Introduction – Quantum state preparation is a foundational challenge in quantum computation and simulation. Given a Hamiltonian H H H
Dissipative state preparation algorithms use controlled interactions with an environment or bath to drive the system toward the desired state. Early heuristic studies provided important evidence that engineered dissipation can successfully prepare pure and thermal quantum states in specific structured settings [27 , 28 , 4 , 29 , 30 , 12 , 18 , 21 , 9 ] . Recently, substantial progress has been made toward systematically designed dissipative algorithms with rigorous performance and complexity guarantees. One direction involves designing Lindblad generators whose stationary states coincide with, or closely approximate, the desired Gibbs or ground states, together with algorithms and analyses establishing end-to-end complexity guarantees in relevant regimes [19 , 8 , 15 , 16 , 11 , 20 , 10 , 3 , 17 , 5 ] . More recently, system-bath interaction algorithms have provided a complementary approach, in which the dissipative dynamics is implemented directly through interactions with a bath that is reset after each round of evolution. A growing body of work has developed such constructions and established increasingly rigorous guarantees on stationary-state accuracy and mixing behavior [14 , 6 , 25 , 13 , 2 , 26 ] .
In recent developments, dissipative algorithms can be designed without explicit knowledge of the target state, requiring only access to Hamiltonian simulation of the form exp ( − i H t ) \exp(-iHt) exp ( − i H t ) N N N O ~ ( N ) \widetilde{\mathcal{O}}(N) O ( N ) [5 , 7 , 24 ] , while a complexity of O ~ ( N 4 ) \widetilde{\mathcal{O}}(N^4) O ( N 4 ) [2 ] with single ancilla bath.
An at least linear dependence on the system size N N N Θ ( N ) \Theta(N) Θ ( N ) T t o t a l = Ω ( N ) T_{\rm total}=\Omega(N) T total = Ω ( N )
Can dissipative algorithms prepare thermal and ground states with sublinear total Hamiltonian simulation time in the system size while retaining local interactions?
In this work, we show that the linear-in-N N N ϵ \epsilon ϵ Θ ( N ) \Theta(N) Θ ( N ) O ~ ( N / ϵ ) \widetilde{\mathcal{O}}(\sqrt{N}/\epsilon) O ( N / ϵ ) O ~ ( polylog ( N / ϵ ) ) \widetilde{\mathcal{O}}(\operatorname{polylog}(N/\epsilon)) O ( polylog ( N / ϵ ))
Although free-fermion systems are classically simulable, establishing efficient quantum state-preparation algorithms for this benchmark remains of significant interest. Free fermions provide a natural testbed for exploring the potential of dissipative state-preparation algorithms [22 ] and have also served as a benchmark for studying their robustness to noise [23 ] . Our results provide rigorous complexity guarantees that are sublinear in the system size N N N
In our construction, we restrict the total Hamiltonian to be quadratic, which is necessary for the system-bath interaction dynamics to preserve the free-fermion structure. Importantly, however, convergence of the algorithm does not require the initial state to be a free-fermion state and holds for arbitrary initial states. This suggests that the underlying mechanism is not intrinsically tied to free-fermion initial data and may admit extensions to more general quantum systems.
Employing an extensive bath, whose size grows in proportion to the system size, has become a common design choice in dissipative algorithms [14 , 25 , 22 ] , yet it is often motivated heuristically, and it remains theoretically unclear whether such a choice can yield a provable improvement in end-to-end complexity. We note that, while increasing the number of bath qubits can enhance the available dissipative transitions, it also introduces additional fluctuations that may drive the system away from the target state. Balancing these competing effects is nontrivial and lies beyond the scope of existing analyses of dissipative algorithms. Consequently, it is not a priori clear whether the advantages of an extensive bath outweigh its potential drawbacks. Our results rigorously demonstrate the advantages of employing an extensive bath over existing system-bath interaction algorithms and their analyses. Furthermore, they establish the use of an extensive bath as a promising design principle for overcoming the linear-in-N N N
System-bath interaction algorithm – Given an N N N H H H 0 < β < ∞ 0<\beta<\infty 0 < β < ∞ σ β ∝ e − β H \sigma_\beta\propto e^{-\beta H} σ β ∝ e − β H \ketbra{\psi_0} . The system-bath interaction algorithm [14 , 6 , 25 , 22 ] prepares the target state by iteratively applying a quantum channel Φ α \Phi_\alpha Φ α
ρ n + 1 = Φ α ( ρ n ) : = E [ Tr E ( U α ( T ) ( ρ n ⊗ ρ E ) U α ( T ) † ) ] . \begin{equation}
\rho_{n+1}
=
\Phi_{\alpha}(\rho_n)
:=
\mathbb{E}\left[
\operatorname{Tr}_E\left(
U^\alpha(T)
\left(\rho_n\otimes\rho_E\right)
U^\alpha(T)^\dagger
\right)
\right]. \tag*{(1)}
\end{equation} ρ n + 1 = Φ α ( ρ n ) := E [ Tr E ( U α ( T ) ( ρ n ⊗ ρ E ) U α ( T ) † ) ] . (1) Here, ρ E \rho_E ρ E U α ( t ) : = T exp ( − i ∫ − T t H α ( s ) d s ) U^\alpha(t)
:=
\mathcal{T}\exp\left(
-i\int_{-T}^{t} H_\alpha(s)\,\mathrm{d}s
\right) U α ( t ) := T exp ( − i ∫ − T t H α ( s ) d s ) t t t
where H E H_E H E H i n t ( t ) H_{\rm int}(t) H int ( t ) α \alpha α H E H_E H E H i n t ( t ) H_{\rm int}(t) H int ( t ) ρ E \rho_E ρ E E [ ⋅ ] \mathbb{E}[\cdot] E [ ⋅ ] (1) is taken over the randomness in the construction of the system-bath interaction algorithm.
In designing a system-bath interaction algorithm, the key is to choose H E H_E H E H i n t ( t ) H_{\rm int}(t) H int ( t ) ρ E \rho_E ρ E Φ α \Phi_\alpha Φ α
where A i A_i A i B i B_i B i f ( t ) f(t) f ( t ) A i ⊗ B i A_i\otimes B_i A i ⊗ B i f ( t ) f(t) f ( t ) T t o t a l = 2 T m T_{\rm total}=2T\,m T total = 2 T m m m m 2 T 2T 2 T
Main results – We assume H = ∑ i , j h i j c i † c j H=\sum_{i,j} h_{ij} c^\dagger_ic_j H = ∑ i , j h ij c i † c j h = ( h i j ) h=(h_{ij}) h = ( h ij ) h h h U U U h = U Λ U † h=U\Lambda U^\dagger h = U Λ U †
H = ∑ k = 1 N λ k ( ∑ j ( U † ) k , j c j ) † ( ∑ j ( U † ) k , j c j ) : = ∑ k = 1 N λ k d k † d k . \begin{equation}
\begin{aligned}
H&=\sum^N_{k=1}\lambda_k\left(\sum_j (U^\dagger)_{k,j}c_j\right)^\dagger\left(\sum_j (U^\dagger)_{k,j}c_j\right)\\
&:=\sum^N_{k=1}\lambda_k d^\dagger_k d_k\,.
\end{aligned} \tag*{(4)}
\end{equation} H = k = 1 ∑ N λ k ( j ∑ ( U † ) k , j c j ) † ( j ∑ ( U † ) k , j c j ) := k = 1 ∑ N λ k d k † d k . (4) Because U U U { d k † , d k } \{d^\dagger_k,d_k\} { d k † , d k }
For free fermions, we use a bath with zero Hamiltonian for thermal state preparation and a bath with a randomly sampled frequency for ground state preparation. We call these the zero-bath and random-bath settings, respectively. In both cases, the construction uses Θ ( N ) \Theta(N) Θ ( N ) H i n t ( t ) H_{\rm int}(t) H int ( t ) (3) consists of Θ ( N ) \Theta(N) Θ ( N ) A i A_i A i B i B_i B i h h h
Theorem 1 (Sublinear thermal state preparation for free fermions). Let Λ = max k ∣ λ k ∣ = O ( 1 ) \Lambda=\max_k|\lambda_k|=\mathcal O(1) Λ = max k ∣ λ k ∣ = O ( 1 ) Φ α \Phi_{\alpha} Φ α T = Θ ~ ( 1 + β ) T=\widetilde{\Theta}(1+\beta) T = Θ ( 1 + β ) m = Θ ~ ( min { N 1 / 2 exp ( β Λ / 2 ) , N } ϵ ) m=\widetilde{\Theta}\left(\frac{\min\{N^{1/2}\exp(\beta\Lambda/2),N\}}{\epsilon}\right) m = Θ ( ϵ m i n { N 1/2 e x p ( β Λ/2 ) , N } ) sup ρ ∥ Φ α m ( ρ ) − σ β ∥ 1 ≤ ϵ \sup_{\rho}\left\|\Phi_{\alpha}^m(\rho)-\sigma_{\beta}\right\|_1\leq \epsilon sup ρ ∥ Φ α m ( ρ ) − σ β ∥ 1 ≤ ϵ β = Θ ( 1 ) \beta=\Theta(1) β = Θ ( 1 ) Λ = Θ ( 1 ) \Lambda=\Theta(1) Λ = Θ ( 1 ) σ β \sigma_\beta σ β ϵ \epsilon ϵ T t o t a l = O ~ ( N 1 / 2 / ϵ ) .
T_{\rm total}=\widetilde{\mathcal{O}}(N^{1/2}/\epsilon)\,.
T total = O ( N 1/2 / ϵ ) . Θ ~ , O ~ \widetilde{\Theta},\widetilde{\mathcal{O}} Θ , O β , N , 1 / ϵ \beta,N,1/\epsilon β , N , 1/ ϵ
The rigorous statement of the theorem is presented in § 1 Theorem 6 , with the proof provided in § 1.2 . The construction of the system-bath interaction algorithm is based on the choice H E = 0 H_E=0 H E = 0 Assumption 3 ), together with a suitably chosen filter function f ( t ) f(t) f ( t ) [11 ] . Specifically, f ( t ) f(t) f ( t )
In contrast to previous system–bath interaction algorithms and analyses [6 , 2 ] , our algorithm employs Θ ( N ) \Theta(N) Θ ( N ) Φ α \Phi_{\alpha} Φ α α \alpha α N N N
To control these errors, we use a transformation that decomposes the total system–bath Hamiltonian into N N N α \alpha α N N N [15 , 11 ] can achieve a similar sublinear-in-N N N Θ ( N ) \Theta(N) Θ ( N )
Our second main result concerns ground state preparation, which is summarized as follows:
Theorem 2 (Polylogarithmic ground state preparation for free fermions). Assume max k ∣ λ k ∣ = O ( 1 ) \max_k|\lambda_k|=\mathcal O(1) max k ∣ λ k ∣ = O ( 1 ) Δ : = min k ∣ λ k ∣ > 0 \Delta:=\min_k|\lambda_k|>0 Δ := min k ∣ λ k ∣ > 0 Φ α \Phi_{\alpha} Φ α T = Θ ~ ( Δ − 1 ) T=\widetilde{\Theta}(\Delta^{-1}) T = Θ ( Δ − 1 ) m = Θ ~ ( Δ − 1 p o l y l o g ( N / ϵ ) ) m=\widetilde{\Theta}\left(\Delta^{-1}\mathrm{polylog}(N/\epsilon)\right) m = Θ ( Δ − 1 polylog ( N / ϵ ) ) sup ρ ∥ Φ α m ( ρ ) − ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ∥ 1 ≤ ϵ \sup_{\rho}\left\|\Phi_{\alpha}^m(\rho)-\ket{\psi_0}\bra{\psi_0}\right\|_1\leq \epsilon sup ρ ∥ Φ α m ( ρ ) − ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ∥ 1 ≤ ϵ ∣ ψ 0 ⟩ \ket{\psi_0} ∣ ψ 0 ⟩ ϵ \epsilon ϵ T t o t a l = O ~ ( Δ − 2 ) T_{\rm total}=\widetilde{\mathcal{O}}(\Delta^{-2}) T total = O ( Δ − 2 )
A detailed version of the above theorem is put in § 2 Theorem 12 with the proof provided in § 2.1 . The construction of the system-bath interaction algorithm is based on a random-bath setting (see Assumption 3 ), where H E H_E H E f ( t ) f(t) f ( t )
We note that achieving the p o l y l o g ( 1 / ϵ ) \mathrm{polylog}(1/\epsilon) polylog ( 1/ ϵ ) α = Θ ~ ( Δ ) \alpha=\widetilde{\Theta}(\sqrt{\Delta}) α = Θ ( Δ ) Φ α \Phi_\alpha Φ α [2 ] . The analysis therefore requires going beyond the conventional Lindblad approximation for open quantum systems. In particular, it is necessary to account for all higher-order terms in the Dyson expansion and establish that their contributions preserve the ground state to the desired accuracy without introducing additional system-size dependencies.
Zero-bath and random-bath settings – In this work, we consider two classes of engineered baths: the zero-bath and random-bath settings.
Assumption 3 (Zero bath and random bath). For the system-bath interaction terms, we choose H i n t ( t ) = f ( t ) ∑ i = 1 N ( c i † ⊗ b i , + † + c i ⊗ b i , − † ) + h . c . . H_{\rm int}(t)=f(t)\sum^N_{i=1}\left(c^\dagger_i\otimes b^\dagger_{i,+}+c_i\otimes b^\dagger_{i,-}\right)+\mathrm{h.c.}\,. H int ( t ) = f ( t ) ∑ i = 1 N ( c i † ⊗ b i , + † + c i ⊗ b i , − † ) + h.c. . { b j , + † , b j , + } j = 1 N \{b^\dagger_{j,+},b_{j,+}\}^N_{j=1} { b j , + † , b j , + } j = 1 N { b j , − † , b j , − } j = 1 N \{b^\dagger_{j,-},b_{j,-}\}^N_{j=1} { b j , − † , b j , − } j = 1 N 2 N 2N 2 N
(Zero-bath) H E = 0 H_E=0 H E = 0 ρ E = ∣ 0 ⟩ + ⟨ 0 ∣ + ⊗ ∣ 0 ⟩ − ⟨ 0 ∣ − \rho_E=\ket{0}_+\bra{0}_+\otimes \ket{0}_-\bra{0}_- ρ E = ∣ 0 ⟩ + ⟨ 0 ∣ + ⊗ ∣ 0 ⟩ − ⟨ 0 ∣ − ∣ 0 ⟩ + \ket{0}_+ ∣ 0 ⟩ + ∣ 0 ⟩ − \ket{0}_- ∣ 0 ⟩ −
(Random-bath) H E = ω ∑ i = 1 N ( b i , + † b i , + + b i , − † b i , − ) H_E=\omega\sum^{N}_{i=1}\left(b^\dagger_{i,+}b_{i,+}+b^\dagger_{i,-}b_{i,-}\right) H E = ω ∑ i = 1 N ( b i , + † b i , + + b i , − † b i , − ) ρ E = [ ( 1 − p b ( ω ) ) ∣ 0 ⟩ ⟨ 0 ∣ + p b ( ω ) ∣ 1 ⟩ ⟨ 1 ∣ ] ⊗ 2 N . \rho_E=
\left[
(1-p_{\rm b}(\omega))|0\rangle\langle0|
+p_{\rm b}(\omega)|1\rangle\langle1|
\right]^{\otimes 2N}. ρ E = [ ( 1 − p b ( ω )) ∣0 ⟩ ⟨ 0∣ + p b ( ω ) ∣1 ⟩ ⟨ 1∣ ] ⊗ 2 N . ω \omega ω μ \mu μ p b ( ω ) ∈ [ 0 , 1 ] p_{\rm b}(\omega)\in[0,1] p b ( ω ) ∈ [ 0 , 1 ]
Although the above assumptions differ only in the choices of H E H_E H E ρ E \rho_E ρ E f ( t ) f(t) f ( t ) f f f Assumption 11 ensures that the Dyson expansion of the quantum channel Φ α \Phi_\alpha Φ α α \alpha α α = Θ ~ ( 1 ) \alpha=\tilde{\Theta}(1) α = Θ ~ ( 1 ) Φ α \Phi_\alpha Φ α Θ ~ ( 1 ) \tilde{\Theta}(1) Θ ~ ( 1 ) Theorem 2 . However, the same advantage does not extend to thermal state preparation: randomness shared across the bath qubits introduces additional fluctuations that obstruct sublinear complexity, as shown in Theorem 5 . By contrast, although the zero-bath setting provides less flexibility in filter design, it avoids the fluctuations induced by a shared random bath frequency. This makes it possible to achieve thermal state preparation with complexity O ~ ( N / ϵ ) \widetilde{\mathcal{O}}(\sqrt{N}/\epsilon) O ( N / ϵ ) Theorem 1 , yielding a polynomial improvement over existing dissipative algorithms.
Lower bounds for thermal state preparation – As established in thm:main_thermal,thm:main_ground , thermal state preparation can be achieved with sublinear total Hamiltonian simulation time, although its system-size dependence remains less favorable than the polylogarithmic scaling achieved for ground state preparation. To explore the possibility of further improvements and identify the limitations of existing choices of system-bath interaction algorithms, we establish: (i). Two lower bounds for thermal state preparation in the zero-bath and random-bath settings, presented in thm:zero_bath_lower_bound_new_1,thm:random_bath_lower_bound , respectively. (ii). A necessary and sufficient condition for achieving improved complexity in the zero-bath setting, presented in Theorem 8 . For the simplicity of presentation, in this section, we assume β = 1 \beta=1 β = 1 ∥ h ∥ ≤ 1 \|h\|\leq 1 ∥ h ∥ ≤ 1 Theorem 8 and technical details in § 1.1 .
We start with zero-bath setting. Our first result shows that the filter functions in Theorem 1 give a lower bound that matches the upper bound in Theorem 1 up to polylogarithmic factors.
Theorem 4 (Lower bound result for zero-bath). For N ≥ 2 N\ge2 N ≥ 2 Assumption 3 . If the filter function f ( t ) f(t) f ( t ) Theorem 1 (Theorem 6 ), then, whenever T − 1 T^{-1} T − 1 α \alpha α ϵ \epsilon ϵ N N N ∫ ∣ t ∣ > T / 2 ∣ f ( t ) ∣ d t = O ( α 4 ) \int_{|t|>T/2}|f(t)|\mathrm{d}t=\mathcal{O}(\alpha^4) ∫ ∣ t ∣ > T /2 ∣ f ( t ) ∣ d t = O ( α 4 ) ϵ \epsilon ϵ T t o t a l = Ω ( N / ϵ ) T_{\rm total}=\Omega(\sqrt{N}/\epsilon) T total = Ω ( N / ϵ )
While Theorem 4 establishes the sharpness of the upper bound in Theorem 1 for the specific filter functions used in Theorem 1 , it won’t rule out the possibility of achieving improved complexity with a different choice of filter function. To complete the whole story, we establish a necessary and sufficient condition for achieving improved complexity in the zero-bath setting for general free fermion Hamiltonians in Theorem 8 of § 1.1 , thereby providing a precise characterization of the design constraints on the filter function f ( t ) f(t) f ( t ) (?) of § 1.1 , we will show that the necessary condition for achieving improved complexity is to satisfy the detailed balance condition for higher order (fourth-order) terms in the interaction strength α \alpha α f ( t ) f(t) f ( t )
Next, we consider the random-bath setting, where the additional random frequency ω \omega ω f ( t ) f(t) f ( t ) f f f
u ( ω ; λ ) = ( 1 − p b ( ω ) ) ∣ f ^ ( λ + ω ) ∣ 2 + p b ( ω ) ∣ f ^ ( ω − λ ) ∣ 2 , d ( ω ; λ ) = ( 1 − p b ( ω ) ) ∣ f ^ ( ω − λ ) ∣ 2 + p b ( ω ) ∣ f ^ ( λ + ω ) ∣ 2 . \begin{equation}
\begin{aligned}
&u(\omega;\lambda)
=\bigl(1-p_{\rm b}(\omega)\bigr)
|\widehat f(\lambda+\omega)|^2
+p_{\rm b}(\omega)
|\widehat f(\omega-\lambda)|^2,\\
&d(\omega;\lambda)
=\bigl(1-p_{\rm b}(\omega)\bigr)
|\widehat f(\omega-\lambda)|^2
+p_{\rm b}(\omega)
|\widehat f(\lambda+\omega)|^2.
\end{aligned} \tag*{(6)}
\end{equation} u ( ω ; λ ) = ( 1 − p b ( ω ) ) ∣ f ( λ + ω ) ∣ 2 + p b ( ω ) ∣ f ( ω − λ ) ∣ 2 , d ( ω ; λ ) = ( 1 − p b ( ω ) ) ∣ f ( ω − λ ) ∣ 2 + p b ( ω ) ∣ f ( λ + ω ) ∣ 2 . (6) Similar to the calculation in the proof of Theorem 2 , to make sure the fixed point is an approximately thermal state when α \alpha α
∫ − ∞ ∞ u ( ω ; λ ) d μ ( ω ) ∫ − ∞ ∞ d ( ω ; λ ) d μ ( ω ) = exp ( − λ ) . \begin{equation}
\frac{\int_{-\infty}^{\infty} u(\omega;\lambda) d\mu(\omega)}{\int_{-\infty}^{\infty} d(\omega;\lambda) d\mu(\omega)} = \exp(-\lambda)\,. \tag*{(7)}
\end{equation} ∫ − ∞ ∞ d ( ω ; λ ) d μ ( ω ) ∫ − ∞ ∞ u ( ω ; λ ) d μ ( ω ) = exp ( − λ ) . (7) Different from zero-bath case, thermal state preparation in this case does not require f ( t ) f(t) f ( t ) ω \omega ω ω \omega ω
Theorem 5. For N ≥ 2 N\ge2 N ≥ 2 Assumption 3 (random-bath) and f ( t ) ∈ L 1 ( R ) f(t)\in L^1(\mathbb{R}) f ( t ) ∈ L 1 ( R ) E ω ( u ( ω ; λ ) ) > 0 \mathbb{E}_{\omega}\left(u(\omega;\lambda)\right)>0 E ω ( u ( ω ; λ ) ) > 0 E ω ( d ( ω ; λ ) ) > 0 \mathbb{E}_{\omega}\left(d(\omega;\lambda)\right)>0 E ω ( d ( ω ; λ ) ) > 0 λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ] λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ] u ( ω ; λ ) ≠ e − λ d ( ω ; λ ) u(\omega;\lambda)\ne e^{-\lambda}d(\omega;\lambda) u ( ω ; λ ) = e − λ d ( ω ; λ ) μ \mu μ T − 1 T^{-1} T − 1 α \alpha α ϵ \epsilon ϵ N N N ∫ ∣ t ∣ > T / 2 ∣ f ( t ) ∣ d t = O ( α 2 ) \int_{|t|>T/2}|f(t)|\mathrm{d}t=\mathcal{O}(\alpha^2) ∫ ∣ t ∣ > T /2 ∣ f ( t ) ∣ d t = O ( α 2 ) ϵ \epsilon ϵ T t o t a l = Ω ( N / ϵ ) T_{\rm total}=\Omega(N/\epsilon) T total = Ω ( N / ϵ )
The rigorous statement and proof of the above theorem are provided in Theorem 13 of § 2 . This result implies that, to overcome the linear-in-N N N f ^ \hat{f} f ^ p b p_{\rm b} p b μ \mu μ f f f p b p_{\rm b} p b μ = δ 0 \mu=\delta_0 μ = δ 0
The fundamental bottleneck identified in Theorem 5 arises from the additional correlations among system qubits induced by the randomness of ω \omega ω H = λ ∑ i = 1 N c i † c i H=\lambda\sum_{i=1}^N c_i^\dagger c_i H = λ ∑ i = 1 N c i † c i (7) is satisfied, the algorithm guarantees only that each single-qubit subsystem has the correct thermal marginal; it does not ensure that the joint state is a product state, as required for the target Gibbs state. In contrast, this obstruction is absent in ground-state preparation, where the target is a pure product state and correct single-qubit marginals uniquely determine the global state. As a result, ground-state preparation can fully exploit the additional flexibility provided by the random bath frequencies, achieving polylogarithmic complexity in both N N N 1 / ϵ 1/\epsilon 1/ ϵ Theorem 2 .
Conclusion and outlook – We have shown that an extensive engineered bath enables dissipative preparation of free-fermion thermal and ground states with sublinear total Hamiltonian simulation time, while retaining local system-bath couplings and convergence from arbitrary initial states. Specifically, the thermal state preparation algorithm achieves O ~ ( N / ϵ ) \widetilde{\mathcal{O}}(\sqrt{N}/\epsilon) O ( N / ϵ ) O ~ ( Δ − 2 p o l y l o g ( N / ϵ ) ) \widetilde{\mathcal{O}}\left(\Delta^{-2}\mathrm{polylog}(N/\epsilon)\right) O ( Δ − 2 polylog ( N / ϵ ) ) Δ \Delta Δ
Our lower bound results also expose distinct obstacles to faster thermal state preparation within the protocols studied. In the zero-bath setting, preserving the target state at second order is insufficient: fourth-order corrections must also satisfy a thermal state preservation condition to improve the N / ϵ \sqrt{N}/\epsilon N / ϵ N N N
Very recently, [1 ] introduced a distinct system-bath approach that achieves a physical evolution time of p o l y l o g ( N / ϵ ) \mathrm{polylog}(N/\epsilon) polylog ( N / ϵ )
Further directions include quantifying the trade-off between bath size and simulation time for intermediate bath sizes, and determining whether suitable bath correlations or alternative coupling structures can circumvent the identified obstructions. Extending the analysis to weakly interacting fermions would test whether the advantage survives. Free fermions thus provide a controlled setting for identifying which features of engineered dissipation can support faster state preparation in interacting systems.
Declaration of AI Assistance
Generative AI tools (ChatGPT 6.0, OpenAI) were used in the development and preparation of this manuscript. The upper-bound results and their proofs were developed independently by the authors, with AI assistance limited to language polishing, checking, and presentation. For the lower-bound results, AI tools contributed ideas and proof strategies, including the central idea underlying the proof of Theorem 4 . All AI-assisted arguments were subsequently reformulated, rigorously developed, and independently verified by the authors. The authors take full responsibility for the correctness and originality of all results, proofs, and conclusions presented in this work.
Acknowledgments
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Accelerated Research in Quantum Computing Centers, Quantum Utility through Advanced Computational Quantum Algorithms, grant no. DE-SC0025572 (L.L.). L.L. is a Simons Investigator in Mathematics. This work was also supported in part by the University of Michigan through a startup grant of Z.D. (Z.D., K.W.) and the Van Loo Postdoctoral Fellowship (K.W.).
Supplementary Material for
Sublinear thermal state and ground state preparation for free fermions via system-bath interaction
Zero-bath setting for thermal state preparation
In this section, we consider the thermal state preparation problem in the zero-bath setting. We will provide a rigorous statement of Theorem 1 for complexity upper bound and Theorem 8 for the complexity lower bound.
We start with the upper bound result. The following theorem provides a rigorous statement of Theorem 1 .
