Super-bath Quantum Eigensolver

Bing-Nan Lu; Mauro Cirio; Tianren Wang; Ying Li; Zongkang Zhang

REVIEW 4 major objections 6 minor 121 references

Super-bath Quantum Eigensolver

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The super-bath quantum eigensolver claims that a quantum computer can prepare a system's ground state in polynomial time whenever some physical bath could do the job, without any detailed knowledge of that bath.

desk verdict Conditional proof with a genuinely new dephasing-plus-super-bath mechanism; the good-bath existence assumption is never instantiated, so treat the polynomial-time claim as a theorem about an unproven precondition. read the letter →

arxiv 2412.19599 v3 pith:WDTNARG7 submitted 2024-12-27 quant-ph

Tianren Wang , Zongkang Zhang , Bing-Nan Lu , Mauro Cirio , Ying Li This is my paper · ORCID

classification quant-ph MSC 81P68 PACS 03.67.Lx

keywords super-bathquantumeigensolvergroundstatepreparationopensystemsLindbladequationGaussianstabilizationdephasingspectraldensitydissipativecomputingnucleargammadecay

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a quantum algorithm, the super-bath quantum eigensolver (SQE), whose central claim is that ground-state preparation can be done in polynomial time using only the existence of an efficient dissipative environment, not a detailed model of it. The algorithm interleaves a Gaussian dephasing channel with simulation of a constructed 'super-bath' in which every system coupling operator sees an identical, standard spectral density. The proof's core is a partial-order statement: if any 'good' sub-bath with positive minimum dissipation power is contained in the super-bath, then the super-bath cools the system almost as well, up to errors controlled by the algorithm's parameters. The authors further argue from nuclear isomer half-life data that nuclei may be a natural application. If correct, the result converts a physical observation—that a system exists in its ground state because of a bath—into a sufficient condition for efficient quantum computation of that ground state.

What carries the argument

The load-bearing object is the super-bath spectral density $J_S(\omega)=J(\omega)\mathbf{1}_N$, a bath in which each of the $N$ system coupling operators $A_\alpha$ couples to an independent copy of the same 'proper' spectral density $J$. A sub-bath $J_{\mathrm{sub}}$ is defined by a constraint factor $b$ bounding its norm relative to $J$ and its relaxation and correlation times relative to $J$'s. The Gaussian stabilization map $G_\sigma\rho=\int dt\,g_\sigma(t)e^{-iHt}\rho e^{iHt}$ prepares approximate stationary states; on stationary states the Redfield and Lindblad dissipation powers coincide, which lets the argument bypass the rotating-wave approximation. Lemma 1 then bounds the energy drop per cooling cycle by $Pt-\epsilon$ with $\epsilon\le Pt/2$ under polynomial parameter scaling.

What would settle it

Numerically evaluate $P_{K,\min}(1/\beta,J)$ for a concrete many-body Hamiltonian and the super-bath spectral density $J_S$: if for every allowed sub-bath with polynomial constraint factor the minimum power over states with ground-state overlap at most $1/2$ is zero or decays faster than polynomially with system size, the energy-drop bound of Lemma 1 fails and Theorem 1's polynomial resource claim does not apply.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1: under the assumption that a sub-bath spectral density $J$ with constraint factor $b$ and minimum power $P_{K,\min}(1/\beta,J)>0$ exists, the SQE observes the ground state with probability at least $1-\kappa$ using $M,t,\sigma = O(\mathrm{Poly}(N,b,r,h,\beta))$ and $K=O(\log(\kappa))$ QPE repetitions. The theorem is proved by showing that in the last round of parameter updates the cooling cycle guarantees $\langle G|\rho_f|G\rangle\ge 1/2$. This establishes a partial order among baths: a super-bath that contains a good sub-bath inherits its dissipation capability, and conversely, if the super-bath is not good then no good bath exists under the stated conditions.

Load-bearing premise

The whole result hinges on the existence of a 'good sub-bath' whose minimum dissipation power is positive for all states with at most half ground-state overlap, with that power and the constraint factor scaling only polynomially with system size; the paper does not construct such a bath for any concrete Hamiltonian.

