}}^{\mathsf{T}} = I_K \pmod8. \tag*{(55)} Z addr = ( B − T ) r + 1 : M , : , G Z addr T = 0 , L Z addr T = I K ( mod 8 ) . (55) For c ∈ R 3 K c \in\mathcal{R}_3^K c ∈ R 3 K a = c Z a d d r ( m o d 8 ) a=cZ_{\mathrm{addr}} \pmod{8} a = c Z addr ( mod 8 ) (20) at h = 3 h=3 h = 3
⨂ j = 1 M ( T a j ⊗ T 2 a j ⊗ T 4 a j ) (56) \bigotimes_{j=1}^{M}\left(T^{a_j}\otimes T^{2a_j}\otimes T^{4a_j}\right) \tag*{(56)} j = 1 ⨂ M ( T a j ⊗ T 2 a j ⊗ T 4 a j ) (56) has phase e 2 π i c x T / 8 e^{2\pi i c x^{\mathsf{T}}/8} e 2 π i c x T /8 E ∣ x ⟩ E|x\rangle E ∣ x ⟩ ⨂ T c ℓ \bigotimes T^{c\ell} ⨂ T c ℓ
Verification of code distances
We certify each distance by verifying detection of all Pauli errors of weight below d d d d d d
For W ( a , b ) = ⨂ j X 8 ( a j ) Z 8 ( b j ) W(a,b)=\bigotimes_j X_8(a_j)Z_8(b_j) W ( a , b ) = ⨂ j X 8 ( a j ) Z 8 ( b j ) (22) gives
Q A † W ( a , b ) Q A = e − π i a A a T / 8 W ( a , b − a A ) , (57) Q_A^\dagger W(a,b)Q_A=e^{-\pi i a A a^{\mathsf{T}}/8}W(a,b-aA), \tag*{(57)} Q A † W ( a , b ) Q A = e − π ia A a T /8 W ( a , b − a A ) , (57) and Lemma S6 requires G ( b − a A ) T = 0 ( m o d 8 ) G(b-aA)^{\mathsf{T}}=0\pmod{8} G ( b − a A ) T = 0 ( mod 8 )
At exact order two, write ( a , b ) = 4 ( α , β ) (a,b)=4(\alpha,\beta) ( a , b ) = 4 ( α , β ) α , β ∈ F 2 M \alpha,\beta\in\mathbb{F}_2^M α , β ∈ F 2 M ( a B − 1 ) r + 1 : M = x ′ − x (aB^{-1})_{r+1:M}=x'-x ( a B − 1 ) r + 1 : M = x ′ − x x , x ′ x,x' x , x ′ { 0 , 4 } \{0,4\} { 0 , 4 } { 0 , 1 , 7 } \{0,1,7\} { 0 , 1 , 7 }
G ( b − a A ) T = 0 , ( a B − 1 ) r + 1 : M = 0 ( m o d 8 ) . G(b-aA)^{\mathsf{T}}=0,\qquad(aB^{-1})_{r+1:M}=0\pmod{8}. G ( b − a A ) T = 0 , ( a B − 1 ) r + 1 : M = 0 ( mod 8 ) . Put G 2 = G ( m o d 2 ) G_2=G\pmod{2} G 2 = G ( mod 2 ) A 2 = A ( m o d 2 ) A_2=A\pmod{2} A 2 = A ( mod 2 ) 4 z = 0 ( m o d 8 ) 4z=0\pmod{8} 4 z = 0 ( mod 8 ) z = 0 ( m o d 2 ) z=0\pmod{2} z = 0 ( mod 2 )
G 2 ( β − α A 2 ) T = G 2 β T + G 2 A 2 α T = 0 ( m o d 2 ) , G_2(\beta-\alpha A_2)^{\mathsf{T}}=G_2\beta^{\mathsf{T}}+G_2A_2\alpha^{\mathsf{T}}=0\pmod{2}, G 2 ( β − α A 2 ) T = G 2 β T + G 2 A 2 α T = 0 ( mod 2 ) , where we used A 2 T = A 2 A_2^{\mathsf{T}}=A_2 A 2 T = A 2 F 2 \mathbb{F}_2 F 2 J α T = 0 J\alpha^{\mathsf{T}}=0 J α T = 0 J = ( ( B − 1 ) : , r + 1 : M ) T ( m o d 2 ) J=((B^{-1})_{:,r+1:M})^{\mathsf{T}}\pmod{2} J = (( B − 1 ) : , r + 1 : M ) T ( mod 2 )
\begin{pmatrix}
G_2A_2 & G_2\\
J & 0
\end{pmatrix}
\begin{pmatrix}
\alpha^{\mathsf{T}}\\
\beta^{\mathsf{T}}
\end{pmatrix}
=0\pmod{2} \tag*{(58)}\end{pmatrix}
By the proof of Lemma S8, the physical Pauli weight is wt ( α ) + wt ( β ) \operatorname{wt}(\alpha)+\operatorname{wt}(\beta) wt ( α ) + wt ( β ) (58) has a nonzero solution with wt ( α ) + wt ( β ) < d \operatorname{wt}(\alpha)+\operatorname{wt}(\beta)<d wt ( α ) + wt ( β ) < d