Theorem 6 (Zero bath Hamiltonian for thermal state preparation). Under Assumption 3 and Λ : = max k ∣ λ k ∣ = O ( 1 ) \Lambda:=\max_k|\lambda_k|=\mathcal{O}(1) Λ := max k ∣ λ k ∣ = O ( 1 ) f ^ ( ν ) = exp ( − 1 + β 2 ν 2 / 4 ) ω ( ν / S ) e − β ν / 4 ,
\hat{f}(\nu)=\exp\left(-\sqrt{1+\beta^2\nu^2}/4\right)\omega(\nu/S)e^{-\beta\nu/4}\,,
f ^ ( ν ) = exp ( − 1 + β 2 ν 2 /4 ) ω ( ν / S ) e − β ν /4 , S = Λ + 1 S=\Lambda+1 S = Λ + 1 ω \omega ω s > 1 s>1 s > 1 ∣ ν ∣ < 1 |\nu|<1 ∣ ν ∣ < 1 ν ≥ 0 \nu\ge0 ν ≥ 0 ω ( ν ) = ω ( − ν ) \omega(\nu)=\omega(-\nu) ω ( ν ) = ω ( − ν ) ω ( ν ) = 1 \omega(\nu)=1 ω ( ν ) = 1 ν ∈ [ − 1 / 2 , 1 / 2 ] \nu\in[-1/2,1/2] ν ∈ [ − 1/2 , 1/2 ] ω ( ν ) = 0 \omega(\nu)=0 ω ( ν ) = 0 ∣ ν ∣ > 1 |\nu|>1 ∣ ν ∣ > 1 T = Θ ~ ( 1 + β ) T=\widetilde{\Theta}(1+\beta) T = Θ ( 1 + β ) α = Θ ~ ( ϵ 1 / 2 / ( log 2 ( 2 + β S ) min { N 1 / 4 exp ( β Λ / 4 ) , N 1 / 2 } ) ) \alpha=\widetilde{\Theta}(\epsilon^{1/2}/(\log^{2}(2+\beta S)\min\{N^{1/4}\exp(\beta\Lambda/4),N^{1/2}\})) α = Θ ( ϵ 1/2 / ( log 2 ( 2 + β S ) min { N 1/4 exp ( β Λ/4 ) , N 1/2 })) ∥ ρ f i x ( Φ α ) − σ β ∥ 1 < ϵ , \left\|\rho_{\rm fix}(\Phi_{\alpha})-\sigma_{\beta}\right\|_1<\epsilon\,, ∥ ρ fix ( Φ α ) − σ β ∥ 1 < ϵ , sup ρ ∥ Φ α m ( ρ ) − σ β ∥ 1 ≤ ϵ , when m = Θ ~ ( min { N 1 / 2 exp ( β Λ / 2 ) , N } ϵ ) . \sup_{\rho}\left\|\Phi_{\alpha}^m(\rho)-\sigma_{\beta}\right\|_1\leq \epsilon,\quad \text{when}\quad m=\widetilde{\Theta}\left(\frac{\min\{N^{1/2}\exp(\beta\Lambda/2),N\}}{\epsilon}\right)\,. sup ρ ∥ Φ α m ( ρ ) − σ β ∥ 1 ≤ ϵ , when m = Θ ( ϵ m i n { N 1/2 e x p ( β Λ/2 ) , N } ) . β = Θ ( 1 ) \beta=\Theta(1) β = Θ ( 1 ) σ β \sigma_\beta σ β ϵ \epsilon ϵ T t o t a l = O ~ ( β N 1 / 2 / ϵ ) T_{\rm total}=\widetilde{\mathcal{O}}(\beta N^{1/2}/\epsilon) T total = O ( β N 1/2 / ϵ ) Θ ~ , O ~ \widetilde{\Theta},\widetilde{\mathcal{O}} Θ , O β , N , 1 / ϵ \beta,N,1/\epsilon β , N , 1/ ϵ
We put the proof of the above theorem in § 1.2 . In Theorem 6 , f f f f ^ ( ν ) exp ( β ν / 2 ) = f ^ ( − ν ) \hat{f}(\nu)\exp(\beta\nu/2)=\hat{f}(-\nu) f ^ ( ν ) exp ( β ν /2 ) = f ^ ( − ν ) [11 ] . According to [11 Section 3] , this guarantees that the approximate Lindbladian operator for Φ α \Phi_{\alpha} Φ α σ β \sigma_\beta σ β H H H
Lower bound for zero-bath setting
In this section, we discuss the lower bound result for thermal state preparation in the zero-bath setting. As mentioned in main text, we assume β = 1 \beta=1 β = 1 ∥ h ∥ ≤ 1 \|h\|\leq 1 ∥ h ∥ ≤ 1 f ( t ) f(t) f ( t )
Condition 1 (Improvement condition for zero-bath setting). Define f ^ ( ν ) = ∫ − ∞ ∞ f ( t ) exp ( i ν t ) d t \hat{f}(\nu)=\int^\infty_{-\infty}f(t)\exp(i\nu t)\mathrm{d}t f ^ ( ν ) = ∫ − ∞ ∞ f ( t ) exp ( i ν t ) d t λ ∈ [ − 1 , 1 ] \lambda\in [-1,1] λ ∈ [ − 1 , 1 ] f f f λ \lambda λ
(Second order thermal state preservation) ∣ f ^ ( λ ) ∣ 2 ∣ f ^ ( λ ) ∣ 2 + ∣ f ^ ( − λ ) ∣ 2 = p ( λ ) : = 1 1 + exp ( λ )
\frac{\left|\hat{f}(\lambda)\right|^2}{\left|\hat{f}(\lambda)\right|^2+\left|\hat{f}(-\lambda)\right|^2}=p(\lambda):=\frac{1}{1+\exp(\lambda)}
∣ f ^ ( λ ) ∣ 2 + ∣ f ^ ( − λ ) ∣ 2 ∣ f ^ ( λ ) ∣ 2 = p ( λ ) := 1 + e x p ( λ ) 1
(Fourth order thermal state preservation) ( 1 − p ( λ ) ) G ( λ ) − p ( λ ) G ( − λ ) = 0 ,
\bigl(1-p(\lambda)\bigr)G(\lambda)
-p(\lambda)G(-\lambda)=0,
( 1 − p ( λ ) ) G ( λ ) − p ( λ ) G ( − λ ) = 0 , G ( λ ) = − 2 Re [ f ^ ( λ ) ‾ J ( λ ) ] G(\lambda)=-2\operatorname{Re}
\left[\overline{\hat{f}(\lambda)}J(\lambda)\right] G ( λ ) = − 2 Re [ f ^ ( λ ) J ( λ ) ] J ( ν ) : = ∫ − ∞ ∞ ∫ − ∞ t 1 ∫ − ∞ t 2 e i ν ( t 1 − t 2 + t 3 ) f ( t 3 ) [ f ( t 1 ) f ( t 2 ) ‾ + f ( t 1 ) ‾ f ( t 2 ) ] d t 3 d t 2 d t 1 . \begin{aligned}
J(\nu):=\int_{-\infty}^{\infty}
\int_{-\infty}^{t_1}
\int_{-\infty}^{t_2}\,
e^{i\nu(t_1-t_2+t_3)}f(t_3)\left[f(t_1)\overline{f(t_2)}
+\overline{f(t_1)}f(t_2)\right]dt_3dt_2dt_1\,.
\end{aligned} J ( ν ) := ∫ − ∞ ∞ ∫ − ∞ t 1 ∫ − ∞ t 2 e i ν ( t 1 − t 2 + t 3 ) f ( t 3 ) [ f ( t 1 ) f ( t 2 ) + f ( t 1 ) f ( t 2 ) ] d t 3 d t 2 d t 1 . (9) is satisfied, (10) is equivalent to Re J ( λ ) f ^ ( λ ) = Re J ( − λ ) f ^ ( − λ ) \operatorname{Re}\frac{J(\lambda)}{\hat f(\lambda)}
=\operatorname{Re}\frac{J(-\lambda)}{\hat f(-\lambda)} Re f ^ ( λ ) J ( λ ) = Re f ^ ( − λ ) J ( − λ )
The following theorem provides a necessary and sufficient condition for achieving improved complexity in the zero-bath setting for general free fermion Hamiltonians:
Theorem 8 (Criterion for improved complexity in the zero-bath setting). Consider the system-bath interaction algorithm under Assumption 3 at inverse temperature β = 1 \beta=1 β = 1 f ∈ L 1 ( R ) f\in L^1(\mathbb R) f ∈ L 1 ( R ) ∣ f ^ ( λ ) ∣ 2 + ∣ f ^ ( − λ ) ∣ 2 > 0 for all λ ∈ [ − 1 , 1 ] , η ( T ) : = ∫ ∣ t ∣ > T / 2 ∣ f ( t ) ∣ d t = O ( e − T s ) |\hat f(\lambda)|^2+|\hat f(-\lambda)|^2>0
\quad\text{for all }\lambda\in[-1,1],
\qquad
\eta(T):=\int_{|t|>T/2}|f(t)|\,\mathrm dt
=\mathcal O(e^{-T^s}) ∣ f ^ ( λ ) ∣ 2 + ∣ f ^ ( − λ ) ∣ 2 > 0 for all λ ∈ [ − 1 , 1 ] , η ( T ) := ∫ ∣ t ∣ > T /2 ∣ f ( t ) ∣ d t = O ( e − T s ) s > 0 s>0 s > 0
Fix C t a i l > 0 C_{\rm tail}>0 C tail > 0 α 0 , T 0 > 0 \alpha_0,T_0>0 α 0 , T 0 > 0 ϵ 0 ∈ ( 0 , 1 ) \epsilon_0\in(0,1) ϵ 0 ∈ ( 0 , 1 ) f f f C t a i l C_{\rm tail} C tail α , T , ϵ \alpha,T,\epsilon α , T , ϵ 0 < α ≤ α 0 , T ≥ T 0 , 0 < ϵ ≤ ϵ 0 , η ( T ) ≤ C t a i l α 4 ,
0<\alpha\le\alpha_0,\qquad
T\ge T_0,\qquad
0<\epsilon\le\epsilon_0,\qquad
\eta(T)\le C_{\rm tail}\alpha^4,
0 < α ≤ α 0 , T ≥ T 0 , 0 < ϵ ≤ ϵ 0 , η ( T ) ≤ C tail α 4 ,
The filter f f f λ \lambda λ (?) for every λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ]
For every N N N H = ∑ i , j h i j c i † c j H=\sum_{i,j}h_{ij}c_i^\dagger c_j H = ∑ i , j h ij c i † c j ∥ h ∥ ≤ 1 \|h\|\le1 ∥ h ∥ ≤ 1 0 < ϵ ≤ ϵ 0 0<\epsilon\le\epsilon_0 0 < ϵ ≤ ϵ 0 α , T \alpha,T α , T m m m sup ρ ∥ Φ α m ( ρ ) − σ 1 ( H ) ∥ 1 ≤ ϵ , T t o t a l = 2 T m = O ~ ( N 1 / 4 ϵ 1 / 2 ) . \sup_\rho\|\Phi_\alpha^m(\rho)-\sigma_1(H)\|_1\le\epsilon,
\qquad
T_{\rm total}=2Tm
=\widetilde{\mathcal O}\!\left(\frac{N^{1/4}}{\epsilon^{1/2}}\right). sup ρ ∥ Φ α m ( ρ ) − σ 1 ( H ) ∥ 1 ≤ ϵ , T total = 2 T m = O ( ϵ 1/2 N 1/4 ) .
Moreover, if condition 1 fails, there exist an energy λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ] c > 0 c>0 c > 0 H = λ ∑ i = 1 N c i † c i H=\lambda\sum_{i=1}^N c_i^\dagger c_i H = λ ∑ i = 1 N c i † c i (11) and achieve ϵ \epsilon ϵ T t o t a l ≥ c N ϵ . T_{\rm total}\ge c\,\frac{\sqrt N}{\epsilon}. T total ≥ c ϵ N .
We put the proof of Theorem 8 in Appendix § 1.3 . In Condition (?) , the second-order thermal state preservation condition (9) is necessary to ensure that the leading O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) α → 0 \alpha\to 0 α → 0 O ( 1 ) \mathcal{O}(1) O ( 1 ) ϵ \epsilon ϵ (10) is necessary and sufficient for the fixed-point error to be O ( α 4 ) \mathcal{O}(\alpha^4) O ( α 4 ) ϵ \epsilon ϵ α = Θ ( ϵ 1 / 4 N − 1 / 8 ) \alpha=\Theta(\epsilon^{1/4}N^{-1/8}) α = Θ ( ϵ 1/4 N − 1/8 ) m = O ~ ( N 1 / 4 / ϵ 1 / 2 ) m=\widetilde{\mathcal O}(N^{1/4}/\epsilon^{1/2}) m = O ( N 1/4 / ϵ 1/2 ) T t o t a l = O ~ ( N 1 / 4 / ϵ 1 / 2 ) T_{\rm total}=\widetilde{\mathcal O}(N^{1/4}/\epsilon^{1/2}) T total = O ( N 1/4 / ϵ 1/2 ) § 1.2 . In addition, for general free-fermion Hamiltonians, we can also transform the system Hamiltonian to a diagonal form and apply the same analysis to H = ∑ i = 1 N λ i n i H=\sum_{i=1}^N\lambda_in_i H = ∑ i = 1 N λ i n i λ i ∈ [ − 1 , 1 ] \lambda_i\in[-1,1] λ i ∈ [ − 1 , 1 ] N / ϵ \sqrt{N}/\epsilon N / ϵ (9) and (10) for all λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ]
Together with Theorem 8 , the following lemma yields Theorem 4 :
Lemma 9. Let 1 < S ≤ 3 1<S\le3 1 < S ≤ 3 w ∈ C c ∞ ( [ − 1 , 1 ] ) w\in C_c^\infty([-1,1]) w ∈ C c ∞ ([ − 1 , 1 ]) ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) x ≥ 0 x\ge0 x ≥ 0 f ^ ( ν ) = F S ( ν ) : = e − ν / 4 e − 1 + ν 2 / 4 w ( ν / S ) \hat f(\nu)=F_S(\nu):=e^{-\nu/4}e^{-\sqrt{1+\nu^2}/4}w(\nu/S) f ^ ( ν ) = F S ( ν ) := e − ν /4 e − 1 + ν 2 /4 w ( ν / S ) (?) (10) on [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
We put the proof of Lemma 9 in Appendix § 1.3 . For the monotone cutoff chosen in Theorem 6 , this lemma and Theorem 8 imply Theorem 4 , as explained at the end of the proof.
Proof of Theorem 6
To prove Theorem 6 , we first have the following proposition:
Proposition 10. Under Assumption 3 , there are CPTP maps Φ α , j : C 2 N × 2 N → C 2 N × 2 N \Phi_{\alpha,j}:\mathbb C^{2^N\times2^N}\to
\mathbb C^{2^N\times2^N} Φ α , j : C 2 N × 2 N → C 2 N × 2 N j j j
[ Φ α , j , Φ α , k ] = 0 ( j ≠ k ) , Φ α = Φ α , 1 ∘ ⋯ ∘ Φ α , N . [\Phi_{\alpha,j},\Phi_{\alpha,k}]=0
\quad(j\ne k),
\qquad \Phi_\alpha
=\Phi_{\alpha,1}\circ\cdots\circ\Phi_{\alpha,N}. [ Φ α , j , Φ α , k ] = 0 ( j = k ) , Φ α = Φ α , 1 ∘ ⋯ ∘ Φ α , N .
Φ α , j \Phi_{\alpha,j} Φ α , j d j d_j d j n j = d j † d j n_j=d^\dagger_jd_j n j = d j † d j Φ α , j ( ρ ) = ∑ r ∈ { 00 , 10 , 01 , 11 } K j , r ρ K j , r † , ∑ r ∈ { 00 , 10 , 01 , 11 } K j , r † K j , r = I ,
\Phi_{\alpha,j}(\rho)
=\sum_{r\in\{00,10,01,11\}}K_{j,r}\rho K_{j,r}^{\dagger},
\qquad
\sum_{r\in\{00,10,01,11\}}K_{j,r}^{\dagger}K_{j,r}=I,
Φ α , j ( ρ ) = ∑ r ∈ { 00 , 10 , 01 , 11 } K j , r ρ K j , r † , ∑ r ∈ { 00 , 10 , 01 , 11 } K j , r † K j , r = I , K j , 00 = a j , 00 ( 0 ) ( I − n j ) + a j , 00 ( 1 ) n j , K j , 10 = a j , 10 d j † , K j , 01 = a j , 01 d j , K j , 11 = a j , 11 ( 0 ) ( I − n j ) + a j , 11 ( 1 ) n j .
\begin{aligned}
K_{j,00}&=a_{j,00}^{(0)}(I-n_j)+a_{j,00}^{(1)}n_j,
&K_{j,10}&=a_{j,10}d_j^\dagger,\\
K_{j,01}&=a_{j,01}d_j,
&K_{j,11}&=a_{j,11}^{(0)}(I-n_j)+a_{j,11}^{(1)}n_j.
\end{aligned}
K j , 00 K j , 01 = a j , 00 ( 0 ) ( I − n j ) + a j , 00 ( 1 ) n j , = a j , 01 d j , K j , 10 K j , 11 = a j , 10 d j † , = a j , 11 ( 0 ) ( I − n j ) + a j , 11 ( 1 ) n j .
In addition, when α ∥ f ∥ L 1 = O ( 1 ) \alpha\|f\|_{L^1}=\mathcal{O}(1) α ∥ f ∥ L 1 = O ( 1 ) L j \mathcal{L}_j L j ∥ Φ α , j − U j ( T ) ∘ exp ( L j α 2 ) ∘ U j ( T ) ∥ 1 ↔ 1 = O ( α 4 min { ∥ f ( t ) ∥ L ∞ 4 T 4 , ∥ f ∥ L 1 4 } ) ,
\left\|\Phi_{\alpha,j}-\mathcal U_j(T)\circ\exp\left(\mathcal L_{j}\alpha^2\right)\circ \mathcal U_j(T)\right\|_{1\leftrightarrow1}=\mathcal{O}\left(\alpha^4\min\left\{\left\|f(t)\right\|^4_{L^\infty}T^4,\|f\|_{L^1}^4\right\}\right)\,,
Φ α , j − U j ( T ) ∘ exp ( L j α 2 ) ∘ U j ( T ) 1 ↔ 1 = O ( α 4 min { ∥ f ( t ) ∥ L ∞ 4 T 4 , ∥ f ∥ L 1 4 } ) , U j ( T ) [ ρ ] = e − i λ j n j T ρ e i λ j n j T \mathcal U_j(T)[\rho]
=e^{-i\lambda_jn_jT}\rho e^{i\lambda_jn_jT} U j ( T ) [ ρ ] = e − i λ j n j T ρ e i λ j n j T L j ( ρ ) = − i [ G j , L a m b , ρ ] + L j , c ρ L j , c † − 1 2 { L j , c † L j , c , ρ } + L j , n ρ L j , n † − 1 2 { L j , n † L j , n , ρ } ,
\begin{aligned}
\mathcal L_j(\rho)
={}&-i[G_{j,\rm Lamb},\rho]+L_{j,\rm c}\rho L_{j,\rm c}^\dagger
-\frac12\left\{L_{j,\rm c}^\dagger L_{j,\rm c},\rho\right\}+L_{j,\rm n}\rho L_{j,\rm n}^\dagger
-\frac12\left\{L_{j,\rm n}^\dagger L_{j,\rm n},\rho\right\},
\end{aligned}
L j ( ρ ) = − i [ G j , Lamb , ρ ] + L j , c ρ L j , c † − 2 1 { L j , c † L j , c , ρ } + L j , n ρ L j , n † − 2 1 { L j , n † L j , n , ρ } , L j , c = ∫ − T T f ( t ) e i λ j t d t d j † , L j , n = ∫ − T T f ( t ) e − i λ j t d t d j ,
L_{j,\rm c}=\int^{T}_{-T}f(t)e^{i\lambda_jt}\mathrm{d}td_j^\dagger,
\qquad
L_{j,\rm n}=\int^{T}_{-T}f(t)e^{-i\lambda_jt}\mathrm{d}td_j,
% \qquad
% \ell_{j,\rm c}:=\int^{T}_{-T}f(t)e^{i\lambda_jt}\mathrm{d}t,
% \qquad
% \ell_{j,\rm n}:=\int^{T}_{-T}f(t)e^{-i\lambda_jt}\mathrm{d}t,
L j , c = ∫ − T T f ( t ) e i λ j t d t d j † , L j , n = ∫ − T T f ( t ) e − i λ j t d t d j , [ G j , L a m b , σ β ] = 0 [G_{j,\rm Lamb},\sigma_\beta]=0 [ G j , Lamb , σ β ] = 0
Proof of Proposition 10 . First, we notice that
c i † = ∑ j U i , j ∗ d j † , c i = ∑ j U i , j d j . c_i^\dagger=\sum_j U^*_{i,j}d_j^\dagger,\qquad c_i=\sum_j U_{i,j}d_j\,. c i † = j ∑ U i , j ∗ d j † , c i = j ∑ U i , j d j . Define two new sets of bath operators:
e j , + † = ∑ i = 1 N U i , j ∗ b i , + † , e j , − † = ∑ i = 1 N U i , j b i , − † . e^\dagger_{j,+} = \sum_{i=1}^N U^*_{i,j}b^\dagger_{i,+},\qquad e^\dagger_{j,-} = \sum_{i=1}^N U_{i,j}b^\dagger_{i,-}\,. e j , + † = i = 1 ∑ N U i , j ∗ b i , + † , e j , − † = i = 1 ∑ N U i , j b i , − † . Because U U U { e j , + † , e j , + } j = 1 N \{e^\dagger_{j,+},e_{j,+}\}^N_{j=1} { e j , + † , e j , + } j = 1 N { e j , − † , e j , − } j = 1 N \{e^\dagger_{j,-},e_{j,-}\}^N_{j=1} { e j , − † , e j , − } j = 1 N { d j † , d j } \{d^\dagger_j,d_j\} { d j † , d j } Assumption 3 , we then have
H i n t ( t ) = ∑ j = 1 N [ f ( t ) ( d j † ⊗ e j , + † + d j ⊗ e j , − † ) + f ( t ) ‾ ( d j ⊗ e j , + + d j † ⊗ e j , − ) ] . H_{\rm int}(t)=\sum_{j=1}^N\left[f(t)\left(d_j^\dagger\otimes e_{j,+}^\dagger+d_j\otimes e_{j,-}^\dagger\right)+\overline{f(t)}\left(d_j\otimes e_{j,+}+d_j^\dagger\otimes e_{j,-}\right)\right]\,. H int ( t ) = j = 1 ∑ N [ f ( t ) ( d j † ⊗ e j , + † + d j ⊗ e j , − † ) + f ( t ) ( d j ⊗ e j , + + d j † ⊗ e j , − ) ] . We then have
H α ( t ) = ∑ j = 1 N H α , j ( t ) , \begin{equation}
H_{\alpha}(t)=\sum_{j=1}^NH_{\alpha,j}(t), \tag*{(17)}
\end{equation} H α ( t ) = j = 1 ∑ N H α , j ( t ) , (17) where
H α , j ( t ) = λ j n j + α [ f ( t ) ( d j † ⊗ e j , + † + d j ⊗ e j , − † ) + f ( t ) ‾ ( d j ⊗ e j , + + d j † ⊗ e j , − ) ] . \begin{equation}
H_{\alpha,j}(t)
=\lambda_jn_j+\alpha\left[f(t)\left(
d_j^\dagger\otimes e_{j,+}^\dagger
+d_j\otimes e_{j,-}^\dagger\right)
+\overline{f(t)}\left(
d_j\otimes e_{j,+}
+d_j^\dagger\otimes e_{j,-}\right)\right]. \tag*{(18)}
\end{equation} H α , j ( t ) = λ j n j + α [ f ( t ) ( d j † ⊗ e j , + † + d j ⊗ e j , − † ) + f ( t ) ( d j ⊗ e j , + + d j † ⊗ e j , − ) ] . (18) Every interaction term in H α , j H_{\alpha,j} H α , j
[ H α , j ( s ) , H α , k ( t ) ] = 0 ( j ≠ k ) . \begin{equation}
[H_{\alpha,j}(s),H_{\alpha,k}(t)]=0
\qquad(j\ne k). \tag*{(19)}
\end{equation} [ H α , j ( s ) , H α , k ( t )] = 0 ( j = k ) . (19) Consequently,
U α ( t ) = ∏ j = 1 N U α , j ( t ) , U α , j ( t ) : = T exp ( − i ∫ − T t H α , j ( s ) d s ) , [ U α , j ( t ) , U α , k ( s ) ] = 0 ( j ≠ k ) . \begin{equation}
U^\alpha(t)=\prod_{j=1}^NU_{\alpha,j}(t),
\quad
U_{\alpha,j}(t):=\mathcal T\exp\left(
-i\int_{-T}^tH_{\alpha,j}(s)\,\mathrm ds\right),\quad [U_{\alpha,j}(t),U_{\alpha,k}(s)]=0\quad(j\ne k)\,. \tag*{(20)}
\end{equation} U α ( t ) = j = 1 ∏ N U α , j ( t ) , U α , j ( t ) := T exp ( − i ∫ − T t H α , j ( s ) d s ) , [ U α , j ( t ) , U α , k ( s )] = 0 ( j = k ) . (20) Now, we claim that the reduced channel Φ α \Phi_\alpha Φ α (1) can be decomposed as
Φ α = Φ α , 1 ∘ ⋯ ∘ Φ α , N , \begin{equation}
\Phi_\alpha=\Phi_{\alpha,1}\circ\cdots\circ\Phi_{\alpha,N}, \tag*{(21)}
\end{equation} Φ α = Φ α , 1 ∘ ⋯ ∘ Φ α , N , (21) where Φ α , j \Phi_{\alpha,j} Φ α , j U α , j ( t ) U_{\alpha,j}(t) U α , j ( t )
Φ α , j ( ρ ) = Tr E [ U α , j ( T ) ( ρ ⊗ ρ E ) U α , j ( T ) † ] . \Phi_{\alpha,j}(\rho)=\operatorname{Tr}_E\left[U_{\alpha,j}(T)(\rho\otimes \rho_E)U_{\alpha,j}(T)^\dagger\right]\,. Φ α , j ( ρ ) = Tr E [ U α , j ( T ) ( ρ ⊗ ρ E ) U α , j ( T ) † ] . To derive (21) , we first note that U α , j ( t ) U_{\alpha,j}(t) U α , j ( t ) j j j U α , j ( t ) U_{\alpha,j}(t) U α , j ( t ) ∣ 0 ⟩ ⟨ 0 ∣ \ket{0}\bra{0} ∣ 0 ⟩ ⟨ 0 ∣ ∣ 1 ⟩ ⟨ 1 ∣ \ket{1}\bra{1} ∣ 1 ⟩ ⟨ 1 ∣ ∣ 1 ⟩ ⟨ 0 ∣ \ket{1}\bra{0} ∣ 1 ⟩ ⟨ 0 ∣ ∣ 0 ⟩ ⟨ 1 ∣ \ket{0}\bra{1} ∣ 0 ⟩ ⟨ 1 ∣ j j j (21) .