Editorial extensions

If this is right

Any physical system that is observed in its ground state because of an efficient low-temperature bath becomes, under the stated conditions, a system whose ground state can be prepared on a quantum computer in polynomial time without modeling the bath's microscopic details.
The algorithm provides a sufficient condition for polynomial-time ground-state preparation that is testable in principle: check whether a candidate sub-bath has positive minimum power with polynomial constraint factor.
The partial order among baths implies that engineering a more complex environment cannot destroy the cooling capability that a good sub-bath provides, as long as the approximation error is controlled by the algorithm's dephasing step.
Nuclear ground-state determination is singled out as a candidate application because measured gamma-decay half-lives of isomers show no strong dependence on mass number, suggesting dissipation power that does not decay exponentially with system size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the theorem's assumption could be certified numerically for small instances by computing $P_{K,\min}$ for concrete Hamiltonians and candidate spectral densities; a positive value would strengthen the case that the algorithmic shortcut is practically realizable, while a vanishing value would demarcate its boundary.
Editorial extension: the same sub-bath partial-order logic could be applied to other dissipative state-preparation tasks, such as preparing low-energy thermal states, by redefining 'good' in terms of mixing or heating rates rather than ground-state dissipation power.
Editorial extension: the nuclear evidence in the paper supports only a scaling trend in isomer half-lives, not that any specific nucleus satisfies the theorem's uniform minimum-power condition; identifying a concrete nucleus and spectral density that provably satisfy it would turn the suggested application into a stated corollary.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know up front. The paper's core is a genuine conditional theorem: if a good sub-bath with uniform positive minimum dissipation power exists, then the super-bath algorithm prepares the ground state in polynomial time, with a detailed error budget. The new mechanism is combining Gaussian stabilization dephasing with a super-bath built from a crude spectral density, which avoids the rotating-wave approximation and requires no detailed knowledge of the physical bath. That is a real contribution, and the partial-order framing is a useful way to organize baths.

The math is mostly solid. The appendices give a step-by-step error analysis: Born-Markov error in trace norm, Redfield-vs-exact bounds, coarse-grained Hamiltonians, and the quasi-steady state argument. The key identity—that Redfield and Lindblad dissipation agree on stationary states—is used carefully, and the Gaussian stabilization approximates a delta-stationary state with controlled error. I traced the structure of Lemma 1 and it holds together; there is no circularity or fitted parameter masquerading as prediction.

Now the soft spots, in proportion. The theorem is conditional on an unproven physical premise: the existence of a good sub-bath with PK,min > 0 uniform over all states with overlap ≤1/2. No concrete Hamiltonian is shown to satisfy this. The nuclear gamma-decay data in Section III is about half-lives scaling gently with mass number, which is a different fact; a single long-lived isomer already gives an exponentially small power for that state, and the authors admit the natural bath can be stuck. They suggest the super-bath bypasses selection rules, but they do not construct a spectral density that provably has PK,min > 0 for any nucleus. So the 'polynomial-time ground state preparation from mere existence of a bath' claim has no demonstrated non-vacuous instance. That is a significant gap, but it is a gap in instantiation, not an internal contradiction.

Two smaller issues. The parameter update loop's stopping rule is heuristic—they stop when the estimated energy is stable, but the theorem bounds the number of batches only in terms of the unknown r=1/PK,min, so the algorithm as stated cannot certify when parameters are in the proper range. And the simulation of the super-bath is assumed to be polynomial-cost (Corollary 1 condition 5) without a concrete cost analysis for a realistic target system.

Who is this for? Researchers working on dissipative ground-state preparation will want to read the error analysis; the partial-order concept can be cited. For the claims about nuclear application, it's an interesting suggestion, not a proven statement. The paper deserves a serious referee—a referee should push for an explicit example or numerical demonstration that the good sub-bath assumption is satisfiable, or at least a discussion of what class of systems would satisfy it.

Referee Report

4 major / 6 minor

Summary. The manuscript proposes a "super-bath quantum eigensolver" (SQE) that prepares ground states by alternating a Gaussian-stabilization dephasing channel with the simulation of a super-bath, and claims polynomial-time ground-state preparation under the assumption that a "good sub-bath" with positive minimum dissipation power exists. The central results are Lemma 1 and Theorem 1, with a step-by-step error analysis in Appendices A-D. Section III uses nuclear gamma-decay half-life data to argue that such a good sub-bath may exist for nuclear systems.