For Q 48 Q_{48} Q 48 Q 90 Q_{90} Q 90 (58) has no nonzero solution with wt ( α ) + wt ( β ) < d \operatorname{wt}(\alpha)+\operatorname{wt}(\beta)<d wt ( α ) + wt ( β ) < d d = 5 d=5 d = 5 d = 7 d=7 d = 7
For exact order four, write ( a , b ) = 2 ( a ′ , b ′ ) (a,b)=2(a',b') ( a , b ) = 2 ( a ′ , b ′ ) a ′ , b ′ ∈ ( Z / 4 Z ) M a',b'\in(\mathbb{Z}/4\mathbb{Z})^M a ′ , b ′ ∈ ( Z /4 Z ) M
G ( b ′ − a ′ A ) T = 0 ( m o d 4 ) . G(b'-a'A)^{\mathsf{T}}=0\pmod{4}. G ( b ′ − a ′ A ) T = 0 ( mod 4 ) . As in the proof of Lemma S9, assign each local pair weight zero at ( 0 , 0 ) (0,0) ( 0 , 0 ) A 3 \mathcal{A}_3 A 3 (37) , and two otherwise; retain only pairs whose total assigned weight is below d d d
At exact order eight, ( a , b ) (a,b) ( a , b )
G ( b − a A ) T = 0 ( m o d 8 ) , ∣ { j : ( a j , b j ) ≠ ( 0 , 0 ) } ∣ < d . G(b-aA)^{\mathsf{T}}=0\pmod{8},\qquad\left|\{j:(a_j,b_j)\ne(0,0)\}\right|<d. G ( b − a A ) T = 0 ( mod 8 ) , ∣ { j : ( a j , b j ) = ( 0 , 0 )} ∣ < d . For both codes, exact verification excludes all solutions under these order and weight restrictions, with d = 5 d = 5 d = 5 d = 7 d = 7 d = 7
Together with the order-two result, these exclusions imply E † P E = 0 E^\dagger P E = 0 E † P E = 0 P P P d d d D 48 ≥ 5 D_{48} \ge5 D 48 ≥ 5 D 90 ≥ 7 D_{90} \ge7 D 90 ≥ 7
Matching upper bounds follow from a common criterion: a Pauli P N P_N P N E N † P N E N E_N^\dagger P_N E_N E N † P N E N D N ≤ wt ( P N ) D_N \le\operatorname{wt}(P_N) D N ≤ wt ( P N )
P 48 = X 9 X 24 X 48 Z 34 Z 40 , (59) P_{48} = X_9X_{24}X_{48}Z_{34}Z_{40}, \tag*{(59)} P 48 = X 9 X 24 X 48 Z 34 Z 40 , (59) P 90 = X 24 X 66 X 69 X 72 Z 25 Z 46 Z 76 . (60) P_{90} = X_{24}X_{66}X_{69}X_{72}Z_{25}Z_{46}Z_{76}. \tag*{(60)} P 90 = X 24 X 66 X 69 X 72 Z 25 Z 46 Z 76 . (60) For the archived matrices, Eq. (23) gives
P N E N = E N Z 2 Z 4 , N ∈ { 48 , 90 } , (61) P_NE_N = E_NZ_2Z_4,\qquad N \in\{48,90\}, \tag*{(61)} P N E N = E N Z 2 Z 4 , N ∈ { 48 , 90 } , (61) where Z 2 Z 4 Z_2Z_4 Z 2 Z 4 D 48 ≤ 5 D_{48} \le5 D 48 ≤ 5 D 90 ≤ 7 D_{90} \le7 D 90 ≤ 7 D 48 = 5 D_{48} = 5 D 48 = 5 D 90 = 7 D_{90} = 7 D 90 = 7
Ancilla-free encoding circuit
For Q 48 Q_{48} Q 48 H H H y ↦ y B y \mapsto yB y ↦ y B Q A Q_A Q A (19) using the 48 data qubits.
The linear map acts on 16 three-qubit registers. Gauss–Jordan elimination of B B B R 3 \mathcal{R}_3 R 3 x ↦ u x m o d 8 x \mapsto ux \bmod8 x ↦ ux mod 8 u u u ( x , y ) ↦ ( x , y + c x m o d 8 ) (x,y) \mapsto(x,y+cx \bmod8) ( x , y ) ↦ ( x , y + c x mod 8 ) § 2.3 of the main text.
Expanding Q A Q_A Q A