Finally, for each Φ α , j \Phi_{\alpha,j} Φ α , j [6 Theorem 11] and [2 Theorem S1] gives (14) . The interaction terms c i † ⊗ b i , + † c^\dagger_i\otimes b^\dagger_{i,+} c i † ⊗ b i , + † c i ⊗ b i , − † c_i\otimes b^\dagger_{i,-} c i ⊗ b i , − † L j , c L_{j,\rm c} L j , c L j , n L_{j,\rm n} L j , n Proposition 10 , respectively. Moreover, the Lamb-shift terms are linear combination of 1 − n j 1-n_j 1 − n j n j n_j n j [ G j , L a m b , n k ] = 0 [G_{j,\rm Lamb},n_k]=0 [ G j , Lamb , n k ] = 0 k k k
Now, we are ready to prove Theorem 6 .
Proof of Theorem 6 . We first note that the choice of f f f Theorem 6 satisfies [11 Assumption 13] (see [11 (3.20)] ). According to [11 Section 3.1] , for any j j j B j B_j B j
L j , K M S ( ρ ) : = − i [ B j , ρ ] + ( L j , c , ∞ ρ L j , c , ∞ † − 1 2 { L j , c , ∞ † L j , c , ∞ , ρ } ) + ( L j , n , ∞ ρ L j , n , ∞ † − 1 2 { L j , n , ∞ † L j , n , ∞ , ρ } ) \mathcal{L}_{j,\rm KMS}(\rho):=-i[B_j,\rho]+\left( L_{j,\rm c,\infty} \rho L_{j,\rm c,\infty}^\dagger - \frac{1}{2} \left\{ L_{j,\rm c,\infty}^\dagger L_{j,\rm c,\infty}, \rho \right\} \right)+\left( L_{j,\rm n,\infty} \rho L_{j,\rm n,\infty}^\dagger - \frac{1}{2} \left\{ L_{j,\rm n,\infty}^\dagger L_{j,\rm n,\infty}, \rho \right\} \right) L j , KMS ( ρ ) := − i [ B j , ρ ] + ( L j , c , ∞ ρ L j , c , ∞ † − 2 1 { L j , c , ∞ † L j , c , ∞ , ρ } ) + ( L j , n , ∞ ρ L j , n , ∞ † − 2 1 { L j , n , ∞ † L j , n , ∞ , ρ } ) satisfies KMS-DBC with respect to the Gibbs state σ β = ∏ j σ β , j \sigma_\beta=\prod_j \sigma_{\beta,j} σ β = ∏ j σ β , j σ β , j = 1 1 + exp ( − β λ j ) ( 1 − n j ) + exp ( − β λ j ) 1 + exp ( − β λ j ) n j \sigma_{\beta,j}=\frac{1}{1+\exp(-\beta\lambda_j)}(1-n_j)+\frac{\exp(-\beta\lambda_j)}{1+\exp(-\beta\lambda_j)}n_j σ β , j = 1 + e x p ( − β λ j ) 1 ( 1 − n j ) + 1 + e x p ( − β λ j ) e x p ( − β λ j ) n j L j , K M S ( σ β ) = 0 \mathcal{L}_{j,\rm KMS}(\sigma_\beta)=0 L j , KMS ( σ β ) = 0
L j , c , ∞ = ∫ − ∞ ∞ f ( t ) exp ( i λ j t ) d j † d t , L j , n , ∞ = ∫ − ∞ ∞ f ( t ) exp ( − i λ j t ) d j d t . L_{j,\rm c,\infty}=\int^{\infty}_{-\infty}f(t)\exp(i\lambda_jt)d^\dagger_j\mathrm{d}t,\quad L_{j,\rm n,\infty}=\int^{\infty}_{-\infty}f(t)\exp(-i\lambda_jt)d_j\mathrm{d}t\,. L j , c , ∞ = ∫ − ∞ ∞ f ( t ) exp ( i λ j t ) d j † d t , L j , n , ∞ = ∫ − ∞ ∞ f ( t ) exp ( − i λ j t ) d j d t . In addition, according to the formula of B j B_{j} B j [11 Theorem 10] , we have [ B j , n k ] = 0 [B_{j},n_k]=0 [ B j , n k ] = 0 k k k
Furthermore, according to [11 Lemma 30, (D3, D13)] , we have ∥ f ( t ) ∥ L ∞ = O ( 1 ) \|f(t)\|_{L^\infty}=\mathcal{O}(1) ∥ f ( t ) ∥ L ∞ = O ( 1 ) ∥ f ( t ) ∥ L 1 = O ( log ( 2 + β S ) ) \|f(t)\|_{L^1}=\mathcal{O}\left(\log(2+\beta S)\right) ∥ f ( t ) ∥ L 1 = O ( log ( 2 + β S ) )
∫ ∣ t ∣ > T ∣ f ( t ) ∣ d t = O ( ( 1 + β ) exp ( − Θ ( ( T / ( 1 + β ) ) 1 / s ) ) ) . \int_{|t|>T}|f(t)|\mathrm{d}t=\mathcal{O}\left((1+\beta)\exp\left(-\Theta\left((T/(1+\beta))^{1/s}\right)\right)\right)\,. ∫ ∣ t ∣ > T ∣ f ( t ) ∣ d t = O ( ( 1 + β ) exp ( − Θ ( ( T / ( 1 + β ) ) 1/ s ) ) ) . Because ∥ d j † ∥ = ∥ d j ∥ = 1 \|d^\dagger_j\|=\|d_j\|=1 ∥ d j † ∥ = ∥ d j ∥ = 1
∥ L j , K M S + i [ B j − G j , L a m b , ⋅ ] − L j ∥ 1 ↔ 1 = O ( ( 1 + β ) exp ( − Θ ( ( T / ( 1 + β ) ) 1 / s ) ) ) . \left\|\mathcal{L}_{j,\rm KMS}+i[B_j-G_{j, \rm Lamb},\cdot]-\mathcal{L}_{j}\right\|_{1\leftrightarrow1}=\mathcal{O}\left((1+\beta)\exp\left(-\Theta\left((T/(1+\beta))^{1/s}\right)\right)\right)\,. ∥ L j , KMS + i [ B j − G j , Lamb , ⋅ ] − L j ∥ 1 ↔ 1 = O ( ( 1 + β ) exp ( − Θ ( ( T / ( 1 + β ) ) 1/ s ) ) ) . Furthermore, according to Proposition 10 (14) , we have
∥ Φ α , j − U j ( T ) ∘ exp ( ( L j , K M S + i [ B j − G j , L a m b , ⋅ ] ) α 2 ) ∘ U j ( T ) ⏟ : = Φ ~ α , j ∥ 1 ↔ 1 = O ( α 2 ( 1 + β ) exp ( − Θ ( ( T / ( 1 + β ) ) 1 / s ) ) + α 4 ∥ f ( t ) ∥ L 1 4 ) = O ( α 2 ( 1 + β ) exp ( − Θ ( ( T / ( 1 + β ) ) 1 / s ) ) + α 4 log 4 ( 2 + β S ) ) \begin{equation}
\begin{aligned}
&\left\|\Phi_{\alpha,j}-\underbrace{\mathcal{U}_j(T)\circ\exp\left(\left(\mathcal{L}_{j,\rm KMS}+i[B_j-G_{j, \rm Lamb},\cdot]\right)\alpha^2\right)\circ \mathcal{U}_j(T)}_{:=\widetilde{\Phi}_{\alpha,j}}\right\|_{1\leftrightarrow1}\\
=&\mathcal{O}\left(\alpha^2(1+\beta)\exp\left(-\Theta\left((T/(1+\beta))^{1/s}\right)\right)+\alpha^4\|f(t)\|_{L^1}^4\right)\\
=&\mathcal{O}\left(\alpha^2(1+\beta)\exp\left(-\Theta\left((T/(1+\beta))^{1/s}\right)\right)+\alpha^4\log^4(2+\beta S)\right)
\end{aligned} \tag*{(22)}
\end{equation} = = Φ α , j − := Φ α , j U j ( T ) ∘ exp ( ( L j , KMS + i [ B j − G j , Lamb , ⋅ ] ) α 2 ) ∘ U j ( T ) 1 ↔ 1 O ( α 2 ( 1 + β ) exp ( − Θ ( ( T / ( 1 + β ) ) 1/ s ) ) + α 4 ∥ f ( t ) ∥ L 1 4 ) O ( α 2 ( 1 + β ) exp ( − Θ ( ( T / ( 1 + β ) ) 1/ s ) ) + α 4 log 4 ( 2 + β S ) ) (22) It is straightforward to check that Φ ~ α , j ( σ β ) = σ β \widetilde{\Phi}_{\alpha,j}(\sigma_{\beta})=\sigma_{\beta} Φ α , j ( σ β ) = σ β j j j Φ ~ α , j \tilde{\Phi}_{\alpha,j} Φ ~ α , j Proposition 10 (12) such that
Φ ~ α , j ( ρ ) = ∑ r ∈ { 00 , 10 , 01 , 11 } K ~ j , r ρ K ~ j , r † , ∑ r ∈ { 00 , 10 , 01 , 11 } K ~ j , r † K ~ j , r = I , \begin{equation}
\widetilde{\Phi}_{\alpha,j}(\rho)
=\sum_{r\in\{00,10,01,11\}}\widetilde{K}_{j,r}\rho \widetilde{K}_{j,r}^{\dagger},
\qquad
\sum_{r\in\{00,10,01,11\}}\widetilde{K}_{j,r}^{\dagger}\widetilde{K}_{j,r}=I, \tag*{(23)}
\end{equation} Φ α , j ( ρ ) = r ∈ { 00 , 10 , 01 , 11 } ∑ K j , r ρ K j , r † , r ∈ { 00 , 10 , 01 , 11 } ∑ K j , r † K j , r = I , (23) where the four Kraus operators have the form
K ~ j , 00 = a ~ j , 00 ( 0 ) ( I − n j ) + a ~ j , 00 ( 1 ) n j , K ~ j , 10 = a ~ j , 10 d j † , K ~ j , 01 = a ~ j , 01 d j , K ~ j , 11 = a ~ j , 11 ( 0 ) ( I − n j ) + a ~ j , 11 ( 1 ) n j . \begin{equation}
\begin{aligned}
\widetilde{K}_{j,00}&=\widetilde{a}_{j,00}^{(0)}(I-n_j)+\widetilde{a}_{j,00}^{(1)}n_j,
&\widetilde{K}_{j,10}&=\widetilde{a}_{j,10}d_j^\dagger,\\
\widetilde{K}_{j,01}&=\widetilde{a}_{j,01}d_j,
&\widetilde{K}_{j,11}&=\widetilde{a}_{j,11}^{(0)}(I-n_j)+\widetilde{a}_{j,11}^{(1)}n_j.
\end{aligned} \tag*{(24)}
\end{equation} K j , 00 K j , 01 = a j , 00 ( 0 ) ( I − n j ) + a j , 00 ( 1 ) n j , = a j , 01 d j , K j , 10 K j , 11 = a j , 10 d j † , = a j , 11 ( 0 ) ( I − n j ) + a j , 11 ( 1 ) n j . (24) According to Proposition 10 and (22) and the eigenvector perturbation theorem, we can set T = Ω ( ( 1 + β ) p o l y l o g ( 2 + β ) ) T=\Omega((1+\beta)\mathrm{polylog}(2+\beta)) T = Ω (( 1 + β ) polylog ( 2 + β )) α = O ~ ( log − 2 ( 2 + β S ) ) \alpha=\widetilde{\mathcal{O}}(\log^{-2}(2+\beta S)) α = O ( log − 2 ( 2 + β S ))
ρ f i x ( Φ α ) = ∏ j = 1 N σ ~ β , j , such that σ ~ β , j = p ~ j n j + ( 1 − p ~ j ) ( I − n j ) , Φ α , j ( σ ~ β , j ) = σ ~ β , j , 0 ≤ p ~ j ≤ 1 \rho_{\rm fix}\left(\Phi_\alpha\right)=\prod^N_{j=1}\widetilde{\sigma}_{\beta,j},\ \text{such that}\quad \widetilde{\sigma}_{\beta,j}=\widetilde{p}_jn_j+(1-\widetilde{p}_j)(I-n_j),\quad \Phi_{\alpha,j}(\widetilde{\sigma}_{\beta,j})=\widetilde{\sigma}_{\beta,j},\quad 0\leq \widetilde{p}_j\leq 1 ρ fix ( Φ α ) = j = 1 ∏ N σ β , j , such that σ β , j = p j n j + ( 1 − p j ) ( I − n j ) , Φ α , j ( σ β , j ) = σ β , j , 0 ≤ p j ≤ 1 This means that the fixed point of Φ α \Phi_\alpha Φ α j j j ρ f i x ( Φ α ) \rho_{\rm fix}\left(\Phi_\alpha\right) ρ fix ( Φ α ) n j n_j n j
Furthermore, for any 1 ≤ j ≤ N 1\leq j\leq N 1 ≤ j ≤ N
R j ( ρ ) = ( ∏ i = 1 j σ β , i ) ( I 1 : j ⊗ T r 1 : j ( ρ ) ) , R ~ j ( ρ ) = ( ∏ i = 1 j σ ~ β , i ) ( I 1 : j ⊗ T r 1 : j ( ρ ) ) , \mathcal{R}_j(\rho)=\left(\prod^j_{i=1}\sigma_{\beta,i}\right)\left(I_{1:j}\otimes \mathrm{Tr}_{1:j}(\rho)\right),\quad \widetilde{\mathcal{R}}_j(\rho)=\left(\prod^j_{i=1}\widetilde{\sigma}_{\beta,i}\right)\left(I_{1:j}\otimes \mathrm{Tr}_{1:j}(\rho)\right)\,, R j ( ρ ) = ( i = 1 ∏ j σ β , i ) ( I 1 : j ⊗ Tr 1 : j ( ρ ) ) , R j ( ρ ) = ( i = 1 ∏ j σ β , i ) ( I 1 : j ⊗ Tr 1 : j ( ρ ) ) , where T r 1 : j \mathrm{Tr}_{1:j} Tr 1 : j j j j I 1 : j I_{1:j} I 1 : j j j j R 0 = i d \mathcal{R}_0=\mathrm{id} R 0 = id R ~ 0 = i d \widetilde{\mathcal{R}}_0=\mathrm{id} R 0 = id
Given any ρ \rho ρ
Φ ~ α , j m ( R j − 1 ( ρ ) ) = ( ∏ i = 1 j − 1 σ β , i ) Φ ~ α , j m ( I 1 : j − 1 ⊗ T r 1 : j − 1 ( ρ ) ) . \widetilde{\Phi}^m_{\alpha,j}\left(\mathcal{R}_{j-1}(\rho)\right)=\left(\prod^{j-1}_{i=1}\sigma_{\beta,i}\right)\widetilde{\Phi}^m_{\alpha,j}\left(I_{1:j-1}\otimes \mathrm{Tr}_{1:j-1}(\rho)\right)\,. Φ α , j m ( R j − 1 ( ρ ) ) = ( i = 1 ∏ j − 1 σ β , i ) Φ α , j m ( I 1 : j − 1 ⊗ Tr 1 : j − 1 ( ρ ) ) . We note that Φ ~ α , j m \widetilde{\Phi}^m_{\alpha,j} Φ α , j m I 1 : j − 1 ⊗ T r 1 : j − 1 ( ρ ) I_{1:j-1}\otimes \mathrm{Tr}_{1:j-1}(\rho) I 1 : j − 1 ⊗ Tr 1 : j − 1 ( ρ ) j j j d j † d^\dagger_j d j † ∣ 1 ⟩ ⟨ 0 ∣ \ket{1}\bra{0} ∣ 1 ⟩ ⟨ 0 ∣ d j d_j d j ∣ 0 ⟩ ⟨ 1 ∣ \ket{0}\bra{1} ∣ 0 ⟩ ⟨ 1 ∣ κ = Ω ( 1 ) \kappa=\Omega(1) κ = Ω ( 1 ) κ < 1 \kappa<1 κ < 1
sup j ∥ Φ ~ α , j m ( R j − 1 ( ⋅ ) ) − R j ∥ 1 ↔ 1 = O ( ϵ N ) , when m = Ω ( α − 2 κ − 1 p o l y l o g ( N / ϵ ) ) . \sup_j\left\|\widetilde{\Phi}^m_{\alpha,j}\left(\mathcal{R}_{j-1}(\cdot)\right)-\mathcal{R}_{j}\right\|_{1\leftrightarrow1}=\mathcal{O}\left(\frac{\epsilon}{N}\right),\quad\text{when}\quad m=\Omega\left(\alpha^{-2}\kappa^{-1}\mathrm{polylog}(N/\epsilon)\right)\,. j sup Φ α , j m ( R j − 1 ( ⋅ ) ) − R j 1 ↔ 1 = O ( N ϵ ) , when m = Ω ( α − 2 κ − 1 polylog ( N / ϵ ) ) . According to the eigenvalue perturbation theorem and (22) , when T = Ω ( ( 1 + β ) p o l y l o g ( 2 + β / κ ) ) T=\Omega((1+\beta)\mathrm{polylog}(2+\beta/\kappa)) T = Ω (( 1 + β ) polylog ( 2 + β / κ )) α = O ~ ( κ 1 / 2 / log 2 ( 2 + β S ) ) \alpha=\widetilde{\mathcal{O}}(\kappa^{1/2}/\log^{2}(2+\beta S)) α = O ( κ 1/2 / log 2 ( 2 + β S ))
sup j ∥ Φ α , j m ( R ~ j − 1 ( ⋅ ) ) − R ~ j ∥ 1 ↔ 1 = O ( ϵ N ) , when m = Ω ( α − 2 κ − 1 p o l y l o g ( N / ϵ ) ) . \sup_j\left\|\Phi^{m}_{\alpha,j}\left(\widetilde{\mathcal{R}}_{j-1}(\cdot)\right)-\widetilde{\mathcal{R}}_{j}\right\|_{1\leftrightarrow1}=\mathcal{O}\left(\frac{\epsilon}{N}\right),\quad\text{when}\quad m=\Omega\left(\alpha^{-2}\kappa^{-1}\mathrm{polylog}(N/\epsilon)\right)\,. j sup Φ α , j m ( R j − 1 ( ⋅ ) ) − R j 1 ↔ 1 = O ( N ϵ ) , when m = Ω ( α − 2 κ − 1 polylog ( N / ϵ ) ) . This implies that
∥ Φ α , N m ∘ ⋯ ∘ Φ α , 1 m ( ρ ) − ρ f i x ( Φ α ) ∥ 1 ≤ ∑ j = 1 N ∥ Φ α , N m ∘ … Φ α , j m ( R ~ j − 1 ( ρ ) ) − Φ α , N m ∘ … Φ α , j + 1 m ( R ~ j ( ρ ) ) ∥ 1 ≤ ∑ j = 1 N ∥ Φ α , j m ( R ~ j − 1 ( ρ ) ) − R ~ j ( ρ ) ∥ 1 = O ( ϵ ) , when m = Ω ( α − 2 κ − 1 p o l y l o g ( N / ϵ ) ) . \begin{equation}
\begin{aligned}
&\left\|\Phi_{\alpha,N}^m\circ \dots \circ \Phi_{\alpha,1}^m(\rho)-\rho_{\rm fix}\left(\Phi_\alpha\right)\right\|_1\\
\leq &\sum^N_{j=1}\left\|\Phi_{\alpha,N}^m\circ \dots \Phi_{\alpha,j}^m\left(\widetilde{\mathcal{R}}_{j-1}(\rho)\right)-\Phi_{\alpha,N}^m\circ \dots \Phi_{\alpha,j+1}^m\left(\widetilde{\mathcal{R}}_{j}(\rho)\right)\right\|_1\\
\leq &\sum^N_{j=1}\left\|\Phi^{m}_{\alpha,j}\left(\widetilde{\mathcal{R}}_{j-1}(\rho)\right)-\widetilde{\mathcal{R}}_{j}(\rho)\right\|_{1}=\mathcal{O}\left(\epsilon\right),\ \text{when}\ m=\Omega\left(\alpha^{-2}\kappa^{-1}\mathrm{polylog}(N/\epsilon)\right)\,.
\end{aligned} \tag*{(25)}
\end{equation} ≤ ≤ Φ α , N m ∘ ⋯ ∘ Φ α , 1 m ( ρ ) − ρ fix ( Φ α ) 1 j = 1 ∑ N Φ α , N m ∘ … Φ α , j m ( R j − 1 ( ρ ) ) − Φ α , N m ∘ … Φ α , j + 1 m ( R j ( ρ ) ) 1 j = 1 ∑ N Φ α , j m ( R j − 1 ( ρ ) ) − R j ( ρ ) 1 = O ( ϵ ) , when m = Ω ( α − 2 κ − 1 polylog ( N / ϵ ) ) . (25) for any density operator ρ \rho ρ
Finally, according to [6 Theorem 8] and (22) , setting T = Θ ( ( 1 + β ) p o l y l o g ( N ( 1 + β ) / ϵ ) ) T=\Theta((1+\beta)\mathrm{polylog}(N(1+\beta)/\epsilon)) T = Θ (( 1 + β ) polylog ( N ( 1 + β ) / ϵ )) α = Θ ~ ( ϵ 1 / 2 / ( log 2 ( 2 + β S ) min { N 1 / 4 exp ( β Λ / 4 ) , N 1 / 2 } ) ) \alpha=\widetilde{\Theta}(\epsilon^{1/2}/(\log^{2}(2+\beta S)\min\{N^{1/4}\exp(\beta\Lambda/4),N^{1/2}\})) α = Θ ( ϵ 1/2 / ( log 2 ( 2 + β S ) min { N 1/4 exp ( β Λ/4 ) , N 1/2 }))
sup j ∥ ( 1 1 + exp ( − β λ j ) 0 0 exp ( − β λ j ) 1 + exp ( − β λ j ) ) − ( 1 − p ~ j 0 0 p ~ j ) ∥ 1 = O ( ϵ max { 1 exp ( β Λ ) N , 1 N } ) . \sup_j\left\|
\begin{pmatrix}
\frac{1}{1+\exp(-\beta\lambda_j)}&0\\
0&\frac{\exp(-\beta\lambda_j)}{1+\exp(-\beta\lambda_j)}
\end{pmatrix}
-
\begin{pmatrix}
1-\widetilde p_j&0\\
0&\widetilde p_j
\end{pmatrix}\right\|_1=\mathcal{O}\left(\epsilon\max\left\{\frac{1}{\sqrt{\exp(\beta \Lambda)N}},\frac{1}{N}\right\}\right)\,. j sup ( 1 + e x p ( − β λ j ) 1 0 0 1 + e x p ( − β λ j ) e x p ( − β λ j ) ) − ( 1 − p j 0 0 p j ) 1 = O ( ϵ max { exp ( β Λ ) N 1 , N 1 } ) . Since Λ : = max k ∣ λ k ∣ = O ( 1 ) \Lambda:=\max_k|\lambda_k|=\mathcal{O}(1) Λ := max k ∣ λ k ∣ = O ( 1 ) ( 1 1 + exp ( − β λ j ) 0 0 exp ( − β λ j ) 1 + exp ( − β λ j ) ) ≥ C I \begin{pmatrix}
\frac{1}{1+\exp(-\beta\lambda_j)}&0\\
0&\frac{\exp(-\beta\lambda_j)}{1+\exp(-\beta\lambda_j)}
\end{pmatrix}\geq CI ( 1 + e x p ( − β λ j ) 1 0 0 1 + e x p ( − β λ j ) e x p ( − β λ j ) ) ≥ C I C = Ω ( exp ( − β Λ ) ) C=\Omega(\exp(-\beta\Lambda)) C = Ω ( exp ( − β Λ )) N N N j j j Lemma 17 , after tuning the constants in T , α T,\alpha T , α
∥ ρ f i x ( Φ α ) − σ β ∥ 1 ≤ ϵ 2 . \left\|\rho_{\rm fix}(\Phi_{\alpha})-\sigma_{\beta}\right\|_1\leq \frac{\epsilon}{2}\,. ∥ ρ fix ( Φ α ) − σ β ∥ 1 ≤ 2 ϵ . Combining this and (25) , we obtain
sup ρ ∥ Φ α m ( ρ ) − σ β ∥ 1 ≤ ϵ , when m = Θ ~ ( min { N 1 / 2 exp ( β Λ / 2 ) , N } ϵ ) . \sup_{\rho}\left\|\Phi_{\alpha}^m(\rho)-\sigma_{\beta}\right\|_1\leq \epsilon,\quad \text{when}\quad m=\widetilde{\Theta}\left(\frac{\min\{N^{1/2}\exp(\beta\Lambda/2),N\}}{\epsilon}\right)\,. ρ sup ∥ Φ α m ( ρ ) − σ β ∥ 1 ≤ ϵ , when m = Θ ( ϵ min { N 1/2 exp ( β Λ/2 ) , N } ) . This concludes the proof of Theorem 6 . ◻
Proof of Theorem 8 and Lemma 9
By Proposition 10 and (13) ,
Φ α = Φ α , 1 ∘ ⋯ ∘ Φ α , N , Φ α , j ( ρ ) = ∑ μ ∈ { 00 , 10 , 01 , 11 } K j , μ ρ K j , μ † . \Phi_\alpha=\Phi_{\alpha,1}\circ\cdots\circ\Phi_{\alpha,N},
\qquad
\Phi_{\alpha,j}(\rho)
=\sum_{\mu\in\{00,10,01,11\}}K_{j,\mu}\rho K_{j,\mu}^\dagger. Φ α = Φ α , 1 ∘ ⋯ ∘ Φ α , N , Φ α , j ( ρ ) = μ ∈ { 00 , 10 , 01 , 11 } ∑ K j , μ ρ K j , μ † . Following the notation in Proposition 10 , we define
a j : = a α , T ( λ j ) = ∣ a j , 10 ∣ 2 , z j : = z α , T ( λ j ) = ∣ a j , 01 ∣ 2 , q j : = q α , T ( λ j ) = a j + z j . \begin{gathered}
a_j:=a_{\alpha,T}(\lambda_j)=|a_{j,10}|^2,
\qquad
z_j:=z_{\alpha,T}(\lambda_j)=|a_{j,01}|^2,
\qquad
q_j:=q_{\alpha,T}(\lambda_j)=a_j+z_j.