Significance. If Theorem 1 is correct, the conceptual contribution is valuable: the algorithm would reduce the need for detailed bath knowledge to an existence assumption, and the paper's partial-order perspective on environments is interesting. The appendix contains a genuine, detailed error analysis rather than a proof sketch, and the derivation from Lemma 1 to Theorem 1 is coherent. However, the main theorem is conditional on an assumption that is not instantiated for any concrete Hamiltonian, and the algorithm as stated has no certified stopping rule, so the practical significance claimed in the abstract and Section III is currently not established.

major comments (4)

[Theorem 1 (Section V, Main result)] The statement of Theorem 1 says "Suppose there exists a sub-bath spectral density J with a constraint factor of b and a minimum power PK,min(1/β,J)" but omits the positivity condition PK,min(1/β,J) > 0 from Eq. (8). If PK,min = 0, then P = 0 in Lemma 1 and the asserted energy-reduction lower bound does not lead to cooling; as stated, the theorem is false. The hypothesis must explicitly require a good sub-bath, i.e., PK,min(1/β,J) > 0.
[Section III and Definition 6/7] The paper never constructs or proves the existence of a good sub-bath for any concrete Hamiltonian. The nuclear gamma-decay data in Fig. 1 concern natural decay half-lives, which are not the same as the uniform minimum power PK,min(1/β,Jg-sub) over all states with ground-state overlap at most 1/2; the authors themselves note that long-lived isomers can make the natural bath inefficient. Thus the polynomial-time claim currently has no demonstrated non-vacuous instance. Either an explicit example of a Hamiltonian and spectral density satisfying the good sub-bath condition should be provided, or the claims of applicability to nuclear ground states should be substantially qualified.
[Section IV, Parameter update; Theorem 1, Eq. (23)] The algorithm's stopping rule is not certified. The text says the parameter-updating loop stops "when confident about Êmin" by tracking stability, but Theorem 1 bounds the number of batches L only in terms of the unknown quantity r = 1/PK,min. The algorithm is not given L, and no lemma proves that the stability heuristic detects the last round or terminates with success probability at least 1 - κ in polynomial time. A certified stopping rule is needed, for example by specifying a sufficient L in terms of a bound on r, or the theorem should be stated as an existence statement about parameter values rather than about the algorithm as written.
[Corollary 1, condition 5] The claimed polynomial complexity also relies on the assumption that the superoperator M(H,T,g^2JS;t) can be implemented at cost Poly(n,1/T,1/g,t,1/epsilon_2). This is not demonstrated: Section VI cites frameworks for bosonic simulation, but does not show that any existing method achieves this scaling for a thermal Gaussian bath with an arbitrary proper spectral density. Since this is a load-bearing assumption for Corollary 1, it should either be proven or explicitly labeled as an additional assumption on the simulation cost model.

minor comments (6)

[Section IV, Input] There is a typo: "The paramers update only involves" should read "The parameters update only involves."
[Section I] The word "implicitely" should be "implicitly."
[Definition 8] The Gaussian energy filter G_sigma(x) is called a "projection," but it is a filtered operator, not a projection in the usual sense; the terminology should be adjusted to avoid confusion.
[Definition 7] The list of conditions for a sub-bath spectral density has a duplicated numbering (two items numbered "2"), and the final item is numbered "2" as well; the numbering should be 1, 2, 3.
[Section V, after Eq. (22)] The sentence "With the gap, QPE cannot distinguish the ground state from the first excited state" appears to be the opposite of what is intended; a polynomial gap is needed so that QPE can distinguish the ground state.
[Eq. (22)] The notation Tr_B(e^{-beta H}) is unclear: the trace should be over the system Hilbert space, not a bath, and the normalization should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is a conditional implication from a hypothesized good sub-bath to polynomial-time super-bath cooling, with independent error analysis.