\end{gathered} a j := a α , T ( λ j ) = ∣ a j , 10 ∣ 2 , z j := z α , T ( λ j ) = ∣ a j , 01 ∣ 2 , q j := q α , T ( λ j ) = a j + z j . Trace preservation gives
∑ μ ∈ { 00 , 11 } ∣ a j , μ ( 0 ) ∣ 2 = 1 − a j , ∑ μ ∈ { 00 , 11 } ∣ a j , μ ( 1 ) ∣ 2 = 1 − z j . \sum_{\mu\in\{00,11\}}|a_{j,\mu}^{(0)}|^2=1-a_j,
\qquad
\sum_{\mu\in\{00,11\}}|a_{j,\mu}^{(1)}|^2=1-z_j. μ ∈ { 00 , 11 } ∑ ∣ a j , μ ( 0 ) ∣ 2 = 1 − a j , μ ∈ { 00 , 11 } ∑ ∣ a j , μ ( 1 ) ∣ 2 = 1 − z j . We first consider the case when the density operator ρ \rho ρ ∣ 0 ⟩ ⟨ 0 ∣ , ∣ 1 ⟩ ⟨ 1 ∣ \ketbra{0}{0},\ketbra{1}{1} ∣ 0 ⟩ ⟨ 0 ∣ , ∣ 1 ⟩ ⟨ 1 ∣ d j d_j d j
v ( r ) : = ( 1 − r ) ∣ 0 ⟩ ⟨ 0 ∣ + r ∣ 1 ⟩ ⟨ 1 ∣ , 0 ≤ r ≤ 1 , ρ r : = ⨂ k = 1 N v ( r k ) , ρ < j : = ⨂ k < j v ( r k ) , ρ > j : = ⨂ k > j v ( r k ) . \begin{gathered}
v(r):=(1-r)\ket{0}\bra{0}+r\ket{1}\bra{1},
\quad 0\le r\le1,
\qquad
\rho_{\mathbf r}:=\bigotimes_{k=1}^Nv(r_k),\\
\rho_{<j}:=\bigotimes_{k<j}v(r_k),
\qquad
\rho_{>j}:=\bigotimes_{k>j}v(r_k).
\end{gathered} v ( r ) := ( 1 − r ) ∣ 0 ⟩ ⟨ 0 ∣ + r ∣ 1 ⟩ ⟨ 1 ∣ , 0 ≤ r ≤ 1 , ρ r := k = 1 ⨂ N v ( r k ) , ρ < j := k < j ⨂ v ( r k ) , ρ > j := k > j ⨂ v ( r k ) . Since
d j = Z ⊗ ( j − 1 ) ⊗ ∣ 0 ⟩ ⟨ 1 ∣ ⊗ I ⊗ ( N − j ) , Z : = ∣ 0 ⟩ ⟨ 0 ∣ − ∣ 1 ⟩ ⟨ 1 ∣ , Z v ( r ) Z = v ( r ) , d_j=Z^{\otimes(j-1)}\otimes\ket{0}\bra{1}\otimes I^{\otimes(N-j)},
\qquad
Z:=\ket{0}\bra{0}-\ket{1}\bra{1},
\qquad
Zv(r)Z=v(r), d j = Z ⊗ ( j − 1 ) ⊗ ∣ 0 ⟩ ⟨ 1 ∣ ⊗ I ⊗ ( N − j ) , Z := ∣ 0 ⟩ ⟨ 0 ∣ − ∣ 1 ⟩ ⟨ 1 ∣ , Z v ( r ) Z = v ( r ) , we obtain, with n j = d j † d j n_j=d_j^\dagger d_j n j = d j † d j
Φ α , j ( ρ r ) = ( 1 − a j ) ( I − n j ) ρ r ( I − n j ) + ( 1 − z j ) n j ρ r n j + a j d j † ρ r d j + z j d j ρ r d j † = ρ < j ⊗ ( ( 1 − a j ) ( 1 − r j ) + z j r j 0 0 a j ( 1 − r j ) + ( 1 − z j ) r j ) ⊗ ρ > j = ρ < j ⊗ v ( r j ′ ) ⊗ ρ > j , r j ′ = a j ( 1 − r j ) + ( 1 − z j ) r j = a j + ( 1 − q j ) r j . \begin{aligned}
\Phi_{\alpha,j}(\rho_{\mathbf r})
&=(1-a_j)(I-n_j)\rho_{\mathbf r}(I-n_j)
+(1-z_j)n_j\rho_{\mathbf r}n_j+a_jd_j^\dagger\rho_{\mathbf r}d_j
+z_jd_j\rho_{\mathbf r}d_j^\dagger\\
&=\rho_{<j}\otimes
\begin{pmatrix}
(1-a_j)(1-r_j)+z_jr_j&0\\
0&a_j(1-r_j)+(1-z_j)r_j
\end{pmatrix}
\otimes\rho_{>j}\\
&=\rho_{<j}\otimes v(r_j')\otimes\rho_{>j},
\qquad
r_j'=a_j(1-r_j)+(1-z_j)r_j=a_j+(1-q_j)r_j.
\end{aligned} Φ α , j ( ρ r ) = ( 1 − a j ) ( I − n j ) ρ r ( I − n j ) + ( 1 − z j ) n j ρ r n j + a j d j † ρ r d j + z j d j ρ r d j † = ρ < j ⊗ ( ( 1 − a j ) ( 1 − r j ) + z j r j 0 0 a j ( 1 − r j ) + ( 1 − z j ) r j ) ⊗ ρ > j = ρ < j ⊗ v ( r j ′ ) ⊗ ρ > j , r j ′ = a j ( 1 − r j ) + ( 1 − z j ) r j = a j + ( 1 − q j ) r j . Consequently,
Φ α ( ⨂ j = 1 N v ( r j ) ) = ⨂ j = 1 N v ( r j ′ ) = ⨂ j = 1 N v ( a α , T ( λ j ) + ( 1 − q α , T ( λ j ) ) r j ) . \begin{equation}
\Phi_\alpha\!\left(\bigotimes_{j=1}^N v(r_j)\right)
=\bigotimes_{j=1}^N v(r_j')
=\bigotimes_{j=1}^N
v\!\left(a_{\alpha,T}(\lambda_j)+(1-q_{\alpha,T}(\lambda_j))r_j\right). \tag*{(26)}
\end{equation} Φ α ( j = 1 ⨂ N v ( r j ) ) = j = 1 ⨂ N v ( r j ′ ) = j = 1 ⨂ N v ( a α , T ( λ j ) + ( 1 − q α , T ( λ j )) r j ) . (26) For sufficiently large T T T α \alpha α (28) below gives q α , T ( λ ) = Θ ( α 2 ) > 0 q_{\alpha,T}(\lambda)=\Theta(\alpha^2)>0 q α , T ( λ ) = Θ ( α 2 ) > 0 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
p α , T ( λ ) : = a α , T ( λ ) q α , T ( λ ) ⟹ a α , T ( λ ) + ( 1 − q α , T ( λ ) ) p α , T ( λ ) = p α , T ( λ ) , p_{\alpha,T}(\lambda):=\frac{a_{\alpha,T}(\lambda)}{q_{\alpha,T}(\lambda)}
\quad\Longrightarrow\quad
a_{\alpha,T}(\lambda)+(1-q_{\alpha,T}(\lambda))p_{\alpha,T}(\lambda)
=p_{\alpha,T}(\lambda), p α , T ( λ ) := q α , T ( λ ) a α , T ( λ ) ⟹ a α , T ( λ ) + ( 1 − q α , T ( λ )) p α , T ( λ ) = p α , T ( λ ) , and
ρ ∗ : = ⨂ j = 1 N v ( p α , T ( λ j ) ) , Φ α ( ρ ∗ ) = ρ ∗ . \begin{equation}
\rho_*:=\bigotimes_{j=1}^N v(p_{\alpha,T}(\lambda_j)),
\qquad
\Phi_\alpha(\rho_*)=\rho_*. \tag*{(27)}
\end{equation} ρ ∗ := j = 1 ⨂ N v ( p α , T ( λ j )) , Φ α ( ρ ∗ ) = ρ ∗ . (27) Let F T ( ν ) = ∫ − T T f ( t ) e i ν t d t F_T(\nu)=\int_{-T}^T f(t)e^{i\nu t}\,\mathrm dt F T ( ν ) = ∫ − T T f ( t ) e i ν t d t J T ( ν ) J_T(\nu) J T ( ν ) J ( ν ) J(\nu) J ( ν ) (?) with all three times restricted to [ − T , T ] [-T,T] [ − T , T ] G T ( ν ) = − 2 Re [ F T ( ν ) ‾ J T ( ν ) ] G_T(\nu)=-2\operatorname{Re}[\overline{F_T(\nu)}J_T(\nu)] G T ( ν ) = − 2 Re [ F T ( ν ) J T ( ν )]
a α , T ( λ ) = α 2 ∣ F T ( λ ) ∣ 2 + α 4 G T ( λ ) + O ( α 6 ) , z α , T ( λ ) = α 2 ∣ F T ( − λ ) ∣ 2 + α 4 G T ( − λ ) + O ( α 6 ) . \begin{equation}
\begin{aligned}
a_{\alpha,T}(\lambda)
&=\alpha^2|F_T(\lambda)|^2+\alpha^4G_T(\lambda)+\mathcal O(\alpha^6),\\
z_{\alpha,T}(\lambda)
&=\alpha^2|F_T(-\lambda)|^2+\alpha^4G_T(-\lambda)+\mathcal O(\alpha^6).
\end{aligned} \tag*{(28)}
\end{equation} a α , T ( λ ) z α , T ( λ ) = α 2 ∣ F T ( λ ) ∣ 2 + α 4 G T ( λ ) + O ( α 6 ) , = α 2 ∣ F T ( − λ ) ∣ 2 + α 4 G T ( − λ ) + O ( α 6 ) . (28) Write q 2 ( λ ) = ∣ f ^ ( λ ) ∣ 2 + ∣ f ^ ( − λ ) ∣ 2 q_2(\lambda)=|\hat f(\lambda)|^2+|\hat f(-\lambda)|^2 q 2 ( λ ) = ∣ f ^ ( λ ) ∣ 2 + ∣ f ^ ( − λ ) ∣ 2 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
p α , T ( λ ) = ∣ f ^ ( λ ) ∣ 2 q 2 ( λ ) + O ( α 2 + η ( T ) ) . \begin{equation}
p_{\alpha,T}(\lambda)
=\frac{|\hat f(\lambda)|^2}{q_2(\lambda)}+\mathcal O(\alpha^2+\eta(T)). \tag*{(29)}
\end{equation} p α , T ( λ ) = q 2 ( λ ) ∣ f ^ ( λ ) ∣ 2 + O ( α 2 + η ( T )) . (29) If (9) holds, define Δ 4 ( λ ) = ( 1 − p ( λ ) ) G ( λ ) − p ( λ ) G ( − λ ) \Delta_4(\lambda)=(1-p(\lambda))G(\lambda)-p(\lambda)G(-\lambda) Δ 4 ( λ ) = ( 1 − p ( λ )) G ( λ ) − p ( λ ) G ( − λ )
p α , T ( λ ) − p ( λ ) = α 2 Δ 4 ( λ ) q 2 ( λ ) + O ( α 4 + η ( T ) ) . \begin{equation}
p_{\alpha,T}(\lambda)-p(\lambda)
=\alpha^2\frac{\Delta_4(\lambda)}{q_2(\lambda)}
+\mathcal O(\alpha^4+\eta(T)). \tag*{(30)}
\end{equation} p α , T ( λ ) − p ( λ ) = α 2 q 2 ( λ ) Δ 4 ( λ ) + O ( α 4 + η ( T )) . (30) All constants in these estimates are independent of N N N
Proof of Theorem 8 . Suppose first that both improvement conditions hold for every λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ] η ( T ) ≤ C t a i l α 4 \eta(T)\le C_{\rm tail}\alpha^4 η ( T ) ≤ C tail α 4 (30) gives
sup ∣ λ ∣ ≤ 1 ∣ p α , T ( λ ) − p ( λ ) ∣ = O ( α 4 ) . \sup_{|\lambda|\le1}|p_{\alpha,T}(\lambda)-p(\lambda)|
=\mathcal O(\alpha^4). ∣ λ ∣ ≤ 1 sup ∣ p α , T ( λ ) − p ( λ ) ∣ = O ( α 4 ) . Using Lemma 17 ,
∥ ρ ∗ − σ 1 ( H ) ∥ 1 ≤ C N α 4 . \begin{equation}
\|\rho_*-\sigma_1(H)\|_1
\le C\sqrt N\,\alpha^4. \tag*{(31)}
\end{equation} ∥ ρ ∗ − σ 1 ( H ) ∥ 1 ≤ C N α 4 . (31) Furthermore, the proof of Theorem 6 shows that there exist constants C , c > 0 C,c>0 C , c > 0
sup ρ ∥ Φ α m ( ρ ) − ρ ∗ ∥ 1 ≤ C N e − c α 2 m . \begin{equation}
\sup_\rho\|\Phi_\alpha^m(\rho)-\rho_*\|_1
\le CN e^{-c\alpha^2m}. \tag*{(32)}
\end{equation} ρ sup ∥ Φ α m ( ρ ) − ρ ∗ ∥ 1 ≤ C N e − c α 2 m . (32) Choose α = Θ ( ϵ 1 / 4 N − 1 / 8 ) \alpha=\Theta(\epsilon^{1/4}N^{-1/8}) α = Θ ( ϵ 1/4 N − 1/8 ) T ≥ T 0 T\ge T_0 T ≥ T 0 η ( T ) ≤ C t a i l α 4 \eta(T)\le C_{\rm tail}\alpha^4 η ( T ) ≤ C tail α 4 T = O ~ ( 1 ) T=\widetilde{\mathcal O}(1) T = O ( 1 ) m = O ( α − 2 log ( N / ϵ ) ) m=\mathcal O(\alpha^{-2}\log(N/\epsilon)) m = O ( α − 2 log ( N / ϵ )) ϵ / 2 \epsilon/2 ϵ /2
T t o t a l = 2 T m = O ~ ( N 1 / 4 / ϵ 1 / 2 ) . T_{\rm total}=2Tm=\widetilde{\mathcal O}(N^{1/4}/\epsilon^{1/2}). T total = 2 T m = O ( N 1/4 / ϵ 1/2 ) . This proves sufficiency for arbitrary free-fermion Hamiltonians in the stated class.
Conversely, fix an energy λ \lambda λ H = λ ∑ j = 1 N n j H=\lambda\sum_{j=1}^N n_j H = λ ∑ j = 1 N n j (29) leaves a nonzero limiting bias. Thus the equilibrium state is Ω ( 1 ) \Omega(1) Ω ( 1 ) Δ 4 ( λ ) ≠ 0 \Delta_4(\lambda)\ne0 Δ 4 ( λ ) = 0 η ( T ) ≤ C t a i l α 4 \eta(T)\le C_{\rm tail}\alpha^4 η ( T ) ≤ C tail α 4 (30) becomes
p α , T ( λ ) − p ( λ ) = α 2 Δ 4 ( λ ) q 2 ( λ ) + O ( α 4 ) . p_{\alpha,T}(\lambda)-p(\lambda)
=\alpha^2\frac{\Delta_4(\lambda)}{q_2(\lambda)}+\mathcal O(\alpha^4). p α , T ( λ ) − p ( λ ) = α 2 q 2 ( λ ) Δ 4 ( λ ) + O ( α 4 ) . Thus, for sufficiently small α 0 \alpha_0 α 0 0 < α ≤ α 0 0<\alpha\le\alpha_0 0 < α ≤ α 0
∣ p α , T ( λ ) − p ( λ ) ∣ ≥ ∣ Δ 4 ( λ ) ∣ 2 q 2 ( λ ) α 2 . |p_{\alpha,T}(\lambda)-p(\lambda)|
\ge\frac{|\Delta_4(\lambda)|}{2q_2(\lambda)}\alpha^2. ∣ p α , T ( λ ) − p ( λ ) ∣ ≥ 2 q 2 ( λ ) ∣ Δ 4 ( λ ) ∣ α 2 . For a fixed p ∈ ( 0 , 1 ) p\in(0,1) p ∈ ( 0 , 1 ) r r r ( 0 , 1 ) (0,1) ( 0 , 1 )
∥ v ( r ) ⊗ N − v ( p ) ⊗ N ∥ 1 ≥ c p min { 1 , N ∣ r − p ∣ } . \begin{equation}
\|v(r)^{\otimes N}-v(p)^{\otimes N}\|_1
\ge c_p\min\{1,\sqrt N\,|r-p|\}. \tag*{(33)}
\end{equation} ∥ v ( r ) ⊗ N − v ( p ) ⊗ N ∥ 1 ≥ c p min { 1 , N ∣ r − p ∣ } . (33) Applying this estimate to (27) yields
∥ ρ ∗ − σ 1 ( H ) ∥ 1 ≥ c min { 1 , N α 2 } . \|\rho_*-\sigma_1(H)\|_1\ge c\min\{1,\sqrt N\,\alpha^2\}. ∥ ρ ∗ − σ 1 ( H ) ∥ 1 ≥ c min { 1 , N α 2 } . Since ρ ∗ \rho_* ρ ∗ ϵ \epsilon ϵ α 2 ≲ ϵ / N \alpha^2\lesssim\epsilon/\sqrt N α 2 ≲ ϵ / N ϵ \epsilon ϵ
Finally, set q : = q α , T ( λ ) ∈ ( 0 , 1 ) q:=q_{\alpha,T}(\lambda)\in(0,1) q := q α , T ( λ ) ∈ ( 0 , 1 ) p ∗ : = p α , T ( λ ) p_*:=p_{\alpha,T}(\lambda) p ∗ := p α , T ( λ )
ρ 0 ( b ) : = v ( b ) ⊗ N , ρ m ( b ) : = Φ α m ( ρ 0 ( b ) ) , b ∈ { 0 , 1 } . \rho_0^{(b)}:=v(b)^{\otimes N},
\qquad
\rho_m^{(b)}:=\Phi_\alpha^m(\rho_0^{(b)}),
\qquad b\in\{0,1\}. ρ 0 ( b ) := v ( b ) ⊗ N , ρ m ( b ) := Φ α m ( ρ 0 ( b ) ) , b ∈ { 0 , 1 } . By (26) ,
ρ m ( b ) = v ( r m ( b ) ) ⊗ N , r m ( b ) = p ∗ + ( b − p ∗ ) ( 1 − q ) m . \rho_m^{(b)}=v(r_m^{(b)})^{\otimes N},
\qquad
r_m^{(b)}=p_*+(b-p_*)(1-q)^m. ρ m ( b ) = v ( r m ( b ) ) ⊗ N , r m ( b ) = p ∗ + ( b − p ∗ ) ( 1 − q ) m . Uniform ϵ \epsilon ϵ
2 ϵ ≥ ∑ b = 0 1 ∥ ρ m ( b ) − σ 1 ( H ) ∥ 1 ≥ ∥ ρ m ( 1 ) − ρ m ( 0 ) ∥ 1 ≥ ∥ Tr 2 , … , N ( ρ m ( 1 ) − ρ m ( 0 ) ) ∥ 1 = ∥ v ( r m ( 1 ) ) − v ( r m ( 0 ) ) ∥ 1 = 2 ∣ r m ( 1 ) − r m ( 0 ) ∣ = 2 ( 1 − q ) m . \begin{aligned}
2\epsilon
&\ge\sum_{b=0}^1\|\rho_m^{(b)}-\sigma_1(H)\|_1
\ge\|\rho_m^{(1)}-\rho_m^{(0)}\|_1\ge\left\|\operatorname{Tr}_{2,\ldots,N}
\bigl(\rho_m^{(1)}-\rho_m^{(0)}\bigr)\right\|_1
=\|v(r_m^{(1)})-v(r_m^{(0)})\|_1\\
&=2|r_m^{(1)}-r_m^{(0)}|=2(1-q)^m.
\end{aligned} 2 ϵ ≥ b = 0 ∑ 1 ∥ ρ m ( b ) − σ 1 ( H ) ∥ 1 ≥ ∥ ρ m ( 1 ) − ρ m ( 0 ) ∥ 1 ≥ Tr 2 , … , N ( ρ m ( 1 ) − ρ m ( 0 ) ) 1 = ∥ v ( r m ( 1 ) ) − v ( r m ( 0 ) ) ∥ 1 = 2∣ r m ( 1 ) − r m ( 0 ) ∣ = 2 ( 1 − q ) m . Hence, using q = Θ ( α 2 ) q=\Theta(\alpha^2) q = Θ ( α 2 )
m ≥ log ( 1 / ϵ ) − log ( 1 − q ) ≳ 1 α 2 log ( 1 / ϵ ) ≳ N ϵ log ( 1 / ϵ ) . m\ge\frac{\log(1/\epsilon)}{-\log(1-q)}
\gtrsim\frac1{\alpha^2}\log(1/\epsilon)
\gtrsim\frac{\sqrt N}{\epsilon}\log(1/\epsilon). m ≥ − log ( 1 − q ) log ( 1/ ϵ ) ≳ α 2 1 log ( 1/ ϵ ) ≳ ϵ N log ( 1/ ϵ ) . This concludes the proof. ◻
Proof of Lemma 9 . Put q ( ν ) = e − 1 + ν 2 / 4 w ( ν / S ) q(\nu)=e^{-\sqrt{1+\nu^2}/4}w(\nu/S) q ( ν ) = e − 1 + ν 2 /4 w ( ν / S ) F S ( ν ) = e − ν / 4 q ( ν ) F_S(\nu)=e^{-\nu/4}q(\nu) F S ( ν ) = e − ν /4 q ( ν ) q q q [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] D ( ν ) = 0 D(\nu)=0 D ( ν ) = 0
D ( ν ) = e ν / 4 Re J ( ν ) − e − ν / 4 Re J ( − ν ) . D(\nu)=e^{\nu/4}\operatorname{Re}J(\nu)-e^{-\nu/4}\operatorname{Re}J(-\nu). D ( ν ) = e ν /4 Re J ( ν ) − e − ν /4 Re J ( − ν ) . We show that D ′ ( 0 ) > 0 D'(0)>0 D ′ ( 0 ) > 0 x > 0 x>0 x > 0 0 ≤ s ≤ 1 / 2 0\le s\le1/2 0 ≤ s ≤ 1/2
M s ( x ) = ∫ R q ( x / 2 + y ) q ( x / 2 − y ) cosh ( s y ) d y , k s = 1 2 − s , c s = 1 2 − s 2 . M_s(x)=\int_{\mathbb R}q(x/2+y)q(x/2-y)\cosh(sy)\,\mathrm dy,
\qquad k_s=\frac12-s,\quad c_s=\frac12-\frac s2. M s ( x ) = ∫ R q ( x /2 + y ) q ( x /2 − y ) cosh ( sy ) d y , k s = 2 1 − s , c s = 2 1 − 2 s . A direct Fourier calculation in the ordered time variables gives
D ′ ( 0 ) = 1 π 2 ∫ 0 ∞ 1 x ∫ 0 1 / 2 M s ( x ) [ − q ′ ( x ) cosh ( c s x ) − k s q ( x ) sinh ( c s x ) ] d s d x . \begin{equation}
D'(0)=\frac1{\pi^2}\int_0^\infty\frac1x
\int_0^{1/2}M_s(x)
\bigl[-q'(x)\cosh(c_sx)-k_sq(x)\sinh(c_sx)\bigr]\,\mathrm ds\,\mathrm dx. \tag*{(34)}
\end{equation} D ′ ( 0 ) = π 2 1 ∫ 0 ∞ x 1 ∫ 0 1/2 M s ( x ) [ − q ′ ( x ) cosh ( c s x ) − k s q ( x ) sinh ( c s x ) ] d s d x . (34) The function M s ( x ) M_s(x) M s ( x ) s s s k s tanh ( c s x ) k_s\tanh(c_sx) k s tanh ( c s x ) cosh ( c s x ) d s \cosh(c_sx)\,\mathrm ds cosh ( c s x ) d s
∫ 0 1 / 2 M s ( x ) k s sinh ( c s x ) d s ∫ 0 1 / 2 M s ( x ) cosh ( c s x ) d s ≤ τ ( x ) : = cosh ( x / 2 ) 2 [ sinh ( x / 2 ) − sinh ( x / 4 ) ] − 2 x \frac{\int_0^{1/2}M_s(x)k_s\sinh(c_sx)\,\mathrm ds}
{\int_0^{1/2}M_s(x)\cosh(c_sx)\,\mathrm ds}
\le\tau(x)
:=\frac{\cosh(x/2)}{2[\sinh(x/2)-\sinh(x/4)]}-\frac2x ∫ 0 1/2 M s ( x ) cosh ( c s x ) d s ∫ 0 1/2 M s ( x ) k s sinh ( c s x ) d s ≤ τ ( x ) := 2 [ sinh ( x /2 ) − sinh ( x /4 )] cosh ( x /2 ) − x 2 whenever the denominator is nonzero. Elementary hyperbolic-function estimates show that τ \tau τ 0 < τ ( x ) < 5 x / 48 0<\tau(x)<5x/48 0 < τ ( x ) < 5 x /48 r = 119 / 5 r=\sqrt{119}/5 r = 119 /5 m ( x ) = x / ( 4 1 + x 2 ) m(x)=x/(4\sqrt{1+x^2}) m ( x ) = x / ( 4 1 + x 2 )
τ ( 5 / 2 ) < 21 / 100 < m ( r ) , τ ( 11 / 4 ) < 223 / 1000 < m ( 5 / 2 ) , τ ( 3 ) < 117 / 500 < m ( 11 / 4 ) . \begin{aligned}
\tau(5/2)&<21/100<m(r),\\
\tau(11/4)&<223/1000<m(5/2),\\
\tau(3)&<117/500<m(11/4).
\end{aligned} τ ( 5/2 ) τ ( 11/4 ) τ ( 3 ) < 21/100 < m ( r ) , < 223/1000 < m ( 5/2 ) , < 117/500 < m ( 11/4 ) . Since m ( x ) ≥ 5 x / 48 m(x)\ge5x/48 m ( x ) ≥ 5 x /48 0 < x ≤ r 0<x\le r 0 < x ≤ r m ( x ) > τ ( x ) m(x)>\tau(x) m ( x ) > τ ( x ) ( 0 , 3 ] (0,3] ( 0 , 3 ] − q ′ ( x ) ≥ m ( x ) q ( x ) -q'(x)\ge m(x)q(x) − q ′ ( x ) ≥ m ( x ) q ( x ) q q q [ − S , S ] [-S,S] [ − S , S ] S ≤ 3 S\le3 S ≤ 3 (34) has nonnegative integrand after integration in s s s 0 < x < S 0<x<S 0 < x < S D ′ ( 0 ) > 0 D'(0)>0 D ′ ( 0 ) > 0 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
To obtain Theorem 4 with the prescribed choice S = ∥ h ∥ + 1 S=\|h\|+1 S = ∥ h ∥ + 1 S = 2 S=2 S = 2 λ \lambda λ N − 1 N-1 N − 1 λ \lambda λ 1 1 1 ∥ h ∥ = 1 \|h\|=1 ∥ h ∥ = 1 N − 1 N-1 N − 1 Ω ( N − 1 / ϵ ) = Ω ( N / ϵ ) \Omega(\sqrt{N-1}/\epsilon)=\Omega(\sqrt N/\epsilon) Ω ( N − 1 / ϵ ) = Ω ( N / ϵ )
State preparation in the random-bath setting
In this section, we consider both thermal state preparation and ground state preparation in the random-bath setting. We first provide a rigorous statement of Theorem 2 , which gives an upper bound on the complexity of ground state preparation, and then provide a rigorous statement of Theorem 5 , which establishes a lower bound on the complexity of thermal state preparation.
To simplify the analysis of the upper bound, we consider a setup that is simpler than the one in Assumption 3 :
Assumption 11 (Random bath Hamiltonian case). Let { b j † , b j } j = 1 N \{b^\dagger_j,b_j\}^N_{j=1} { b j † , b j } j = 1 N H E = ω ∑ i = 1 N b i † b i , B E , i = b i † , ρ E : ground state of H E , H_E=\omega\sum^{N}_{i=1}b^\dagger_{i}b_{i},\quad
B_{E,i}=b^\dagger_{i},\quad
\rho_E:\,\text{ground state of }H_E, H E = ω ∑ i = 1 N b i † b i , B E , i = b i † , ρ E : ground state of H E , H i n t ( t ) = f ( t ) ∑ i = 1 N ( c i † ⊗ B E , i + c i ⊗ B E , i † ) . H_{\rm int}(t)
=f(t)\sum_{i=1}^N
\left(c^\dagger_i\otimes B_{E,i}+c_i\otimes B_{E,i}^\dagger\right)\,. H int ( t ) = f ( t ) ∑ i = 1 N ( c i † ⊗ B E , i + c i ⊗ B E , i † ) . ω \omega ω [ − S , S ] [-S,S] [ − S , S ] f ( t ) = e − t 2 / ( 4 σ 2 ) σ 2 π f(t)=\frac{e^{-t^2/(4\sigma^2)}}{\sqrt{\sigma\sqrt{2\pi}}} f ( t ) = σ 2 π e − t 2 / ( 4 σ 2 )
The choice of f ( t ) f(t) f ( t ) Assumption 11 is inspired by [6 , 2 ] . Although Assumption 11 is a simplified version of Assumption 3 , involving only N N N f ( t ) f(t) f ( t ) Assumption 3 would lead to the same complexity scaling as in the following theorem:
Theorem 12. Assume max k ∣ λ k ∣ = O ( 1 ) \max_k|\lambda_k|=\mathcal O(1) max k ∣ λ k ∣ = O ( 1 ) Δ : = min k ∣ λ k ∣ > 0 \Delta:=\min_k|\lambda_k|>0 Δ := min k ∣ λ k ∣ > 0 σ = Θ ( Δ − 1 p o l y l o g ( N / ϵ ) ) , T = Θ ( Δ − 1 p o l y l o g ( N / ϵ ) ) , \sigma=\Theta(\Delta^{-1}\mathrm{polylog}(N/\epsilon)),\qquad
T=\Theta(\Delta^{-1}\mathrm{polylog}(N/\epsilon)), σ = Θ ( Δ − 1 polylog ( N / ϵ )) , T = Θ ( Δ − 1 polylog ( N / ϵ )) , α σ = Θ ( 1 ) \alpha\sqrt{\sigma}=\Theta(1) α σ = Θ ( 1 ) S > max j ∣ λ j ∣ S>\max_j|\lambda_j| S > max j ∣ λ j ∣ S = Θ ( 1 ) S=\Theta(1) S = Θ ( 1 ) ρ \rho ρ ∥ Φ α m ρ − ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ∥ 1 ≤ ϵ , m = Θ ( 1 Δ p o l y l o g ( N / ϵ ) ) . \|\Phi^m_{\alpha}\rho-\ketbra{\psi_0}{\psi_0}\|_1\leq \epsilon,\qquad
m=\Theta\left(\frac{1}{\Delta}\mathrm{polylog}(N/\epsilon)\right)\,. ∥ Φ α m ρ − ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ∥ 1 ≤ ϵ , m = Θ ( Δ 1 polylog ( N / ϵ ) ) . ∣ ψ 0 ⟩ \ket{\psi_0} ∣ ψ 0 ⟩ ϵ \epsilon ϵ T t o t a l = O ~ ( Δ − 2 ) . T_{\rm total}=\widetilde{\mathcal{O}}(\Delta^{-2})\,. T total = O ( Δ − 2 ) .
We put the proof in § 2.1 . Here, we suppress additional polylogarithmic factors in 1 / Δ 1/\Delta 1/Δ
Next, we provide a rigorous statement of Theorem 5 . We note that the lower bound is established under a more general assumption than Assumption 11 :
Theorem 13. For N ≥ 2 N\ge2 N ≥ 2 Assumption 3 and f ( t ) ∈ L 1 ( R ) f(t)\in L^1(\mathbb{R}) f ( t ) ∈ L 1 ( R ) E ω ( u ( ω ; λ ) ) > 0 \mathbb{E}_{\omega}\left(u(\omega;\lambda)\right)>0 E ω ( u ( ω ; λ ) ) > 0 E ω ( d ( ω ; λ ) ) > 0 \mathbb{E}_{\omega}\left(d(\omega;\lambda)\right)>0 E ω ( d ( ω ; λ ) ) > 0 λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ] λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ] ∫ − ∞ ∞ ∣ u ( ω ; λ ) − exp ( − λ ) d ( ω ; λ ) ∣ 2 d μ ( ω ) > 0
\int^\infty_{-\infty}\left|u(\omega;\lambda)-\exp(-\lambda)d(\omega;\lambda)\right|^2\mathrm{d}\mu(\omega)>0
∫ − ∞ ∞ ∣ u ( ω ; λ ) − exp ( − λ ) d ( ω ; λ ) ∣ 2 d μ ( ω ) > 0 T 0 , α 0 , ϵ 0 , κ , C T_0,\alpha_0,\epsilon_0,\kappa,C T 0 , α 0 , ϵ 0 , κ , C T > T 0 T>T_0 T > T 0 α < α 0 \alpha<\alpha_0 α < α 0 ϵ < ϵ 0 \epsilon<\epsilon_0 ϵ < ϵ 0 ∫ ∣ t ∣ > T / 2 ∣ f ( t ) ∣ d t ≤ κ α 2 \int_{|t|>T/2}|f(t)|\mathrm{d}t\leq \kappa \alpha^2 ∫ ∣ t ∣ > T /2 ∣ f ( t ) ∣ d t ≤ κ α 2 sup H = λ ∑ i = 1 N n i , ∣ λ ∣ ≤ 1 inf { τ | sup ρ sup m ≥ τ ∥ Φ α m ( ρ ) − σ 1 ( H ) ∥ 1 ≤ ϵ } ≥ C N ϵ . \sup_{H=\lambda\sum_{i=1}^Nn_i,|\lambda|\leq 1}\inf\left\{\tau\middle|\sup_{\rho}\sup_{m\geq \tau}\|\Phi^m_{\alpha}(\rho)-\sigma_1(H)\|_1\leq \epsilon\right\}\geq \frac{CN}{\epsilon}\,. sup H = λ ∑ i = 1 N n i , ∣ λ ∣ ≤ 1 inf { τ sup ρ sup m ≥ τ ∥ Φ α m ( ρ ) − σ 1 ( H ) ∥ 1 ≤ ϵ } ≥ ϵ C N . ϵ \epsilon ϵ T t o t a l = Ω ( N / ϵ ) T_{\rm total}=\Omega(N/\epsilon) T total = Ω ( N / ϵ )
As shown in § 2.2 , although E ω ( Φ α ) \mathrm{E}_{\omega}(\Phi_{\alpha}) E ω ( Φ α ) [2 ] , it is not guaranteed that the fixed point of Φ α \Phi_{\alpha} Φ α O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) O ( N α 2 ) \mathcal{O}(N\alpha^2) O ( N α 2 ) N N N > 0 >0 > 0 = 0 =0 = 0 (35) , thus helps to eliminate two-qubit correlations in the O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) α 2 \alpha^2 α 2
We first show a proposition similar to Proposition 10 for the random bath case. The proof is similar to that of Proposition 10 .
Proposition 14. Under Assumption 11 , there are CPTP maps Φ α , j ( ω ) : C 2 N × 2 N → C 2 N × 2 N \Phi_{\alpha,j}(\omega):\mathbb C^{2^N\times2^N}\to
\mathbb C^{2^N\times2^N} Φ α , j ( ω ) : C 2 N × 2 N → C 2 N × 2 N ( d j † , d j ) (d^\dagger_j,d_j) ( d j † , d j ) ω \omega ω
For ω ≠ 0 \omega\neq 0 ω = 0 [ Φ α , j ( ω ) , Φ α , k ( ω ) ] = 0 ( j ≠ k ) [\Phi_{\alpha,j}(\omega),\Phi_{\alpha,k}(\omega)]=0
\quad(j\ne k) [ Φ α , j ( ω ) , Φ α , k ( ω )] = 0 ( j = k )
Φ α = E ω ( Φ α ( ω ) ) : = E ω ( Φ α , 1 ( ω ) ∘ ⋯ ∘ Φ α , N ( ω ) ) . \Phi_\alpha
=\mathbb{E}_{\omega}\left(\Phi_\alpha\left(\omega\right)\right):=\mathbb{E}_{\omega}\left(\Phi_{\alpha,1}(\omega)\circ\cdots\circ\Phi_{\alpha,N}(\omega)\right). Φ α = E ω ( Φ α ( ω ) ) := E ω ( Φ α , 1 ( ω ) ∘ ⋯ ∘ Φ α , N ( ω ) ) .
For ω ≠ 0 \omega\neq 0 ω = 0 Φ α , j ( ω ) [ ρ ] = T r B j ( U α , j ( ω ) ( ρ ⊗ ρ B ( ω ) ) U α , j † ( ω ) ) ,
\Phi_{\alpha,j}(\omega)[\rho]=\mathrm{Tr}_{B_j}\left(U_{\alpha,j}(\omega)\left(\rho\otimes\rho_{B}(\omega)\right)U_{\alpha,j}^\dagger(\omega)\right)\,,
Φ α , j ( ω ) [ ρ ] = Tr B j ( U α , j ( ω ) ( ρ ⊗ ρ B ( ω ) ) U α , j † ( ω ) ) , U α , j ( ω ) : = T exp ( − i ∫ − T T H α , j , ω ( s ) d s ) U_{\alpha,j}(\omega):=\mathcal T\exp\left(
-i\int_{-T}^T H_{\alpha,j,\omega}(s)\,\mathrm ds\right) U α , j ( ω ) := T exp ( − i ∫ − T T H α , j , ω ( s ) d s ) e j = ∑ k U k , j b k , e_j = \sum_k U_{k,j} b_k, e j = ∑ k U k , j b k , H α , j , ω ( t ) = λ j d j † d j + ω e j † e j + α f ( t ) ( d j † ⊗ e j † + d j ⊗ e j ) . H_{\alpha, j,\omega}(t) = \lambda_j d_j^\dagger d_j + \omega e_j^\dagger e_j + \alpha f(t) (d_j^\dagger \otimes e_j^\dagger + d_j \otimes e_j). H α , j , ω ( t ) = λ j d j † d j + ω e j † e j + α f ( t ) ( d j † ⊗ e j † + d j ⊗ e j ) .
More specifically, Φ α , j = U j ( T ) ∘ Ψ α , j ∘ U j ( T )
\Phi_{\alpha, j} = \mathcal{U}_j(T)\circ \Psi_{\alpha,j} \circ \mathcal{U}_j(T)
Φ α , j = U j ( T ) ∘ Ψ α , j ∘ U j ( T ) U j ( T ) [ ρ ] = exp ( − i λ j d j † d j T ) ρ exp ( i λ j d j † d j T ) \mathcal{U}_j(T)[\rho]=\exp(-i\lambda_jd_j^\dagger d_j T)\rho\exp(i\lambda_jd_j^\dagger d_jT) U j ( T ) [ ρ ] = exp ( − i λ j d j † d j T ) ρ exp ( i λ j d j † d j T ) Ψ α , j ( ρ ) = ∑ n , m ≥ 0 ( − 1 ) n ( i α ) n + m ∫ ∫ − T ≤ s n ≤ ⋯ ≤ s 1 ≤ T , − T ≤ τ m ≤ ⋯ ≤ τ 1 ≤ T f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) ∑ α 1 , … , α n ∈ { − 1 , 1 } β 1 , … , β m ∈ { − 1 , 1 } e − i ( ω + λ j ) ∑ q = 1 m β q τ q + i ( ω + λ j ) ∑ p = 1 n α p s p d s 1 ⋯ d s n d τ 1 ⋯ d τ m d α 1 , j ⋯ d α n , j ρ d − β m , j ⋯ d − β 1 , j T r E j ( e α 1 , j ⋯ e α n , j ρ B ( ω ) e − β m , j ⋯ e − β 1 , j ) .
\begin{aligned}
\Psi_{\alpha, j}(\rho)& = \sum_{n,m\geq 0}(-1)^n(i\alpha)^{n+m}\int \int_{-T\leq s_n\leq \cdots \leq s_1 \leq T, -T\leq \tau_m\leq \cdots\leq \tau_1 \leq T}f(s_1)\cdots f(s_n) f(\tau_1)\cdots f(\tau_m)\\
&\sum_{\substack{\alpha_1,\ldots,\alpha_n\in\{-1,1\}\\\beta_1,\ldots,\beta_m\in\{-1,1\}}}
e^{-i(\omega+\lambda_j)\sum_{q=1}^m\beta_q\tau_q
+i(\omega+\lambda_j)\sum_{p=1}^{n}\alpha_p s_p}\mathrm{d}s_1 \cdots \mathrm{d}s_n \mathrm{d}\tau_1\cdots \mathrm{d}\tau_m \\
&d_{\alpha_1,j}\cdots d_{\alpha_n,j} \rho d_{-\beta_m,j} \cdots d_{-\beta_1,j}\mathrm{Tr}_{E_j}(e_{\alpha_1,j}\cdots e_{\alpha_n,j}\rho_B(\omega) e_{-\beta_m,j} \cdots e_{-\beta_1,j} ) \,.\\
\end{aligned}
Ψ α , j ( ρ ) = n , m ≥ 0 ∑ ( − 1 ) n ( i α ) n + m ∫ ∫ − T ≤ s n ≤ ⋯ ≤ s 1 ≤ T , − T ≤ τ m ≤ ⋯ ≤ τ 1 ≤ T f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) α 1 , … , α n ∈ { − 1 , 1 } β 1 , … , β m ∈ { − 1 , 1 } ∑ e − i ( ω + λ j ) ∑ q = 1 m β q τ q + i ( ω + λ j ) ∑ p = 1 n α p s p d s 1 ⋯ d s n d τ 1 ⋯ d τ m d α 1 , j ⋯ d α n , j ρ d − β m , j ⋯ d − β 1 , j Tr E j ( e α 1 , j ⋯ e α n , j ρ B ( ω ) e − β m , j ⋯ e − β 1 , j ) . e η , j = { e j † η = 1 e j η = − 1 e_{\eta,j} = \begin{cases}
e_j^\dagger & \eta = 1\\ e_j & \eta= -1
\end{cases} e η , j = { e j † e j η = 1 η = − 1 d η , j = { d j † η = 1 d j η = − 1 d_{\eta,j} = \begin{cases}
d_j^\dagger & \eta = 1\\ d_j & \eta= -1
\end{cases} d η , j = { d j † d j η = 1 η = − 1
Proof of Proposition 14 . Under Assumption 11 , the total Hamiltonian is
H α ( t ) = H + H E + α H i n t ( t ) = ∑ k = 1 N λ k d k † d k + ω ∑ k = 1 N b k † b k + α f ( t ) ∑ k = 1 N ( c k † ⊗ B E , k + c k ⊗ B E , k † ) . \begin{equation}
\begin{aligned}
H_\alpha(t) &= H + H_E + \alpha H_{\rm int}(t)\\
& = \sum_{k = 1}^N \lambda_k d_k^\dagger d_k + \omega \sum_{k = 1}^N b_k^\dagger b_k + \alpha f(t)\sum_{k=1}^N (c_k^\dagger \otimes B_{E,k} + c_k \otimes B_{E,k}^\dagger)\,.
\end{aligned} \tag*{(36)}
\end{equation} H α ( t ) = H + H E + α H int ( t ) = k = 1 ∑ N λ k d k † d k + ω k = 1 ∑ N b k † b k + α f ( t ) k = 1 ∑ N ( c k † ⊗ B E , k + c k ⊗ B E , k † ) . (36) After applying the unitary transformation, the system Hamiltonian is diagonalized in terms of d k , d k † d_k, d_k^\dagger d k , d k † e k , e k † e_k, e_k^\dagger e k , e k †
e j = ∑ k U k , j b k , e j † = ∑ k U k j ∗ b k † . \begin{equation}
e_j = \sum_k U_{k,j} b_k, \quad e_j^\dagger = \sum_k U_{kj}^* b_k^\dagger\,. \tag*{(37)}
\end{equation} e j = k ∑ U k , j b k , e j † = k ∑ U k j ∗ b k † . (37) Therefore the interaction term is decomposed as a summation of one mode terms so that the total Hamiltonian can be rewritten as
H α ( t ) = ∑ k = 1 N λ k d k † d k + ω ∑ k = 1 N e k † e k + α f ( t ) ∑ k = 1 N ( d k † ⊗ e k † + d k ⊗ e k ) = ∑ k H α , k , ω ( t ) . \begin{equation}
\begin{aligned}
H_\alpha(t) = \sum_{k = 1}^N \lambda_k d_k^\dagger d_k + \omega \sum_{k = 1}^N e_k^\dagger e_k + \alpha f(t)\sum_{k=1}^N (d_k^\dagger \otimes e_k^\dagger + d_k \otimes e_k)= \sum_k H_{\alpha, k,\omega}(t)\,.\\
\end{aligned} \tag*{(38)}
\end{equation} H α ( t ) = k = 1 ∑ N λ k d k † d k + ω k = 1 ∑ N e k † e k + α f ( t ) k = 1 ∑ N ( d k † ⊗ e k † + d k ⊗ e k ) = k ∑ H α , k , ω ( t ) . (38) Here
H α , j , ω ( t ) = λ j d j † d j + ω e j † e j + α f ( t ) ( d j † ⊗ e j † + d j ⊗ e j ) . \begin{equation}
H_{\alpha, j,\omega}(t) = \lambda_j d_j^\dagger d_j + \omega e_j^\dagger e_j + \alpha f(t) (d_j^\dagger \otimes e_j^\dagger + d_j \otimes e_j). \tag*{(39)}
\end{equation} H α , j , ω ( t ) = λ j d j † d j + ω e j † e j + α f ( t ) ( d j † ⊗ e j † + d j ⊗ e j ) . (39) Those single mode Hamiltonian commutes with each other [ H α , k , ω ( t ) , H α , j , ω ( s ) ] = 0 [H_{\alpha, k,\omega}(t), H_{\alpha, j,\omega}(s)] = 0 [ H α , k , ω ( t ) , H α , j , ω ( s )] = 0 k ≠ j k \neq j k = j U α , j U_{\alpha,j} U α , j
U α ( T ) = T ( exp ( − i ∫ − T T H α ( s ) d s ) ) = ∏ k T ( exp ( − i ∫ − T T H α , k , ω ( s ) d s ) ) = ∏ k U α , k ( T ) \begin{equation}
\begin{aligned}
U^\alpha(T) &= \mathcal{T}\left(\exp\left(-i\int_{-T}^T H_{\alpha}(s)\mathrm{d}s\right)\right)
= \prod_k \mathcal{T}\left(\exp\left(-i\int_{-T}^T H_{\alpha,k,\omega}(s)\mathrm{d}s\right)\right) = \prod_kU_{\alpha, k }(T)
\end{aligned} \tag*{(40)}
\end{equation} U α ( T ) = T ( exp ( − i ∫ − T T H α ( s ) d s ) ) = k ∏ T ( exp ( − i ∫ − T T H α , k , ω ( s ) d s ) ) = k ∏ U α , k ( T ) (40) The induced quantum channel becomes
Φ α ( ρ ) = E ω [ Φ α , 1 ( ω ) ∘ Φ α , 2 ( ω ) ∘ ⋯ ∘ Φ α , N ( ω ) ( ρ ) ] . \begin{equation}
\Phi_\alpha(\rho) = \mathbb{E}_\omega[\Phi_{\alpha, 1}(\omega) \circ \Phi_{\alpha, 2}(\omega)\circ \cdots \circ \Phi_{\alpha, N}(\omega) (\rho)]. \tag*{(41)}
\end{equation} Φ α ( ρ ) = E ω [ Φ α , 1 ( ω ) ∘ Φ α , 2 ( ω ) ∘ ⋯ ∘ Φ α , N ( ω ) ( ρ )] . (41) Let S η , j = { d j † η = + 1 d j η = − 1 S_{\eta,j} = \begin{cases}
d_j^\dagger & \eta = +1\\ d_j & \eta = -1\\
\end{cases} S η , j = { d j † d j η = + 1 η = − 1 B η , j = { e j † η = + 1 e j η = − 1 . B_{\eta,j} = \begin{cases}
e_j^\dagger & \eta = +1 \\ e_j & \eta = -1\\
\end{cases}. B η , j = { e j † e j η = + 1 η = − 1 .