full rationale

The central claim is explicitly conditional: Theorem 1 assumes the existence of a sub-bath spectral density J with constraint factor b and positive minimum power PK,min(1/β,J), and concludes polynomial bounds for L, K, M, t, and σ. The proof of Lemma 1 derives the per-cycle energy decrease bound from the definition of PK,min via a chain of independently bounded approximations: Redfield-vs-exact evolution (Lemma 2, Corollary 2), higher-order Dyson terms, Gaussian stabilization to a δ-stationary state (Lemma 4), coarse-grained Hamiltonian approximation (Lemma 5), and a bound on the excitation part of the complementary bath (Lemma 6). Each error term is controlled by explicit parameter inequalities in Appendix D; the result is not obtained by renaming the assumption as the conclusion. The quantity PK,min is the minimum instantaneous Redfield dissipation rate over states with ground-state overlap at most 1/2; the theorem's conclusion concerns the probability of observing the ground state after QPE batches, which is a different statement bridged by the error analysis. No fitted numerical parameter is relabeled as a prediction: the NUBASE/NuDat half-life data in Sec. III is used only as motivational evidence that some nuclear baths may have favorable dissipation scaling, not as an input to the algorithm or as validation of a fitted value. The authors' self-citations ([15], [16], [22], [28], [47]) appear in background enumerations of projection, variational, and dissipative algorithms and are not load-bearing for Lemma 1 or Theorem 1. The two genuine limitations — the unproven existence of a good sub-bath for any concrete Hamiltonian and the heuristic parameter-update stopping rule — affect the applicability and end-to-end certification of the polynomial claim, but they are not circularity: the theorem honestly states its condition, and the derivation from that condition is self-contained. No circular step satisfying the quoted-evidence standard was found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The theorem is conditional on the existence of a good sub-bath with positive minimum power. The free parameters are mostly algorithmic tuning parameters with proven existence of valid ranges, plus the unspecified quantities r and b. The nuclear data does not fix these values. The paper's contribution is the conditional proof, not a verified application.

free parameters (4)

beta (inverse temperature supremum) = Set by Eq. (22): 1/beta such that the thermal weight of the ground state is 3/4
The temperature 1/beta is a parameter of the theorem that must be attained for the algorithm to be efficient. It depends on the spectrum of H and is not derived from first principles.
b (constraint factor) = O(Poly(n)) assumed
The constraint factor of the sub-bath Jg-sub enters all complexity bounds. The theorem only succeeds when b is polynomial, which is assumed without a constructive recipe.
PK,min(1/beta, Jg-sub) (minimum power) = Positive, but only existence is assumed
The entire algorithm is conditioned on this quantity being strictly positive and not decaying faster than polynomial in n. The paper does not compute it for any concrete system.
g, t, sigma, delta (algorithmic parameters) = Updated by halving/doubling rules in Appendix D 3
These parameters determine the cooling cycle. The paper proves there exist ranges satisfying the inequalities, but the initial values and update factors require computing timescales tauR and tauB of the chosen J.

assumptions (5)

domain assumption The system H has a spectral gap of size at least 1/beta = O(Poly(n)) between the ground state and the first excited state.
Invoked in Corollary 1 and in the context of QPE; the gap is needed for QPE to distinguish the ground state. Not proven for target systems.
standard math The interaction Hamiltonian is of the form H_I = sum A_alpha tensor B_alpha with dimensionless A_alpha satisfying ||A_alpha|| <= 1.
Standard setup of open quantum systems, stated in Appendix A. It is an assumption on the structure of the bath coupling.
domain assumption The bath is Gaussian and bosonic, initially in a thermal state.
The entire formalism relies on the bath correlation function description, Eqs. (A4-A7). The algorithm's simulation of the super-bath inherits this assumption.
standard math The exact open-system evolution satisfies the Born-Markov error bound given in Lemma 2 (from Nathan and Rudner).
This is a published rigorous bound, cited as Ref. [65], and adapted in Appendix A 2b. It is a key tool, not proved in full from scratch in this paper.
domain assumption For the chosen super-bath spectral density J, the time scales tauR and tauB are computable and the parameter choice t = tauR(T, JS)/g satisfies the constraints.
The algorithm assumes it can compute tauR and tauB for the chosen J, and that they satisfy the polynomial bounds of Definition 5. This is true for the super-Ohmic example in Appendix E, but for a realistic system it is a modeling task.

invented entities (2)