Φ α , j ( ρ ) = U j ( T ) ∘ Ψ α , j ∘ U j ( T ) ( ρ ) . \begin{equation}
\Phi_{\alpha, j}(\rho) = \mathcal{U}_{j}(T) \circ \Psi_{\alpha, j}\circ \mathcal{U}_{j}(T) (\rho). \tag*{(42)}
\end{equation} Φ α , j ( ρ ) = U j ( T ) ∘ Ψ α , j ∘ U j ( T ) ( ρ ) . (42) where
Ψ α , j ( ρ ) = ∑ n , m ≥ 0 ( − 1 ) n ( i α ) n + m ∫ ∫ − T ≤ s n ≤ ⋯ ≤ s 1 ≤ T , − T ≤ τ m ≤ ⋯ ≤ τ 1 ≤ T d s 1 ⋯ d s n d τ 1 ⋯ d τ m ∑ α i , β j ∈ { − 1 , 1 } S α 1 , j ( s 1 ) ⋯ S α n , j ( s n ) ρ S β m , j † ( τ m ) ⋯ S β 1 , j † ( τ 1 ) f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) T r E j ( B α 1 , j ( s 1 ) ⋯ B α n , j ( s n ) ρ B ( ω ) B β m , j † ( τ m ) ⋯ B β 1 , j † ( τ 1 ) ) , \begin{equation}
\begin{aligned}
\Psi_{\alpha, j}(\rho)& = \sum_{n,m\geq 0}(-1)^n(i\alpha)^{n+m}\int \int_{-T\leq s_n\leq \cdots \leq s_1 \leq T, -T\leq \tau_m\leq \cdots\leq \tau_1 \leq T}\mathrm{d}s_1 \cdots \mathrm{d}s_n \mathrm{d}\tau_1\cdots \mathrm{d}\tau_m \\
&\sum_{\alpha_i,\beta_j\in\{-1,1\}}S_{\alpha_1,j}(s_1)\cdots S_{\alpha_n,j}(s_n) \rho S_{\beta_m,j}^\dag(\tau_m) \cdots S_{\beta_1,j}^\dag (\tau_1)\\
& f(s_1)\cdots f(s_n) f(\tau_1)\cdots f(\tau_m)\mathrm{Tr}_{E_j}(B_{\alpha_1,j}(s_1)\cdots B_{\alpha_n,j}(s_n)\rho_B(\omega) B_{\beta_m,j}^\dag(\tau_m) \cdots B_{\beta_1,j}^\dag (\tau_1))\,, \\
\end{aligned} \tag*{(43)}
\end{equation} Ψ α , j ( ρ ) = n , m ≥ 0 ∑ ( − 1 ) n ( i α ) n + m ∫ ∫ − T ≤ s n ≤ ⋯ ≤ s 1 ≤ T , − T ≤ τ m ≤ ⋯ ≤ τ 1 ≤ T d s 1 ⋯ d s n d τ 1 ⋯ d τ m α i , β j ∈ { − 1 , 1 } ∑ S α 1 , j ( s 1 ) ⋯ S α n , j ( s n ) ρ S β m , j † ( τ m ) ⋯ S β 1 , j † ( τ 1 ) f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) Tr E j ( B α 1 , j ( s 1 ) ⋯ B α n , j ( s n ) ρ B ( ω ) B β m , j † ( τ m ) ⋯ B β 1 , j † ( τ 1 )) , (43) The Heisenbergy picture of the system and bath coupling operator can be characterized as
S η , j ( t ) = e i η λ j t S η , j , B η , j ( t ) = e i η ω t B η , j . \begin{equation}
S_{\eta, j}(t) = e^{i\eta\lambda_j t}S_{\eta, j}, \quad B_{\eta, j}(t) = e^{i\eta\omega t}B_{\eta,j}\,. \tag*{(44)}
\end{equation} S η , j ( t ) = e i η λ j t S η , j , B η , j ( t ) = e i η ω t B η , j . (44) Therefore, the expression can be further simplified to
Ψ α , j ( ρ ) = ∑ n , m ≥ 0 ( − 1 ) n ( i α ) n + m ∫ ∫ − T ≤ s n ≤ ⋯ ≤ s 1 ≤ T , − T ≤ τ m ≤ ⋯ ≤ τ 1 ≤ T f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) ∑ α 1 , … , α n ∈ { − 1 , 1 } β 1 , … , β m ∈ { − 1 , 1 } e − i ( ω + λ j ) ∑ q = 1 m β q τ q + i ( ω + λ j ) ∑ p = 1 n α p s p d s 1 ⋯ d s n d τ 1 ⋯ d τ m S α 1 , j ⋯ S α n , j ρ S β m , j † ⋯ S β 1 , j † T r E j ( B α 1 , j ⋯ B α n , j ρ B ( ω ) B β m , j † ⋯ B β 1 , j † ) . \begin{equation}
\begin{aligned}
\Psi_{\alpha, j}(\rho)& = \sum_{n,m\geq 0}(-1)^n(i\alpha)^{n+m}\int \int_{-T\leq s_n\leq \cdots \leq s_1 \leq T, -T\leq \tau_m\leq \cdots\leq \tau_1 \leq T}f(s_1)\cdots f(s_n) f(\tau_1)\cdots f(\tau_m)\\
&\sum_{\substack{\alpha_1,\ldots,\alpha_n\in\{-1,1\}\\\beta_1,\ldots,\beta_m\in\{-1,1\}}}
e^{-i(\omega+\lambda_j)\sum_{q=1}^m\beta_q\tau_q
+i(\omega+\lambda_j)\sum_{p=1}^{n}\alpha_p s_p}\mathrm{d}s_1 \cdots \mathrm{d}s_n \mathrm{d}\tau_1\cdots \mathrm{d}\tau_m \\
&S_{\alpha_1,j}\cdots S_{\alpha_n,j} \rho S_{\beta_m,j}^\dag \cdots S_{\beta_1,j}^\dag\mathrm{Tr}_{E_j}(B_{\alpha_1,j}\cdots B_{\alpha_n,j}\rho_B(\omega) B_{\beta_m,j}^\dag \cdots B_{\beta_1,j}^\dag ) \,.\\
\end{aligned} \tag*{(45)}
\end{equation} Ψ α , j ( ρ ) = n , m ≥ 0 ∑ ( − 1 ) n ( i α ) n + m ∫ ∫ − T ≤ s n ≤ ⋯ ≤ s 1 ≤ T , − T ≤ τ m ≤ ⋯ ≤ τ 1 ≤ T f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) α 1 , … , α n ∈ { − 1 , 1 } β 1 , … , β m ∈ { − 1 , 1 } ∑ e − i ( ω + λ j ) ∑ q = 1 m β q τ q + i ( ω + λ j ) ∑ p = 1 n α p s p d s 1 ⋯ d s n d τ 1 ⋯ d τ m S α 1 , j ⋯ S α n , j ρ S β m , j † ⋯ S β 1 , j † Tr E j ( B α 1 , j ⋯ B α n , j ρ B ( ω ) B β m , j † ⋯ B β 1 , j † ) . (45) ◻
Specifically, the initial bath state is
ρ B ( ω ) = { ∣ 0 ⟩ ⟨ 0 ∣ if ω > 0 ∣ 1 ⟩ ⟨ 1 ∣ if ω < 0 . \begin{equation}
\rho_B(\omega)=\left\{\begin{aligned}\ket{0}\bra{0}\ \text{if}\ \omega>0\\
\ket{1}\bra{1}\ \text{if}\ \omega<0
\end{aligned}\right.\,. \tag*{(46)}
\end{equation} ρ B ( ω ) = { ∣ 0 ⟩ ⟨ 0 ∣ if ω > 0 ∣ 1 ⟩ ⟨ 1 ∣ if ω < 0 . (46) Define the number operator n j = d j † d j n_j=d_j^\dagger d_j n j = d j † d j j j j
Φ ~ α , j = U j ( T ) ∘ Ψ ~ α , j ∘ U j ( T ) \begin{equation}
\widetilde{\Phi}_{\alpha,j} =\mathcal{U}_j(T)\circ \widetilde{\Psi}_{\alpha,j}\circ \mathcal{U}_j(T) \tag*{(47)}
\end{equation} Φ α , j = U j ( T ) ∘ Ψ α , j ∘ U j ( T ) (47) where Ψ ~ j \widetilde{\Psi}_j Ψ j Ψ j \Psi_j Ψ j
Lemma 15. Under the settings and results of Proposition 14 ,
For ω ≠ 0 \omega\neq 0 ω = 0 Φ α , j † ( ω ) [ n k ] = Φ ~ α , j † ( ω ) [ n k ] = n k \Phi^\dagger_{\alpha,j}(\omega)[n_k]=\widetilde{\Phi}^\dagger_{\alpha,j}(\omega)[n_k]=n_k Φ α , j † ( ω ) [ n k ] = Φ α , j † ( ω ) [ n k ] = n k k ≠ j k\neq j k = j
For ω ≠ 0 \omega\neq 0 ω = 0 ∥ Φ ~ α , j ( ω ) − Φ α , j ( ω ) ∥ 1 ↔ 1 = O ( α 2 σ 2 T exp ( − T 2 / ( 4 σ 2 ) ) ) \left\|\widetilde{\Phi}_{\alpha,j}(\omega)-\Phi_{\alpha,j}(\omega)\right\|_{1\leftrightarrow 1}=\mathcal{O}\left(\frac{\alpha^2\sigma^2}{T}\exp(-T^2/(4\sigma^2))\right) Φ α , j ( ω ) − Φ α , j ( ω ) 1 ↔ 1 = O ( T α 2 σ 2 exp ( − T 2 / ( 4 σ 2 )) )
When σ = Ω ( Δ − 1 ) \sigma=\Omega(\Delta^{-1}) σ = Ω ( Δ − 1 ) α σ = O ( 1 ) \alpha\sqrt{\sigma}=\mathcal{O}(1) α σ = O ( 1 ) S > max j ∣ λ j ∣ S>\max_j|\lambda_j| S > max j ∣ λ j ∣ S = Θ ( 1 ) S=\Theta(1) S = Θ ( 1 ) C , C 0 > 0 C,C_0>0 C , C 0 > 0
If λ j > 0 \lambda_j>0 λ j > 0 ( E ω ( Φ ~ α , j ( ω ) ) ) † [ n j ] ⪯ ( 1 − C α 2 ) n j + ∑ i = 1 ∞ ( C 0 α σ ) 2 i − 1 ( 2 i − 1 ) ! exp ( − σ 2 Δ 2 ( 2 i − 1 ) ) I \left(\mathbb{E}_{\omega}\left(\widetilde{\Phi}_{\alpha,j}(\omega)\right)\right)^\dagger[n_j]\preceq \left(1-C\alpha^2\right)n_j+\sum^\infty_{i=1}\frac{\left(C_0\alpha\sqrt{\sigma}\right)^{2i-1}}{(2i-1)!}\exp\left(-\frac{\sigma^2\Delta^2}{(2i-1)}\right) I ( E ω ( Φ α , j ( ω ) ) ) † [ n j ] ⪯ ( 1 − C α 2 ) n j + ∑ i = 1 ∞ ( 2 i − 1 )! ( C 0 α σ ) 2 i − 1 exp ( − ( 2 i − 1 ) σ 2 Δ 2 ) I
If λ j < 0 \lambda_j<0 λ j < 0 ( E ω ( Φ ~ α , j ( ω ) ) ) † [ I − n j ] ⪯ ( 1 − C α 2 ) ( I − n j ) + ∑ i = 1 ∞ ( C 0 α σ ) 2 i − 1 ( 2 i − 1 ) ! exp ( − σ 2 Δ 2 ( 2 i − 1 ) ) I \left(\mathbb{E}_{\omega}\left(\widetilde{\Phi}_{\alpha,j}(\omega)\right)\right)^\dagger[I-n_j]\preceq \left(1-C\alpha^2\right)(I-n_j)+\sum^\infty_{i=1}\frac{\left(C_0\alpha\sqrt{\sigma}\right)^{2i-1}}{(2i-1)!}\exp\left(-\frac{\sigma^2\Delta^2}{(2i-1)}\right) I ( E ω ( Φ α , j ( ω ) ) ) † [ I − n j ] ⪯ ( 1 − C α 2 ) ( I − n j ) + ∑ i = 1 ∞ ( 2 i − 1 )! ( C 0 α σ ) 2 i − 1 exp ( − ( 2 i − 1 ) σ 2 Δ 2 ) I
The proof of Lemma 15 is similar to that of [2 Theorem S14] . For completeness, we provide the proof in § 4 .
Now, we are ready to prove Theorem 12 .
Proof of Theorem 12 . In the proof, we consider the decay of the total number operator N ^ = ∑ j = 1 N ( n j 1 λ j > 0 + ( I − n j ) 1 λ j < 0 ) \hat{N}=\sum_{j=1}^N \left(n_j\textbf{1}_{\lambda_j>0}+(I-n_j)\textbf{1}_{\lambda_j<0}\right) N ^ = ∑ j = 1 N ( n j 1 λ j > 0 + ( I − n j ) 1 λ j < 0 ) Lemma 15 , for each j j j m > 0 m>0 m > 0
( Φ α m ) † ( n j ) = ( ( E ω ( Φ α , j ( ω ) ) ) m ) † [ n j ] = ( ( E ω ( Φ ~ α , j ( ω ) ) ) m ) † [ n j ] + O ( m α 2 σ 2 exp ( − T 2 / ( 4 σ 2 ) ) / T ) . \left(\Phi^m_{\alpha}\right)^\dagger\left(n_j\right)=\left(\left(\mathbb{E}_{\omega}\left(\Phi_{\alpha,j}(\omega)\right)\right)^m\right)^\dagger[n_j]=\left(\left(\mathbb{E}_{\omega}\left(\widetilde{\Phi}_{\alpha,j}(\omega)\right)\right)^m\right)^\dagger[n_j]+\mathcal{O}\left(m\alpha^2\sigma^2\exp(-T^2/(4\sigma^2))/T\right)\,. ( Φ α m ) † ( n j ) = ( ( E ω ( Φ α , j ( ω ) ) ) m ) † [ n j ] = ( ( E ω ( Φ α , j ( ω ) ) ) m ) † [ n j ] + O ( m α 2 σ 2 exp ( − T 2 / ( 4 σ 2 )) / T ) . This implies that
T r ( Φ α m [ ρ ] N ^ ) = ∑ j = 1 N [ T r ( ( E ω ( Φ α , j ( ω ) ) ) m [ ρ ] n j ) 1 λ j > 0 + T r ( ( E ω ( Φ α , j ( ω ) ) ) m [ ρ ] ( I − n j ) ) 1 λ j < 0 ] = ∑ j = 1 N [ T r ( ( E ω ( Φ ~ α , j ( ω ) ) ) m [ ρ ] n j ) 1 λ j > 0 + T r ( ( E ω ( Φ ~ α , j ( ω ) ) ) m [ ρ ] ( I − n j ) ) 1 λ j < 0 ] + O ( m N α 2 σ 2 exp ( − T 2 / ( 4 σ 2 ) ) / T ) ≤ ( 1 − C α 2 ) m T r ( ρ N ^ ) + ∑ j = 1 N ∑ i = 1 ∞ m ( C 0 α σ ) 2 i − 1 ( 2 i − 1 ) ! exp ( − σ 2 Δ 2 ( 2 i − 1 ) ) + O ( m N α 2 σ 2 exp ( − T 2 / ( 4 σ 2 ) ) / T ) . \begin{aligned}
&\mathrm{Tr}\left(\Phi^m_{\alpha}[\rho]\hat{N}\right)=\sum^N_{j=1}\left[\mathrm{Tr}\left(\left(\mathbb{E}_{\omega}\left(\Phi_{\alpha,j}(\omega)\right)\right)^m[\rho]n_j\right)\textbf{1}_{\lambda_j>0}+\mathrm{Tr}\left(\left(\mathbb{E}_{\omega}\left(\Phi_{\alpha,j}(\omega)\right)\right)^m[\rho](I-n_j)\right)\textbf{1}_{\lambda_j<0}\right]\\
=&\sum^N_{j=1}\left[\mathrm{Tr}\left(\left(\mathbb{E}_{\omega}\left(\widetilde{\Phi}_{\alpha,j}(\omega)\right)\right)^m[\rho]n_j\right)\textbf{1}_{\lambda_j>0}+\mathrm{Tr}\left(\left(\mathbb{E}_{\omega}\left(\widetilde{\Phi}_{\alpha,j}(\omega)\right)\right)^m[\rho]\left(I-n_j\right)\right)\textbf{1}_{\lambda_j<0}\right]+\mathcal{O}\left(mN\alpha^2\sigma^2\exp(-T^2/(4\sigma^2))/T\right)\\
\leq &\left(1-C\alpha^2\right)^m\mathrm{Tr}\left(\rho\hat{N}\right)+\sum^N_{j=1}\sum^\infty_{i=1}m\frac{\left(C_0\alpha\sqrt{\sigma}\right)^{2i-1}}{(2i-1)!}\exp\left(-\frac{\sigma^2\Delta^2}{(2i-1)}\right)+\mathcal{O}\left(mN\alpha^2\sigma^2\exp(-T^2/(4\sigma^2))/T\right)\\
\end{aligned}\,. = ≤ Tr ( Φ α m [ ρ ] N ^ ) = j = 1 ∑ N [ Tr ( ( E ω ( Φ α , j ( ω ) ) ) m [ ρ ] n j ) 1 λ j > 0 + Tr ( ( E ω ( Φ α , j ( ω ) ) ) m [ ρ ] ( I − n j ) ) 1 λ j < 0 ] j = 1 ∑ N [ Tr ( ( E ω ( Φ α , j ( ω ) ) ) m [ ρ ] n j ) 1 λ j > 0 + Tr ( ( E ω ( Φ α , j ( ω ) ) ) m [ ρ ] ( I − n j ) ) 1 λ j < 0 ] + O ( m N α 2 σ 2 exp ( − T 2 / ( 4 σ 2 )) / T ) ( 1 − C α 2 ) m Tr ( ρ N ^ ) + j = 1 ∑ N i = 1 ∑ ∞ m ( 2 i − 1 )! ( C 0 α σ ) 2 i − 1 exp ( − ( 2 i − 1 ) σ 2 Δ 2 ) + O ( m N α 2 σ 2 exp ( − T 2 / ( 4 σ 2 )) / T ) . By Fuchs-van de Graaf inequality, for any density matrix ρ \rho ρ
∥ ρ − ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ∥ 1 ≤ 2 1 − ⟨ ψ 0 ∣ ρ ∣ ψ 0 ⟩ ≤ 2 T r ( ρ N ^ ) . \|\rho-\ketbra{\psi_0}{\psi_0}\|_1\leq 2\sqrt{1-\bra{\psi_0}\rho\ket{\psi_0}}\leq 2\sqrt{\mathrm{Tr}(\rho\hat{N})}\,. ∥ ρ − ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ∥ 1 ≤ 2 1 − ⟨ ψ 0 ∣ ρ ∣ ψ 0 ⟩ ≤ 2 Tr ( ρ N ^ ) . Combining this with T r ( ρ N ^ ) ≤ N \mathrm{Tr}\left(\rho\hat{N}\right)\leq N Tr ( ρ N ^ ) ≤ N
∥ Φ α m ρ − ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ∥ 1 ≤ 2 ( 1 − C α 2 ) m / 2 N + 2 ∑ j = 1 N ∑ i = 1 ∞ m ( C 0 α σ ) 2 i − 1 ( 2 i − 1 ) ! exp ( − σ 2 Δ 2 ( 2 i − 1 ) ) + O ( m N α 2 σ 2 exp ( − T 2 / ( 4 σ 2 ) ) / T ) \|\Phi^m_{\alpha}\rho-\ketbra{\psi_0}{\psi_0}\|_1\leq 2\left(1-C\alpha^2\right)^{m/2}\sqrt{N}+2\sqrt{\sum^N_{j=1}\sum^\infty_{i=1}m\frac{\left(C_0\alpha\sqrt{\sigma}\right)^{2i-1}}{(2i-1)!}\exp\left(-\frac{\sigma^2\Delta^2}{(2i-1)}\right)+\mathcal{O}\left(mN\alpha^2\sigma^2\exp(-T^2/(4\sigma^2))/T\right)} ∥ Φ α m ρ − ∣ ψ 0 ⟩ ⟨ ψ 0 ∣ ∥ 1 ≤ 2 ( 1 − C α 2 ) m /2 N + 2 j = 1 ∑ N i = 1 ∑ ∞ m ( 2 i − 1 )! ( C 0 α σ ) 2 i − 1 exp ( − ( 2 i − 1 ) σ 2 Δ 2 ) + O ( m N α 2 σ 2 exp ( − T 2 / ( 4 σ 2 )) / T ) Plugging in the choices of σ \sigma σ T T T α \alpha α
We establish the lower bound for H = λ ∑ j = 1 N n j H=\lambda\sum_{j=1}^N n_j H = λ ∑ j = 1 N n j ω \omega ω d = ∣ 0 ⟩ ⟨ 1 ∣ d=\ket{0}\bra{1} d = ∣ 0 ⟩ ⟨ 1 ∣ n = d † d n=d^\dagger d n = d † d b + b_+ b + b − b_- b − Assumption 3 . The reduced channel is
Φ α , ω ( 1 ) ( ρ ) = Tr B [ U ( ρ ⊗ σ B ) U † ] , \begin{equation}
\Phi^{(1)}_{\alpha,\omega}(\rho)
=\operatorname{Tr}_B\!\left[U(\rho\otimes\sigma_B)U^\dagger\right], \tag*{(48)}
\end{equation} Φ α , ω ( 1 ) ( ρ ) = Tr B [ U ( ρ ⊗ σ B ) U † ] , (48) where σ B = σ b ⊗ 2 \sigma_B=\sigma_b^{\otimes2} σ B = σ b ⊗ 2
σ b = ( 1 − p b ( ω ) ) ∣ 0 ⟩ ⟨ 0 ∣ + p b ( ω ) ∣ 1 ⟩ ⟨ 1 ∣ . \sigma_b=(1-p_{\rm b}(\omega))\ket{0}\bra{0}
+p_{\rm b}(\omega)\ket{1}\bra{1}\,. σ b = ( 1 − p b ( ω )) ∣ 0 ⟩ ⟨ 0 ∣ + p b ( ω ) ∣ 1 ⟩ ⟨ 1 ∣ . and U = T exp ( − i ∫ − T T H α , ω ( 1 ) ( t ) d t ) U=\mathcal T\exp(-i\int_{-T}^T H^{(1)}_{\alpha,\omega}(t)\,\mathrm dt) U = T exp ( − i ∫ − T T H α , ω ( 1 ) ( t ) d t )
H α , ω ( 1 ) ( t ) = λ n + ω ( n b , + + n b , − ) + α ( d † ⊗ F ( t ) + d ⊗ F ( t ) † ) , F ( t ) = f ( t ) b + † + f ( t ) ‾ b − , \begin{equation}
\begin{aligned}
H^{(1)}_{\alpha,\omega}(t)
&=\lambda n+\omega(n_{b,+}+n_{b,-})+\alpha\left(d^\dagger\otimes F(t)+d\otimes F(t)^\dagger\right),\\
F(t)&=f(t)b_+^\dagger+\overline{f(t)}b_-,
\end{aligned} \tag*{(49)}
\end{equation} H α , ω ( 1 ) ( t ) F ( t ) = λn + ω ( n b , + + n b , − ) + α ( d † ⊗ F ( t ) + d ⊗ F ( t ) † ) , = f ( t ) b + † + f ( t ) b − , (49) where n b , ± = b ± † b ± n_{b,\pm}=b_\pm^\dagger b_\pm n b , ± = b ± † b ± d † ∣ 0 ⟩ = ∣ 1 ⟩ d^\dagger\ket{0}=\ket{1} d † ∣ 0 ⟩ = ∣ 1 ⟩ d ∣ 1 ⟩ = ∣ 0 ⟩ d\ket{1}=\ket{0} d ∣ 1 ⟩ = ∣ 0 ⟩ ∣ 0 ⟩ ⊗ ∣ χ ⟩ \ket{0}\otimes\ket{\chi} ∣ 0 ⟩ ⊗ ∣ χ ⟩ ∣ 1 ⟩ ⊗ F ( t ) ∣ χ ⟩ \ket{1}\otimes F(t)\ket{\chi} ∣ 1 ⟩ ⊗ F ( t ) ∣ χ ⟩ ∣ 1 ⟩ ⊗ ∣ χ ⟩ \ket{1}\otimes\ket{\chi} ∣ 1 ⟩ ⊗ ∣ χ ⟩ ∣ 0 ⟩ ⊗ F ( t ) † ∣ χ ⟩ \ket{0}\otimes F(t)^\dagger\ket{\chi} ∣ 0 ⟩ ⊗ F ( t ) † ∣ χ ⟩ ∣ χ ⟩ \ket{\chi} ∣ χ ⟩
Let U r s = ( ⟨ r ∣ ⊗ I B ) U ( ∣ s ⟩ ⊗ I B ) U_{rs}=(\bra{r}\otimes I_B)U(\ket{s}\otimes I_B) U r s = ( ⟨ r ∣ ⊗ I B ) U ( ∣ s ⟩ ⊗ I B ) r , s ∈ { 0 , 1 } r,s\in\{0,1\} r , s ∈ { 0 , 1 } U U U
U ( ∣ 0 ⟩ ⊗ ∣ χ ⟩ ) = ∣ 0 ⟩ ⊗ U 00 ∣ χ ⟩ + ∣ 1 ⟩ ⊗ U 10 ∣ χ ⟩ , U ( ∣ 1 ⟩ ⊗ ∣ χ ⟩ ) = ∣ 0 ⟩ ⊗ U 01 ∣ χ ⟩ + ∣ 1 ⟩ ⊗ U 11 ∣ χ ⟩ . \begin{equation}
\begin{aligned}
U(\ket{0}\otimes\ket{\chi})
&=\ket{0}\otimes U_{00}\ket{\chi}
+\ket{1}\otimes U_{10}\ket{\chi},\\
U(\ket{1}\otimes\ket{\chi})
&=\ket{0}\otimes U_{01}\ket{\chi}
+\ket{1}\otimes U_{11}\ket{\chi}.
\end{aligned} \tag*{(50)}
\end{equation} U ( ∣ 0 ⟩ ⊗ ∣ χ ⟩ ) U ( ∣ 1 ⟩ ⊗ ∣ χ ⟩ ) = ∣ 0 ⟩ ⊗ U 00 ∣ χ ⟩ + ∣ 1 ⟩ ⊗ U 10 ∣ χ ⟩ , = ∣ 0 ⟩ ⊗ U 01 ∣ χ ⟩ + ∣ 1 ⟩ ⊗ U 11 ∣ χ ⟩ . (50) Taking the bath trace gives
Φ α , ω ( 1 ) ( ∣ s ⟩ ⟨ s ∣ ) = ∑ r , r ′ = 0 1 Tr B ( U r s σ B U r ′ s † ) ∣ r ⟩ ⟨ r ′ ∣ . \begin{equation}
\Phi^{(1)}_{\alpha,\omega}({\ket{s}\bra{s}})
=\sum_{r,r'=0}^1
\operatorname{Tr}_B(U_{rs}\sigma_BU_{r's}^\dagger)
\ket{r}\bra{r'}. \tag*{(51)}
\end{equation} Φ α , ω ( 1 ) ( ∣ s ⟩ ⟨ s ∣ ) = r , r ′ = 0 ∑ 1 Tr B ( U r s σ B U r ′ s † ) ∣ r ⟩ ⟨ r ′ ∣ . (51) The conserved parity ( − 1 ) n + n b , + + n b , − (-1)^{n+n_{b,+}+n_{b,-}} ( − 1 ) n + n b , + + n b , − Tr B ( U r s σ B U r ′ s † ) = 0 \operatorname{Tr}_B(U_{rs}\sigma_BU_{r's}^\dagger)=0 Tr B ( U r s σ B U r ′ s † ) = 0 r ≠ r ′ r\ne r' r = r ′ ∑ r U r s † U r s = I B \sum_rU_{rs}^\dagger U_{rs}=I_B ∑ r U r s † U r s = I B
Φ α , ω ( 1 ) ( ∣ 0 ⟩ ⟨ 0 ∣ ) = ( 1 − a ( ω ) ) ∣ 0 ⟩ ⟨ 0 ∣ + a ( ω ) ∣ 1 ⟩ ⟨ 1 ∣ , Φ α , ω ( 1 ) ( ∣ 1 ⟩ ⟨ 1 ∣ ) = z ( ω ) ∣ 0 ⟩ ⟨ 0 ∣ + ( 1 − z ( ω ) ) ∣ 1 ⟩ ⟨ 1 ∣ , \begin{equation}
\begin{aligned}
\Phi^{(1)}_{\alpha,\omega}({\ket{0}\bra{0}})
&=(1-a(\omega)){\ket{0}\bra{0}}+a(\omega){\ket{1}\bra{1}},\\
\Phi^{(1)}_{\alpha,\omega}({\ket{1}\bra{1}})
&=z(\omega){\ket{0}\bra{0}}+(1-z(\omega)){\ket{1}\bra{1}},
\end{aligned} \tag*{(52)}
\end{equation} Φ α , ω ( 1 ) ( ∣ 0 ⟩ ⟨ 0 ∣ ) Φ α , ω ( 1 ) ( ∣ 1 ⟩ ⟨ 1 ∣ ) = ( 1 − a ( ω )) ∣ 0 ⟩ ⟨ 0 ∣ + a ( ω ) ∣ 1 ⟩ ⟨ 1 ∣ , = z ( ω ) ∣ 0 ⟩ ⟨ 0 ∣ + ( 1 − z ( ω )) ∣ 1 ⟩ ⟨ 1 ∣ , (52) where
a ( ω ) = Tr B ( U 10 σ B U 10 † ) = ⟨ 1 ∣ Φ α , ω ( 1 ) ( ∣ 0 ⟩ ⟨ 0 ∣ ) ∣ 1 ⟩ , z ( ω ) = Tr B ( U 01 σ B U 01 † ) = ⟨ 0 ∣ Φ α , ω ( 1 ) ( ∣ 1 ⟩ ⟨ 1 ∣ ) ∣ 0 ⟩ . \begin{equation}
\begin{aligned}
a(\omega)&=\operatorname{Tr}_B(U_{10}\sigma_BU_{10}^\dagger)
=\bra{1}\Phi^{(1)}_{\alpha,\omega}({\ket{0}\bra{0}})\ket{1},\\
z(\omega)&=\operatorname{Tr}_B(U_{01}\sigma_BU_{01}^\dagger)
=\bra{0}\Phi^{(1)}_{\alpha,\omega}({\ket{1}\bra{1}})\ket{0}.
\end{aligned} \tag*{(53)}
\end{equation} a ( ω ) z ( ω ) = Tr B ( U 10 σ B U 10 † ) = ⟨ 1 ∣ Φ α , ω ( 1 ) ( ∣ 0 ⟩ ⟨ 0 ∣ ) ∣ 1 ⟩ , = Tr B ( U 01 σ B U 01 † ) = ⟨ 0 ∣ Φ α , ω ( 1 ) ( ∣ 1 ⟩ ⟨ 1 ∣ ) ∣ 0 ⟩ . (53) In particular, 0 ≤ a ( ω ) , z ( ω ) ≤ 1 0\le a(\omega),z(\omega)\le1 0 ≤ a ( ω ) , z ( ω ) ≤ 1 α \alpha α F T ( ν ) = ∫ − T T f ( t ) e i ν t d t F_T(\nu)=\int_{-T}^T f(t)e^{i\nu t}\,\mathrm dt F T ( ν ) = ∫ − T T f ( t ) e i ν t d t r = p b ( ω ) r=p_{\rm b}(\omega) r = p b ( ω )
a ( ω ) = α 2 a 2 ( ω ) + O ( α 4 ) , a 2 ( ω ) = ( 1 − r ) ∣ F T ( λ + ω ) ∣ 2 + r ∣ F T ( ω − λ ) ∣ 2 , z ( ω ) = α 2 z 2 ( ω ) + O ( α 4 ) , z 2 ( ω ) = ( 1 − r ) ∣ F T ( ω − λ ) ∣ 2 + r ∣ F T ( λ + ω ) ∣ 2 . \begin{equation}
\begin{aligned}
a(\omega)=\alpha^2a_2(\omega)+\mathcal O(\alpha^4),\quad a_2(\omega)=(1-r)\left|F_T(\lambda+\omega)\right|^2+r\left|F_T(\omega-\lambda)\right|^2,\\
z(\omega)=\alpha^2z_2(\omega)+\mathcal O(\alpha^4),\quad z_2(\omega)=(1-r)\left|F_T(\omega-\lambda)\right|^2+r\left|F_T(\lambda+\omega)\right|^2.
\end{aligned} \tag*{(54)}
\end{equation} a ( ω ) = α 2 a 2 ( ω ) + O ( α 4 ) , a 2 ( ω ) = ( 1 − r ) ∣ F T ( λ + ω ) ∣ 2 + r ∣ F T ( ω − λ ) ∣ 2 , z ( ω ) = α 2 z 2 ( ω ) + O ( α 4 ) , z 2 ( ω ) = ( 1 − r ) ∣ F T ( ω − λ ) ∣ 2 + r ∣ F T ( λ + ω ) ∣ 2 . (54) The remainders in (54) are uniform in ω \omega ω T T T 2 ∣ f ( t ) ∣ 2|f(t)| 2∣ f ( t ) ∣ f ∈ L 1 ( R ) f\in L^1(\mathbb R) f ∈ L 1 ( R ) (53) are Born probabilities for occupation measurements. If X t ∈ { 0 , 1 } X_t\in\{0,1\} X t ∈ { 0 , 1 }
P ( X t + 1 = 1 ∣ X t = 0 , ω ) = a ( ω ) , P ( X t + 1 = 0 ∣ X t = 1 , ω ) = z ( ω ) . \begin{equation}
\begin{aligned}
\mathbb P(X_{t+1}=1\mid X_t=0,\omega)&=a(\omega),\\
\mathbb P(X_{t+1}=0\mid X_t=1,\omega)&=z(\omega).
\end{aligned} \tag*{(55)}
\end{equation} P ( X t + 1 = 1 ∣ X t = 0 , ω ) P ( X t + 1 = 0 ∣ X t = 1 , ω ) = a ( ω ) , = z ( ω ) . (55) On diagonal states, the action of the quantum channel is described by the scalar occupation-probability map T ω \mathcal T_\omega T ω p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] v ( p ) = ( 1 − p ) ∣ 0 ⟩ ⟨ 0 ∣ + p ∣ 1 ⟩ ⟨ 1 ∣ ∈ D ( C 2 ) v(p)=(1-p)\ket{0}\bra{0}+p\ket{1}\bra{1}\in\mathcal D(\mathbb C^2) v ( p ) = ( 1 − p ) ∣ 0 ⟩ ⟨ 0 ∣ + p ∣ 1 ⟩ ⟨ 1 ∣ ∈ D ( C 2 ) p p p (52) ,
Φ α , ω ( 1 ) ( v ( p ) ) = [ ( 1 − p ) ( 1 − a ( ω ) ) + p z ( ω ) ] ∣ 0 ⟩ ⟨ 0 ∣ + [ ( 1 − p ) a ( ω ) + p ( 1 − z ( ω ) ) ] ∣ 1 ⟩ ⟨ 1 ∣ = v ( T ω ( p ) ) , \begin{equation}
\begin{aligned}
\Phi^{(1)}_{\alpha,\omega}(v(p))=[(1-p)(1-a(\omega))+pz(\omega)]\ket{0}\bra{0}+[(1-p)a(\omega)+p(1-z(\omega))]\ket{1}\bra{1}=v(\mathcal T_\omega(p)),
\end{aligned} \tag*{(56)}
\end{equation} Φ α , ω ( 1 ) ( v ( p )) = [( 1 − p ) ( 1 − a ( ω )) + p z ( ω )] ∣ 0 ⟩ ⟨ 0 ∣ + [( 1 − p ) a ( ω ) + p ( 1 − z ( ω ))] ∣ 1 ⟩ ⟨ 1 ∣ = v ( T ω ( p )) , (56) where
T ω ( p ) = ( 1 − p ) a ( ω ) + p ( 1 − z ( ω ) ) = a ( ω ) + ( 1 − q ( ω ) ) p , q ( ω ) = a ( ω ) + z ( ω ) . \begin{equation}
\begin{aligned}
\mathcal T_\omega(p)=(1-p)a(\omega)+p(1-z(\omega))=a(\omega)+(1-q(\omega))p,\quad q(\omega)&=a(\omega)+z(\omega).
\end{aligned} \tag*{(57)}
\end{equation} T ω ( p ) = ( 1 − p ) a ( ω ) + p ( 1 − z ( ω )) = a ( ω ) + ( 1 − q ( ω )) p , q ( ω ) = a ( ω ) + z ( ω ) . (57) For independent frequencies ω t ∼ μ \omega_t\sim\mu ω t ∼ μ
P t + 1 = T ω t ( P t ) , P 0 = p ∈ [ 0 , 1 ] . P_{t+1}=\mathcal T_{\omega_t}(P_t),\qquad P_0=p\in[0,1]. P t + 1 = T ω t ( P t ) , P 0 = p ∈ [ 0 , 1 ] . Here P t P_t P t
E ω [ Φ α , ω ( 1 ) ( v ( p ) ) ] = v ( E ω [ T ω ( p ) ] ) . \mathbb E_\omega\!\left[\Phi^{(1)}_{\alpha,\omega}(v(p))\right]
=v\!\left(\mathbb E_\omega[\mathcal T_\omega(p)]\right). E ω [ Φ α , ω ( 1 ) ( v ( p )) ] = v ( E ω [ T ω ( p )] ) . For sufficiently small α \alpha α T T T 0 ≤ q ( ω ) < 1 0\le q(\omega)<1 0 ≤ q ( ω ) < 1 E ω [ q ( ω ) ] > 0 \mathbb E_\omega[q(\omega)]>0 E ω [ q ( ω )] > 0 ν \nu ν P t P_t P t [ 0 , 1 ] [0,1] [ 0 , 1 ] P ∞ ∼ ν P_\infty\sim\nu P ∞ ∼ ν ω ∼ μ \omega\sim\mu ω ∼ μ
T ω ( P ∞ ) ∼ ν . \begin{equation}
\mathcal{T}_\omega(P_\infty)\sim \nu. \tag*{(58)}
\end{equation} T ω ( P ∞ ) ∼ ν . (58) The stationary distribution can also be characterized as the limiting distribution of P t P_t P t t → ∞ t\rightarrow\infty t → ∞ lim τ → ∞ T ω − 1 ∘ T ω − 2 ∘ ⋯ ∘ T ω − τ ( p ) \lim_{\tau\rightarrow \infty} \mathcal{T}_{\omega_{-1}}\circ \mathcal{T}_{\omega_{-2}}\circ \cdots \circ \mathcal{T}_{\omega_{-\tau}}(p) lim τ → ∞ T ω − 1 ∘ T ω − 2 ∘ ⋯ ∘ T ω − τ ( p ) p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] P ∞ P_\infty P ∞
Expectation: μ P ∞ = E [ P ∞ ] = E ω [ a ( ω ) ] E ω [ q ( ω ) ] . \mu_{P_\infty}=\mathbb{E}[P_\infty] = \frac{\mathbb{E}_{\omega} [a(\omega)]}{\mathbb{E}_{\omega}[q(\omega)]}. μ P ∞ = E [ P ∞ ] = E ω [ q ( ω )] E ω [ a ( ω )] .
Variance: V a r [ P ∞ ] = E ω [ ( a ( ω ) − μ P ∞ q ( ω ) ) 2 ] E ω [ 2 q ( ω ) − q 2 ( ω ) ] \mathrm{Var}[P_\infty] = \frac{\mathbb{E}_{\omega}[(a(\omega) -\mu_{P_\infty}q(\omega))^2]}{\mathbb{E}_{\omega} [2q(\omega)- q^2(\omega)]} Var [ P ∞ ] = E ω [ 2 q ( ω ) − q 2 ( ω )] E ω [( a ( ω ) − μ P ∞ q ( ω ) ) 2 ]
This finishes the discussion of the single-mode case.
For N N N ( − 1 ) n j + n b , j , + + n b , j , − (-1)^{n_j+n_{b,j,+}+n_{b,j,-}} ( − 1 ) n j + n b , j , + + n b , j , − λ j = λ \lambda_j=\lambda λ j = λ T ω \mathcal T_\omega T ω ω \omega ω
Φ α , ω ( v ( p ) ⊗ N ) = ⨂ j = 1 N Φ α , ω ( 1 ) ( v ( p ) ) = v ( T ω ( p ) ) ⊗ N . \begin{equation}
\begin{aligned}
\Phi_{\alpha,\omega}(v(p)^{\otimes N})=\bigotimes_{j=1}^N\Phi^{(1)}_{\alpha,\omega}(v(p))=v(\mathcal T_\omega(p))^{\otimes N}.
\end{aligned} \tag*{(59)}
\end{equation} Φ α , ω ( v ( p ) ⊗ N ) = j = 1 ⨂ N Φ α , ω ( 1 ) ( v ( p )) = v ( T ω ( p ) ) ⊗ N . (59) Starting from v ( p ) ⊗ N v(p)^{\otimes N} v ( p ) ⊗ N v ( P t ) ⊗ N v(P_t)^{\otimes N} v ( P t ) ⊗ N
Φ α t ( v ( p ) ⊗ N ) = E [ v ( P t ) ⊗ N ] . \Phi_\alpha^t(v(p)^{\otimes N})
=\mathbb E[v(P_t)^{\otimes N}]. Φ α t ( v ( p ) ⊗ N ) = E [ v ( P t ) ⊗ N ] . The same frequency is shared by all modes at each iteration. Consequently, averaging over frequency histories can create classical correlations even though the state is a product conditional on the frequency history.
Define the stationary density operator
ρ ∗ = E P ∞ ∼ ν [ v ( P ∞ ) ⊗ N ] . \begin{equation}
\rho_*=\mathbb E_{P_\infty\sim\nu}[v(P_\infty)^{\otimes N}]. \tag*{(60)}
\end{equation} ρ ∗ = E P ∞ ∼ ν [ v ( P ∞ ) ⊗ N ] . (60) Using (59) and averaging over both P ∞ P_\infty P ∞
Φ α ( ρ ∗ ) = E P ∞ , ω [ v ( T ω ( P ∞ ) ) ⊗ N ] = E P ∞ ∼ ν [ v ( P ∞ ) ⊗ N ] = ρ ∗ . \begin{equation}
\begin{aligned}
\Phi_\alpha(\rho_*)=\mathbb E_{P_\infty,\omega}
\!\left[v(\mathcal T_\omega(P_\infty))^{\otimes N}\right]=\mathbb E_{P_\infty\sim\nu}[v(P_\infty)^{\otimes N}]=\rho_*.
\end{aligned} \tag*{(61)}
\end{equation} Φ α ( ρ ∗ ) = E P ∞ , ω [ v ( T ω ( P ∞ ) ) ⊗ N ] = E P ∞ ∼ ν [ v ( P ∞ ) ⊗ N ] = ρ ∗ . (61) Let the binomial function b N , k ( p ) = ( N k ) p k ( 1 − p ) N − k , b_{N,k}(p) = \binom{N}{k} p^k (1-p)^{N-k}, b N , k ( p ) = ( k N ) p k ( 1 − p ) N − k , p β ( λ ) = 1 1 + e β λ p_\beta(\lambda) = \frac{1}{1+ e^{\beta \lambda}} p β ( λ ) = 1 + e β λ 1 \ketbra{1} in the Gibbs state. The difference between the fixed point ρ ∗ \rho_* ρ ∗
∥ ρ ∗ − σ β ∥ 1 = ∑ k = 0 N ∣ E [ b N , k ( P ∞ ) ] − b N , k ( p β ) ∣ \begin{equation}
\begin{aligned}
\|\rho_* - \sigma_\beta\|_1 &= \sum_{k = 0}^N |\mathbb{E}[b_{N,k}(P_\infty)] - b_{N,k}(p_\beta)|\\
\end{aligned} \tag*{(62)}
\end{equation} ∥ ρ ∗ − σ β ∥ 1 = k = 0 ∑ N ∣ E [ b N , k ( P ∞ )] − b N , k ( p β ) ∣ (62) We introduce the following lemma to bound the difference between the corresponding binomial probabilities.
Lemma 16. Given a random variable P ∈ [ 0 , 1 ] P \in [0,1] P ∈ [ 0 , 1 ] p ∈ ( 0 , 1 ) p \in(0,1) p ∈ ( 0 , 1 ) C > 0 C>0 C > 0 E ∣ P − E P ∣ 4 ≤ C V a r ( P ) 2 ,
\mathbb{E}|P - \mathbb{E}P|^4 \leq C \mathrm{Var}(P)^2\,,
E ∣ P − E P ∣ 4 ≤ C Var ( P ) 2 , c > 0 c>0 c > 0 p p p C C C N ≥ 2 N\geq 2 N ≥ 2 ∑ k = 0 N ∣ E [ b N , k ( P ) ] − b N , k ( p ) ∣ ≥ c min { 1 , N ( V a r ( P ) + ( E ( P ) − p ) 2 ) } .
\sum_{k = 0}^N |\mathbb{E}[b_{N,k}(P)] - b_{N,k}(p)|\geq c \min \{1, N (\mathrm{Var}(P) + (\mathbb{E}(P) - p)^2)\}\,.
∑ k = 0 N ∣ E [ b N , k ( P )] − b N , k ( p ) ∣ ≥ c min { 1 , N ( Var ( P ) + ( E ( P ) − p ) 2 )} .
Proof of Lemma 16 . We may assume C ≥ 1 C\ge1 C ≥ 1 0 < θ ≤ 1 / N 0<\theta\leq 1/\sqrt{N} 0 < θ ≤ 1/ N
h ( p ) ≔ ∑ k = 0 N e i k θ b N , k ( p ) = ( 1 + ( e i θ − 1 ) p ) N . \begin{equation}
h(p) \coloneqq \sum_{k = 0}^N e^{ik\theta}b_{N,k}(p) = (1+(e^{i\theta} - 1)p)^N\,. \tag*{(63)}
\end{equation} h ( p ) : = k = 0 ∑ N e ik θ b N , k ( p ) = ( 1 + ( e i θ − 1 ) p ) N . (63) It corresponds to the generating function of P P P
∣ E [ h ( P ) ] − h ( p ) ∣ = ∣ ∑ k = 0 N e i k θ ( E [ b N , k ( P ) ] − b N , k ( p ) ) ∣ ≤ ∑ k = 0 N ∣ E [ b N , k ( P ) ] − b N , k ( p ) ∣ . \begin{equation}
|\mathbb{E}[h(P)] - h(p)| = \left|\sum_{k = 0}^N e^{ik\theta} \left(\mathbb{E}[b_{N,k}(P)] - b_{N,k}\left(p\right)\right)\right|\leq \sum_{k = 0}^N \left|\mathbb{E}[b_{N,k}(P)] - b_{N,k}\left(p\right)\right|\,. \tag*{(64)}
\end{equation} ∣ E [ h ( P )] − h ( p ) ∣ = k = 0 ∑ N e ik θ ( E [ b N , k ( P )] − b N , k ( p ) ) ≤ k = 0 ∑ N ∣ E [ b N , k ( P )] − b N , k ( p ) ∣ . (64) We therefore derive a lower bound using Taylor expansion of the function h h h
h ( q ) = h ( p ) + h ′ ( p ) ( q − p ) + h ′ ′ ( p ) 2 ! ( q − p ) 2 + 1 2 ∫ p q ( q − s ) 2 h ′ ′ ′ ( s ) d s \begin{equation}
h(q) = h(p) + h'(p)(q-p) + \frac{h''(p)}{2!}(q-p)^2 +\frac{1}{2}\int_p^q (q-s)^2 h'''(s)\mathrm{d}s \tag*{(65)} %\frac{h'''(\xi)}{3!}(q-p)^3
\end{equation} h ( q ) = h ( p ) + h ′ ( p ) ( q − p ) + 2 ! h ′′ ( p ) ( q − p ) 2 + 2 1 ∫ p q ( q − s ) 2 h ′′′ ( s ) d s (65) where
∣ h ′ ′ ′ ( s ) ∣ = ∣ N ( N − 1 ) ( N − 2 ) ( 1 + ( e i θ − 1 ) s ) N − 3 ( e i θ − 1 ) 3 ∣ ≤ N 3 θ 3 . \begin{equation}
|h'''(s)| = |N(N-1)(N-2) (1+(e^{i\theta} - 1)s)^{N-3}(e^{i\theta} - 1)^3|\leq N^3\theta^3\,. \tag*{(66)}
\end{equation} ∣ h ′′′ ( s ) ∣ = ∣ N ( N − 1 ) ( N − 2 ) ( 1 + ( e i θ − 1 ) s ) N − 3 ( e i θ − 1 ) 3 ∣ ≤ N 3 θ 3 . (66) We decompose the difference into the leading term and the remainder
E [ h ( P ) ] − h ( p ) = M + R \mathbb{E}[h(P)] - h(p) = M + R E [ h ( P )] − h ( p ) = M + R The contribution from the first and second-order terms satisfies
∣ M ∣ = ∣ h ′ ( p ) ( E [ P ] − p ) + h ′ ′ ( p ) 2 ! E [ ( P − p ) 2 ] ∣ = ∣ h ′ ( p ) ( ( E [ P ] − p ) + h ′ ′ ( p ) 2 ! h ′ ( p ) E [ ( P − p ) 2 ] ) ∣ ≥ ∣ h ′ ( p ) ∣ 2 I m ( h ′ ′ ( p ) h ′ ( p ) ) E [ ( P − p ) 2 ] ≥ N 2 θ 2 32 E [ ( P − p ) 2 ] . \begin{equation}
\begin{aligned}
|M| &= |h'(p)(\mathbb{E}[P]-p) + \frac{h''(p)}{2!}\mathbb{E}[(P-p)^2]| = \left|h'(p)\left((\mathbb{E}[P]-p) + \frac{h''(p)}{2! h'(p)}\mathbb{E}[(P-p)^2]\right)\right|\\
&\geq \frac{| h'(p)|}{2} \mathrm{Im}\left(\frac{h''(p)}{h'(p)}\right)\mathbb{E}[(P-p)^2]\geq \frac{N^2\theta^2}{32} \mathbb{E}[(P-p)^2]\,.
\end{aligned} \tag*{(67)}
\end{equation} ∣ M ∣ = ∣ h ′ ( p ) ( E [ P ] − p ) + 2 ! h ′′ ( p ) E [( P − p ) 2 ] ∣ = h ′ ( p ) ( ( E [ P ] − p ) + 2 ! h ′ ( p ) h ′′ ( p ) E [( P − p ) 2 ] ) ≥ 2 ∣ h ′ ( p ) ∣ Im ( h ′ ( p ) h ′′ ( p ) ) E [( P − p ) 2 ] ≥ 32 N 2 θ 2 E [( P − p ) 2 ] . (67) For the remainder term, we have
∣ R ∣ ≤ N 3 θ 3 6 E [ ∣ P − p ∣ 3 ] \begin{equation}
|R|\leq \frac{N^3\theta^3}{6} \mathbb{E}[|P-p|^3] \tag*{(68)} %\leq\frac{N^3\theta^3}{6}\mathbb{E}[(P-p)^2]^{3/2}.
\end{equation} ∣ R ∣ ≤ 6 N 3 θ 3 E [ ∣ P − p ∣ 3 ] (68) Using ( a + b ) 4 ≤ 8 ( a 4 + b 4 ) (a+b)^4 \le 8(a^4 + b^4) ( a + b ) 4 ≤ 8 ( a 4 + b 4 )
E [ ( P − p ) 4 ] ≤ 8 ( C V a r ( P ) 2 + ( E P − p ) 4 ) ≤ 8 C ( E [ ( P − p ) 2 ] ) 2 , \begin{equation}
\mathbb{E}[(P - p)^4] \leq 8 \left( C\mathrm{Var}(P)^2 + (\mathbb{E}P -p)^4\right)\leq 8C \left(\mathbb{E}[\left(P-p\right)^2]\right)^2, \tag*{(69)}
\end{equation} E [( P − p ) 4 ] ≤ 8 ( C Var ( P ) 2 + ( E P − p ) 4 ) ≤ 8 C ( E [ ( P − p ) 2 ] ) 2 , (69) By Cauchy–Schwarz, E [ ∣ P − p ∣ 3 ] ≤ E [ ( P − p ) 2 ] E [ ( P − p ) 4 ] ≤ 8 C E [ ( P − p ) 2 ] 3 / 2 \mathbb E[|P-p|^3]\leq\sqrt{\mathbb E[(P-p)^2]\mathbb E[(P-p)^4]}\leq\sqrt{8C}\,\mathbb E[(P-p)^2]^{3/2} E [ ∣ P − p ∣ 3 ] ≤ E [( P − p ) 2 ] E [( P − p ) 4 ] ≤ 8 C E [( P − p ) 2 ] 3/2
∣ R ∣ ≤ 8 C N 3 θ 3 6 E [ ( P − p ) 2 ] 3 / 2 . \begin{equation}
|R|\leq \frac{\sqrt{8C}\,N^3\theta^3}{6}\mathbb{E}[(P-p)^2]^{3/2}\,. \tag*{(70)}
\end{equation} ∣ R ∣ ≤ 6 8 C N 3 θ 3 E [( P − p ) 2 ] 3/2 . (70) Set S = E [ ( P − p ) 2 ] S=\mathbb E[(P-p)^2] S = E [( P − p ) 2 ] S = 0 S=0 S = 0
∣ E [ h ( P ) ] − h ( p ) ∣ ≥ N 2 θ 2 S ( 1 32 − 8 C 6 N θ S ) . \begin{equation}
\left|\mathbb{E}[h(P)] - h(p)\right|\geq N^2\theta^2 S\left(\frac1{32}-\frac{\sqrt{8C}}6N\theta\sqrt S\right). \tag*{(71)}
\end{equation} ∣ E [ h ( P )] − h ( p ) ∣ ≥ N 2 θ 2 S ( 32 1 − 6 8 C N θ S ) . (71) Choose θ = c C min { N − 1 / 2 , ( N S ) − 1 } \theta=c_C\min\{N^{-1/2},(N\sqrt S)^{-1}\} θ = c C min { N − 1/2 , ( N S ) − 1 } 0 < c C ≤ min { 1 , 3 / ( 32 8 C ) } 0<c_C\leq\min\{1,3/(32\sqrt{8C})\} 0 < c C ≤ min { 1 , 3/ ( 32 8 C )} 1 / 64 1/64 1/64
∣ E [ h ( P ) ] − h ( p ) ∣ ≳ min { 1 , N E [ ( P − p ) 2 ] } . \begin{equation}
\left|\mathbb{E}[h(P)] - h(p)\right|\gtrsim \min\{1, N \mathbb{E}[(P-p)^2]\}\,. \tag*{(72)}
\end{equation} ∣ E [ h ( P )] − h ( p ) ∣ ≳ min { 1 , N E [( P − p ) 2 ]} . (72) Finally, since E [ ( P − p ) 2 ] = V a r ( P ) + ( E ( P ) − p ) 2 \mathbb{E}[(P-p)^2] = \mathrm{Var}(P) + (\mathbb{E}(P) - p)^2 E [( P − p ) 2 ] = Var ( P ) + ( E ( P ) − p ) 2 (64) and (72) . ◻
Now, we are ready to prove Theorem 13 :
Proof. Fix λ ∈ [ − 1 , 1 ] \lambda\in[-1,1] λ ∈ [ − 1 , 1 ] (35) , and set H = λ ∑ j = 1 N n j H=\lambda\sum_{j=1}^Nn_j H = λ ∑ j = 1 N n j p 1 = ( 1 + e λ ) − 1 p_1=(1+e^\lambda)^{-1} p 1 = ( 1 + e λ ) − 1 Q ( ω ) = u ( ω ; λ ) + d ( ω ; λ ) Q(\omega)=u(\omega;\lambda)+d(\omega;\lambda) Q ( ω ) = u ( ω ; λ ) + d ( ω ; λ ) Q ‾ = E ω [ Q ( ω ) ] > 0 \overline Q=\mathbb E_\omega[Q(\omega)]>0 Q = E ω [ Q ( ω )] > 0 p 0 = E ω [ u ( ω ; λ ) ] / Q ‾ p_0=\mathbb E_\omega[u(\omega;\lambda)]/\overline Q p 0 = E ω [ u ( ω ; λ )] / Q
sup ν ∣ F T ( ν ) − f ^ ( ν ) ∣ ≤ ∫ ∣ t ∣ > T ∣ f ( t ) ∣ d t ≤ κ α 2 . \sup_\nu|F_T(\nu)-\widehat f(\nu)|
\leq\int_{|t|>T}|f(t)|\,\mathrm dt\leq\kappa\alpha^2. ν sup ∣ F T ( ν ) − f ( ν ) ∣ ≤ ∫ ∣ t ∣ > T ∣ f ( t ) ∣ d t ≤ κ α 2 . Together with (54) , this yields, uniformly in ω \omega ω T T T
a ( ω ) = α 2 u ( ω ; λ ) + O ( α 4 ) , z ( ω ) = α 2 d ( ω ; λ ) + O ( α 4 ) , q ‾ : = E ω [ q ( ω ) ] = α 2 Q ‾ + O ( α 4 ) = Θ ( α 2 ) . \begin{equation}
\begin{aligned}
a(\omega)&=\alpha^2u(\omega;\lambda)+\mathcal O(\alpha^4),\quad z(\omega)=\alpha^2d(\omega;\lambda)+\mathcal O(\alpha^4),\\
\overline q:=\mathbb E_\omega[q(\omega)]
&=\alpha^2\overline Q+\mathcal O(\alpha^4)=\Theta(\alpha^2).
\end{aligned} \tag*{(73)}
\end{equation} a ( ω ) q := E ω [ q ( ω )] = α 2 u ( ω ; λ ) + O ( α 4 ) , z ( ω ) = α 2 d ( ω ; λ ) + O ( α 4 ) , = α 2 Q + O ( α 4 ) = Θ ( α 2 ) . (73) In particular, μ P ∞ = E ω [ a ( ω ) ] / q ‾ = p 0 + O ( α 2 ) \mu_{P_\infty}=\mathbb E_\omega[a(\omega)]/\overline q=p_0+\mathcal O(\alpha^2) μ P ∞ = E ω [ a ( ω )] / q = p 0 + O ( α 2 )
If p 0 ≠ p 1 p_0\ne p_1 p 0 = p 1 α \alpha α
∥ ρ ∗ − σ 1 ( H ) ∥ 1 ≥ 2 ∣ μ P ∞ − p 1 ∣ ≥ ∣ p 0 − p 1 ∣ > 0 , \|\rho_*-\sigma_1(H)\|_1
\geq2|\mu_{P_\infty}-p_1|\geq|p_0-p_1|>0, ∥ ρ ∗ − σ 1 ( H ) ∥ 1 ≥ 2∣ μ P ∞ − p 1 ∣ ≥ ∣ p 0 − p 1 ∣ > 0 , by measuring one system occupation. Since ρ ∗ \rho_* ρ ∗ ϵ \epsilon ϵ p 0 = p 1 p_0=p_1 p 0 = p 1 (7) . In this case,
μ P ∞ − p 1 = O ( α 2 ) . \begin{equation}
\mu_{P_\infty}-p_1=\mathcal O(\alpha^2). \tag*{(74)}
\end{equation} μ P ∞ − p 1 = O ( α 2 ) . (74) Var ( P ∞ ) = α 2 E ω [ ( u ( ω ; λ ) − p 1 Q ( ω ) ) 2 ] 2 Q ‾ + O ( α 4 ) = Θ ( α 2 ) . \begin{equation}
\operatorname{Var}(P_\infty)
=\alpha^2\frac{\mathbb E_\omega[(u(\omega;\lambda)-p_1Q(\omega))^2]}{2\overline Q}
+\mathcal O(\alpha^4)=\Theta(\alpha^2). \tag*{(75)}
\end{equation} Var ( P ∞ ) = α 2 2 Q E ω [( u ( ω ; λ ) − p 1 Q ( ω ) ) 2 ] + O ( α 4 ) = Θ ( α 2 ) . (75) Indeed, u − p 1 Q = ( u − e − λ d ) / ( 1 + e − λ ) u-p_1Q=(u-e^{-\lambda}d)/(1+e^{-\lambda}) u − p 1 Q = ( u − e − λ d ) / ( 1 + e − λ ) (35) .