Super-bath spectral density JS(omega) = J(omega) 1_N
purpose: A synthetic environment that replaces the physical bath, coupling each system operator to an independent copy of a chosen bath spectrum.
The super-bath is a theoretical construction for the algorithm. Its physical realizability as a simulation subroutine is assumed through Corollary 1 condition 5, but there is no experimental demonstration. It is not an empirical entity.
Good sub-bath Jg-sub with constraint factor b
purpose: The assumed physical environment whose existence is sufficient for the algorithm's efficiency.
The existence of such a sub-bath is the central existence assumption. The paper suggests nuclear gamma decay as a candidate but does not identify a concrete Jg-sub for any realistic Hamiltonian. No independent falsifiable prediction is tied to this entity.

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Cite this review

Pith. "Pith review of Super-bath Quantum Eigensolver." pith.science (2026). https://pith.science/paper/WDTNARG7

@misc{pith2026241219599,
  author       = {Pith},
  title        = {Pith review of: Super-bath Quantum Eigensolver},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDTNARG7}},
  note         = {Machine review of arXiv:2412.19599}
}

abstract

The simulation of the dynamics of a system coupled to a low-temperature environment is a promising application of quantum computers to determine ground-state properties of physical systems. However, this approach requires not only the $\textit{existence}$ of an environment that allows the system to dissipate energy and evolve to its ground state, but also the $\textit{detailed knowledge}$ of the properties of the bath. In this paper, we propose a polynomial-time algorithm for ground state preparation which only relies on the $\textit{existence}$ of a physical bath which achieves the same task, while a detailed description of the environment may remain $\textit{unknown}$. In particular, we show that this ``super-bath quantum eigensolver algorithm'' prepares the ground state of the system by combining a Gaussian stabilization dephasing procedure with the simulation of the interaction between the system and a super-bath which only requires minimal knowledge of the physical environment. Based on our algorithmic framework, we establish a partial order relation among environments. Supported by experimental lifetime data of nuclear metastable states, we suggest that our algorithm is applicable to determine nuclear ground states in polynomial time. These results highlight the potential advantage of quantum computing in addressing ground state problems in real-world physical systems.

Figures

Figures reproduced from arXiv: 2412.19599 by the authors.

**Figure 2.** FIG. 2: Schematic diagram of the super-bath quantum eigensolver. The quantum computer receives fixed parameters ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: (a) The coupling between the system and the super [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reference graph

Works this paper leans on

121 extracted references · 53 canonical work pages

[1]

For all ω, Jsub(ω) is positive semi-definite
[2]

For all ω, ∥Jsub(ω)∥∞ ≤ bJ(ω); (17)
[3]

(19) In this definition, the first condition states that the sub-bath spectral density is a valid spectral density

For all T >0, τR(T, Jsub) ≥ b−1τR(T, J), (18) τB(T, Jsub) ≤ bτB(T, J). (19) In this definition, the first condition states that the sub-bath spectral density is a valid spectral density. The second condition corresponds to assume the existence of a decomposition of the super bath JS into a sub-bath Jsub and a complementary bath JC in terms of an efficienc...
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The spectrum range of the Hamiltonian is h = O(Poly(n))
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(22) is 1/β = O(Poly(n))

The temperature in Eq. (22) is 1/β = O(Poly(n))
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There exists a good sub-bath Jg-sub with a constraint factor b satisfying 1/PK,min(1/β, Jg-sub) and b = 10 O(Poly(n))
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The operator e−iHt can be implemented with an additive error ϵ1 at the time (qubit) cost Poly(n, t,1/ϵ1)
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quantum detailed balance

The superoperator M(H, T, g2JS; t) can be imple- mented with an additive error ϵ2 at the time (qubit) cost Poly(n, 1/T, 1/g, t,1/ϵ2). The time for attaining the ground state depends on the energy dissipation power and the initial energy of the system. Condition 1 introduces a bound on the initial en- ergy of the system. Under condition 2, we can reach a s...

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Here, δl,l′ is the Kronecker delta, and δ(ω − ω′) is the Dirac delta function

Gaussian boson bath In the case of a Gaussian boson bath, the bath Hamiltonian is in the form HB = X l Z ∞ 0 dωωb † l (ω)bl(ω), (A4) where [bl(ω), b† l′(ω′)] = δl,l′δ(