To verify the fourth-moment hypothesis of Lemma 16 , write X = P ∞ − μ P ∞ X=P_\infty-\mu_{P_\infty} X = P ∞ − μ P ∞ A = 1 − q ( ω ) A=1-q(\omega) A = 1 − q ( ω ) B = a ( ω ) − μ P ∞ q ( ω ) B=a(\omega)-\mu_{P_\infty}q(\omega) B = a ( ω ) − μ P ∞ q ( ω ) X X X A X + B = d X AX+B\overset{d}=X A X + B = d X E ω [ B ] = 0 \mathbb E_\omega[B]=0 E ω [ B ] = 0
( 1 − E ω [ A 4 ] ) E [ X 4 ] = E ω [ B 4 ] + 6 E ω [ A 2 B 2 ] E [ X 2 ] + 4 E ω [ A 3 B ] E [ X 3 ] . \begin{equation}
\begin{aligned}
(1-\mathbb E_\omega[A^4])\mathbb E[X^4]=\mathbb E_\omega[B^4]
+6\mathbb E_\omega[A^2B^2]\mathbb E[X^2]+4\mathbb E_\omega[A^3B]\mathbb E[X^3].
\end{aligned} \tag*{(76)}
\end{equation} ( 1 − E ω [ A 4 ]) E [ X 4 ] = E ω [ B 4 ] + 6 E ω [ A 2 B 2 ] E [ X 2 ] + 4 E ω [ A 3 B ] E [ X 3 ] . (76) Here ∥ q ∥ ∞ , ∥ B ∥ ∞ = O ( α 2 ) \|q\|_\infty,\|B\|_\infty=\mathcal O(\alpha^2) ∥ q ∥ ∞ , ∥ B ∥ ∞ = O ( α 2 ) E [ X 2 ] = Θ ( α 2 ) \mathbb E[X^2]=\Theta(\alpha^2) E [ X 2 ] = Θ ( α 2 ) ∣ E [ X 3 ] ∣ ≤ E [ X 2 ] |\mathbb E[X^3]|\leq\mathbb E[X^2] ∣ E [ X 3 ] ∣ ≤ E [ X 2 ] ∣ X ∣ ≤ 1 |X|\leq1 ∣ X ∣ ≤ 1 E ω [ A 3 B ] = E ω [ ( A 3 − 1 ) B ] = O ( α 4 ) \mathbb E_\omega[A^3B]=\mathbb E_\omega[(A^3-1)B]=\mathcal O(\alpha^4) E ω [ A 3 B ] = E ω [( A 3 − 1 ) B ] = O ( α 4 ) 1 − E ω [ A 4 ] = Θ ( α 2 ) 1-\mathbb E_\omega[A^4]=\Theta(\alpha^2) 1 − E ω [ A 4 ] = Θ ( α 2 )
E [ X 4 ] = O ( α 4 ) = O ( Var ( P ∞ ) 2 ) . \mathbb E[X^4]=\mathcal O(\alpha^4)
=\mathcal O\!\left(\operatorname{Var}(P_\infty)^2\right). E [ X 4 ] = O ( α 4 ) = O ( Var ( P ∞ ) 2 ) . Applying Lemma 16 with p = p 1 p=p_1 p = p 1 N ≥ 2 N\ge2 N ≥ 2
∥ ρ ∗ − σ 1 ( H ) ∥ 1 ≥ c min { 1 , N ( Var ( P ∞ ) + ( μ P ∞ − p 1 ) 2 ) } ≥ c ′ min { 1 , N α 2 } . \begin{equation}
\begin{aligned}
\|\rho_*-\sigma_1(H)\|_1\geq c\min\{1,N(\operatorname{Var}(P_\infty)+(\mu_{P_\infty}-p_1)^2)\}\geq c'\min\{1,N\alpha^2\}.
\end{aligned} \tag*{(77)}
\end{equation} ∥ ρ ∗ − σ 1 ( H ) ∥ 1 ≥ c min { 1 , N ( Var ( P ∞ ) + ( μ P ∞ − p 1 ) 2 )} ≥ c ′ min { 1 , N α 2 } . (77) Since ρ ∗ \rho_* ρ ∗ α 2 ≲ ϵ / N \alpha^2\lesssim\epsilon/N α 2 ≲ ϵ / N ϵ \epsilon ϵ
Finally, applying the averaged occupation update to the initial states v ( 0 ) ⊗ N v(0)^{\otimes N} v ( 0 ) ⊗ N v ( 1 ) ⊗ N v(1)^{\otimes N} v ( 1 ) ⊗ N
Tr [ n 1 Φ α m ( v ( 1 ) ⊗ N ) ] − Tr [ n 1 Φ α m ( v ( 0 ) ⊗ N ) ] = ( 1 − q ‾ ) m . \operatorname{Tr}\!\left[n_1\Phi_\alpha^m(v(1)^{\otimes N})\right]
-\operatorname{Tr}\!\left[n_1\Phi_\alpha^m(v(0)^{\otimes N})\right]
=(1-\overline q)^m. Tr [ n 1 Φ α m ( v ( 1 ) ⊗ N ) ] − Tr [ n 1 Φ α m ( v ( 0 ) ⊗ N ) ] = ( 1 − q ) m . If both outputs are within trace norm ϵ \epsilon ϵ σ 1 ( H ) \sigma_1(H) σ 1 ( H ) ( 1 − q ‾ ) m ≤ ϵ (1-\overline q)^m\leq\epsilon ( 1 − q ) m ≤ ϵ q ‾ = Θ ( α 2 ) \overline q=\Theta(\alpha^2) q = Θ ( α 2 )
m ≥ log ( 1 / ϵ ) − log ( 1 − q ‾ ) ≳ 1 α 2 log ( 1 / ϵ ) ≳ N ϵ log ( 1 / ϵ ) . m\geq\frac{\log(1/\epsilon)}{-\log(1-\overline q)}
\gtrsim\frac1{\alpha^2}\log(1/\epsilon)
\gtrsim\frac{N}{\epsilon}\log(1/\epsilon). m ≥ − log ( 1 − q ) log ( 1/ ϵ ) ≳ α 2 1 log ( 1/ ϵ ) ≳ ϵ N log ( 1/ ϵ ) . This proves the stated worst-case iteration lower bound. Since T > T 0 > 0 T>T_0>0 T > T 0 > 0 Ω ( N / ϵ ) \Omega(N/\epsilon) Ω ( N / ϵ )
Tensorization of product-state errors
Lemma 17. Let { ρ j , σ j } j = 1 N \{\rho_j,\sigma_j\}_{j=1}^N { ρ j , σ j } j = 1 N σ j ≥ C I \sigma_j\geq C I σ j ≥ C I C > 0 C>0 C > 0 ∥ ⨂ j = 1 N ρ j − ⨂ j = 1 N σ j ∥ 1 ≤ min { ( 2 C ∑ j = 1 N ∥ ρ j − σ j ∥ 1 2 ) 1 / 2 , ∑ j = 1 N ∥ ρ j − σ j ∥ 1 } . \left\|\bigotimes_{j=1}^N\rho_j-\bigotimes_{j=1}^N\sigma_j\right\|_1
\leq \min\left\{\left(\frac{2}{C}\sum_{j=1}^N\|\rho_j-\sigma_j\|_1^2\right)^{1/2},\sum^N_{j=1}\|\rho_j-\sigma_j\|_1\right\}. ⨂ j = 1 N ρ j − ⨂ j = 1 N σ j 1 ≤ min { ( C 2 ∑ j = 1 N ∥ ρ j − σ j ∥ 1 2 ) 1/2 , ∑ j = 1 N ∥ ρ j − σ j ∥ 1 } . ϵ \epsilon ϵ ∥ ρ j − σ j ∥ 1 ≤ C / 2 ϵ / N \|\rho_j-\sigma_j\|_1\leq \sqrt{C/2}\,\epsilon/\sqrt{N} ∥ ρ j − σ j ∥ 1 ≤ C /2 ϵ / N j j j
Proof. Let ρ = ⨂ j ρ j \rho=\bigotimes_j\rho_j ρ = ⨂ j ρ j σ = ⨂ j σ j \sigma=\bigotimes_j\sigma_j σ = ⨂ j σ j
ρ − σ = ∑ j = 1 N ( ⨂ k = 1 j − 1 ρ k ⊗ ( ρ j − σ j ) ⊗ ⨂ k = j + 1 N σ k ) , \rho-\sigma=\sum_{j=1}^N \left(\bigotimes_{k=1}^{j-1}\rho_k\otimes(\rho_j-\sigma_j)\otimes\bigotimes_{k=j+1}^{N}\sigma_k\right), ρ − σ = j = 1 ∑ N k = 1 ⨂ j − 1 ρ k ⊗ ( ρ j − σ j ) ⊗ k = j + 1 ⨂ N σ k , and then apply the triangle inequality and the submultiplicativity of the trace norm:
∥ ρ − σ ∥ 1 ≤ ∑ j = 1 N ∥ ρ j − σ j ∥ 1 . \|\rho-\sigma\|_1\leq \sum_{j=1}^N \|\rho_j-\sigma_j\|_1. ∥ ρ − σ ∥ 1 ≤ j = 1 ∑ N ∥ ρ j − σ j ∥ 1 . By quantum Pinsker’s inequality, additivity of relative entropy, and monotonicity of the Petz Rényi divergence in its order,
1 2 ∥ ρ − σ ∥ 1 2 ≤ D ( ρ ∥ σ ) = ∑ j D ( ρ j ∥ σ j ) ≤ ∑ j log T r ( ρ j 2 σ j − 1 ) . \frac{1}{2}\|\rho-\sigma\|_1^2
\leq D(\rho\|\sigma)
=\sum_jD(\rho_j\|\sigma_j)
\leq\sum_j\log\mathrm{Tr}(\rho_j^2\sigma_j^{-1}). 2 1 ∥ ρ − σ ∥ 1 2 ≤ D ( ρ ∥ σ ) = j ∑ D ( ρ j ∥ σ j ) ≤ j ∑ log Tr ( ρ j 2 σ j − 1 ) . Since σ j ≥ C I \sigma_j\geq C I σ j ≥ C I
log T r ( ρ j 2 σ j − 1 ) = log ( 1 + T r ( ( ρ j − σ j ) 2 σ j − 1 ) ) ≤ C − 1 ∥ ρ j − σ j ∥ 1 2 . \log\mathrm{Tr}(\rho_j^2\sigma_j^{-1})
=\log\left(1+\mathrm{Tr}\left((\rho_j-\sigma_j)^2\sigma_j^{-1}\right)\right)
\leq C^{-1}\|\rho_j-\sigma_j\|_1^2. log Tr ( ρ j 2 σ j − 1 ) = log ( 1 + Tr ( ( ρ j − σ j ) 2 σ j − 1 ) ) ≤ C − 1 ∥ ρ j − σ j ∥ 1 2 . Combining these inequalities proves the result. ◻
Proof of Lemma 15 . The error from the infinite time window follows from [2 ] [Lemma S3]. It is sufficient to consider the adjoint channel,
Φ ~ α , j † ( O ) = U j † ( T ) ∘ Ψ ~ α , j † ∘ U j † ( T ) ( O ) , \begin{equation}
\widetilde\Phi_{\alpha, j}^\dagger(O) = \mathcal{U}_{j}^\dagger(T) \circ \widetilde\Psi_{\alpha, j}^\dagger\circ \mathcal{U}_{j}^\dagger(T) (O), \tag*{(78)}
\end{equation} Φ α , j † ( O ) = U j † ( T ) ∘ Ψ α , j † ∘ U j † ( T ) ( O ) , (78) where we obtain Ψ ~ α , j † ( O ) \widetilde\Psi_{\alpha,j}^\dagger(O) Ψ α , j † ( O ) T → ∞ T\to\infty T → ∞
Ψ α , j † ( O ) = ∑ n , m ≥ 0 ( − 1 ) n ( i α ) n + m ∫ ∫ − T ≤ s n ≤ ⋯ ≤ s 1 ≤ T , − T ≤ τ m ≤ ⋯ ≤ τ 1 ≤ T d s 1 ⋯ d s n d τ 1 ⋯ d τ m ∑ α i , β j ∈ { − 1 , 1 } S β m , j † ⋯ S β 1 , j † O S α 1 , j ⋯ S α n , j T r E j ( B α 1 , j ⋯ B α n , j ρ B ( ω ) B β m , j † ⋯ B β 1 , j † ) f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) e − i ( ω + λ j ) ∑ q = 1 m β q τ q + i ( ω + λ j ) ∑ p = 1 n α p s p . \begin{equation}
\begin{aligned}
\Psi_{\alpha, j}^\dagger(O)
&= \sum_{n,m\geq 0}(-1)^n(i\alpha)^{n+m}\int \int_{-T\leq s_n\leq \cdots \leq s_1 \leq T, -T\leq \tau_m\leq \cdots\leq \tau_1 \leq T}\mathrm{d}s_1 \cdots \mathrm{d}s_n \mathrm{d}\tau_1\cdots \mathrm{d}\tau_m \\
&\sum_{\alpha_i,\beta_j\in\{-1,1\}} S_{\beta_m,j}^\dag \cdots S_{\beta_1,j}^\dag O S_{\alpha_1,j}\cdots S_{\alpha_n,j}\mathrm{Tr}_{E_j}(B_{\alpha_1,j}\cdots B_{\alpha_n,j}\rho_B(\omega) B_{\beta_m,j}^\dag \cdots B_{\beta_1,j}^\dag )\\
& f(s_1)\cdots f(s_n) f(\tau_1)\cdots f(\tau_m)e^{-i(\omega+\lambda_j)\sum_{q=1}^{m}\beta_q\tau_q
+i(\omega+\lambda_j)\sum_{p=1}^{n}\alpha_p s_p}\,. \\
\end{aligned} \tag*{(79)}
\end{equation} Ψ α , j † ( O ) = n , m ≥ 0 ∑ ( − 1 ) n ( i α ) n + m ∫ ∫ − T ≤ s n ≤ ⋯ ≤ s 1 ≤ T , − T ≤ τ m ≤ ⋯ ≤ τ 1 ≤ T d s 1 ⋯ d s n d τ 1 ⋯ d τ m α i , β j ∈ { − 1 , 1 } ∑ S β m , j † ⋯ S β 1 , j † O S α 1 , j ⋯ S α n , j Tr E j ( B α 1 , j ⋯ B α n , j ρ B ( ω ) B β m , j † ⋯ B β 1 , j † ) f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) e − i ( ω + λ j ) ∑ q = 1 m β q τ q + i ( ω + λ j ) ∑ p = 1 n α p s p . (79) By substituting the corresponding number operator n j = d j † d j n_j = d_j^\dagger d_j n j = d j † d j
p j ( ω ) : = ∣ ∑ r = 1 ∞ ( − i α ) 2 r − 1 ∫ t 1 < ⋯ < t 2 r − 1 e i ( λ j + ω ) ∑ ℓ = 1 2 r − 1 ( − 1 ) ℓ + 1 t ℓ ∏ ℓ = 1 2 r − 1 f ( t ℓ ) d t 1 ⋯ d t 2 r − 1 ∣ 2 , \begin{equation}
p_{j}(\omega):=\left|\sum_{r=1}^{\infty}(-i\alpha)^{2r-1}\int_{t_1<\cdots<t_{2r-1}}e^{i(\lambda_j+\omega)\sum_{\ell=1}^{2r-1}(-1)^{\ell+1}t_\ell}\prod_{\ell=1}^{2r-1}f(t_\ell)\,\mathrm{d}t_1\cdots\mathrm{d}t_{2r-1}\right|^2 , \tag*{(80)}
\end{equation} p j ( ω ) := r = 1 ∑ ∞ ( − i α ) 2 r − 1 ∫ t 1 < ⋯ < t 2 r − 1 e i ( λ j + ω ) ∑ ℓ = 1 2 r − 1 ( − 1 ) ℓ + 1 t ℓ ℓ = 1 ∏ 2 r − 1 f ( t ℓ ) d t 1 ⋯ d t 2 r − 1 2 , (80) When ω > 0 \omega>0 ω > 0 \rho_B(\omega) = \ketbra{0} ,
Ψ ~ α , j † ( n j ) = n j + ∑ n , m ≥ 0 odd , n + m ≥ 2 ( − 1 ) n ( i α ) n + m ∫ ∫ s n ≤ ⋯ ≤ s 1 , τ m ≤ ⋯ ≤ τ 1 d s 1 ⋯ d s n d τ 1 ⋯ d τ m f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) e − i ( ω + λ j ) ∑ q = 1 m ( − 1 ) q + 1 τ q + i ( ω + λ j ) ∑ p = 1 n ( − 1 ) p + 1 s p ( I − n j ) = ( 1 − p j ( ω ) ) n j + p j ( ω ) I . \begin{equation}
\begin{aligned}
\widetilde\Psi_{\alpha, j}^\dagger(n_j)
& = n_j + \sum_{n,m\geq 0\text{ odd}, n+m\geq2}(-1)^n(i\alpha)^{n+m}\int \int_{ s_n\leq \cdots \leq s_1, \tau_m\leq \cdots\leq \tau_1 }\mathrm{d}s_1 \cdots \mathrm{d}s_n \mathrm{d}\tau_1\cdots \mathrm{d}\tau_m\\
&f(s_1)\cdots f(s_n) f(\tau_1)\cdots f(\tau_m)
e^{-i(\omega+\lambda_j)\sum_{q=1}^{m}(-1)^{q+1}\tau_q
+i(\omega+\lambda_j)\sum_{p=1}^{n}(-1)^{p+1}s_p}(I-n_j)\\
&= (1-p_{j}(\omega)) n_j + p_j(\omega) I \,. \\
\end{aligned} \tag*{(81)}
\end{equation} Ψ α , j † ( n j ) = n j + n , m ≥ 0 odd , n + m ≥ 2 ∑ ( − 1 ) n ( i α ) n + m ∫ ∫ s n ≤ ⋯ ≤ s 1 , τ m ≤ ⋯ ≤ τ 1 d s 1 ⋯ d s n d τ 1 ⋯ d τ m f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) e − i ( ω + λ j ) ∑ q = 1 m ( − 1 ) q + 1 τ q + i ( ω + λ j ) ∑ p = 1 n ( − 1 ) p + 1 s p ( I − n j ) = ( 1 − p j ( ω )) n j + p j ( ω ) I . (81) When ω < 0 \omega<0 ω < 0 \rho_B(\omega) = \ketbra{1} ,
Ψ ~ α , j † ( n j ) = n j + ∑ n , m ≥ 0 , n + m ≥ 2 , n , m even ( − 1 ) n ( i α ) n + m ∫ ∫ s n ≤ ⋯ ≤ s 1 , τ m ≤ ⋯ ≤ τ 1 d s 1 ⋯ d s n d τ 1 ⋯ d τ m f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) e − i ( ω + λ j ) ∑ q = 1 m ( − 1 ) q + 1 τ q + i ( ω + λ j ) ∑ p = 1 n ( − 1 ) p + 1 s p n j = ( 1 − p j ( ω ) ) n j . \begin{equation}
\begin{aligned}
\widetilde\Psi_{\alpha, j}^\dagger(n_j) &= n_j + \sum_{n,m\geq 0, n+m\geq2, n,m \text{ even}}(-1)^n(i\alpha)^{n+m}\int \int_{ s_n\leq \cdots \leq s_1 , \tau_m\leq \cdots\leq \tau_1 }\mathrm{d}s_1 \cdots \mathrm{d}s_n \mathrm{d}\tau_1\cdots \mathrm{d}\tau_m\\
& f(s_1)\cdots f(s_n) f(\tau_1)\cdots f(\tau_m) e^{-i(\omega+\lambda_j)\sum_{q=1}^{m}(-1)^{q+1}\tau_q
+i(\omega+\lambda_j)\sum_{p=1}^{n}(-1)^{p+1}s_p} n_j\\
&=(1-p_{j}(\omega)) n_j\,. \\
\end{aligned} \tag*{(82)}
\end{equation} Ψ α , j † ( n j ) = n j + n , m ≥ 0 , n + m ≥ 2 , n , m even ∑ ( − 1 ) n ( i α ) n + m ∫ ∫ s n ≤ ⋯ ≤ s 1 , τ m ≤ ⋯ ≤ τ 1 d s 1 ⋯ d s n d τ 1 ⋯ d τ m f ( s 1 ) ⋯ f ( s n ) f ( τ 1 ) ⋯ f ( τ m ) e − i ( ω + λ j ) ∑ q = 1 m ( − 1 ) q + 1 τ q + i ( ω + λ j ) ∑ p = 1 n ( − 1 ) p + 1 s p n j = ( 1 − p j ( ω )) n j . (82) Furthermore, Φ ~ α , j † ( n j ) = Ψ ~ α , j † ( n j ) \widetilde{\Phi}_{\alpha,j}^\dagger(n_j) =\widetilde{\Psi}^\dagger_{\alpha,j}(n_j) Φ α , j † ( n j ) = Ψ α , j † ( n j ) ω \omega ω g ( ω ) = ( 2 S ) − 1 1 [ − S , S ] ( ω ) g(\omega)=(2S)^{-1}\mathbf{1}_{[-S,S]}(\omega) g ( ω ) = ( 2 S ) − 1 1 [ − S , S ] ( ω )
E ω ( Φ ~ α , j † ( ω ) ) ( n j ) = ( 1 − ∫ R g ( ω ) p j ( ω ) d ω ) n j + ∫ R + g ( ω ) p j ( ω ) d ω I \begin{equation}
\mathbb{E}_\omega(\widetilde{\Phi}^\dagger_{\alpha,j}(\omega))(n_j) = \left(1-\int_{\mathbb{R}} g(\omega) p_{j}(\omega)\mathrm{d}\omega\right) n_j + \int_{\mathbb{R}^+} g(\omega)p_j(\omega)\mathrm{d}\omega I \tag*{(83)}
\end{equation} E ω ( Φ α , j † ( ω )) ( n j ) = ( 1 − ∫ R g ( ω ) p j ( ω ) d ω ) n j + ∫ R + g ( ω ) p j ( ω ) d ω I (83) For λ j > 0 \lambda_j>0 λ j > 0 [2 ] with ∥ f ∥ L 1 = 2 3 / 4 π 1 / 4 σ \|f\|_{L^1}=2^{3/4}\pi^{1/4}\sqrt{\sigma} ∥ f ∥ L 1 = 2 3/4 π 1/4 σ
∫ R + g ( ω ) p j ( ω ) d ω ≤ ∫ R + g ( ω ) ∣ ∑ r ≥ 1 α 2 r − 1 ( 2 3 / 4 π 1 / 4 σ ) 2 r − 1 ( 2 r − 1 ) ! e − σ 2 2 r − 1 ( λ j + ω ) 2 ∣ 2 d ω \begin{equation}
\int_{\mathbb{R}^+} g(\omega)p_j(\omega)\mathrm{d}\omega\leq \int_{\mathbb{R}^+} g(\omega) \left|\sum_{r\geq 1} \alpha^{2r-1} \frac{(2^{3/4}\pi^{1/4}\sqrt{\sigma})^{2r-1}}{(2r-1)!}e^{-\frac{\sigma^2}{2r-1}(\lambda_j + \omega)^2}\right|^2\mathrm{d}\omega \tag*{(84)}
\end{equation} ∫ R + g ( ω ) p j ( ω ) d ω ≤ ∫ R + g ( ω ) r ≥ 1 ∑ α 2 r − 1 ( 2 r − 1 )! ( 2 3/4 π 1/4 σ ) 2 r − 1 e − 2 r − 1 σ 2 ( λ j + ω ) 2 2 d ω (84) On J j = [ − λ j , − λ j + ( 2 σ ) − 1 ] ⊂ [ − S , 0 ] J_j=[-\lambda_j,-\lambda_j+(2\sigma)^{-1}]\subset[-S,0] J j = [ − λ j , − λ j + ( 2 σ ) − 1 ] ⊂ [ − S , 0 ] 2 3 / 4 π 1 / 4 e − 1 / 4 α σ 2^{3/4}\pi^{1/4}e^{-1/4}\alpha\sqrt{\sigma} 2 3/4 π 1/4 e − 1/4 α σ O ( ( α σ ) 3 ) \mathcal O((\alpha\sqrt{\sigma})^3) O (( α σ ) 3 ) p j ( ω ) ≥ c α 2 σ p_j(\omega)\ge c\alpha^2\sigma p j ( ω ) ≥ c α 2 σ J j J_j J j α σ \alpha\sqrt{\sigma} α σ c > 0 c>0 c > 0
∫ R − g ( ω ) p j ( ω ) d ω ≥ C α 2 \begin{equation}
\int_{\mathbb{R}^-} g(\omega) p_{j}(\omega)\mathrm{d}\omega\ge C\alpha^2 \tag*{(85)}
\end{equation} ∫ R − g ( ω ) p j ( ω ) d ω ≥ C α 2 (85) Since λ j + ω ≥ Δ \lambda_j+\omega\ge\Delta λ j + ω ≥ Δ ω > 0 \omega>0 ω > 0 α σ \alpha\sqrt{\sigma} α σ
E ω ( Φ ~ α , j † ( ω ) ) ( n j ) ≤ ( 1 − C α 2 ) n j + ∑ r ≥ 1 ( O ( α σ ) ) 2 r − 1 ( 2 r − 1 ) ! e − σ 2 2 r − 1 Δ 2 I . \begin{equation}
\mathbb{E}_\omega(\widetilde{\Phi}^\dagger_{\alpha,j}(\omega))(n_j) \leq (1- C\alpha^2)n_j + \sum_{r\geq 1}\frac{(\mathcal{O}(\alpha \sqrt{\sigma}))^{2r-1}}{(2r-1)!} e^{-\frac{\sigma^2}{2r-1}\Delta^2}I\,. \tag*{(86)}
\end{equation} E ω ( Φ α , j † ( ω )) ( n j ) ≤ ( 1 − C α 2 ) n j + r ≥ 1 ∑ ( 2 r − 1 )! ( O ( α σ ) ) 2 r − 1 e − 2 r − 1 σ 2 Δ 2 I . (86) For λ j < 0 \lambda_j<0 λ j < 0 I − n j I-n_j I − n j
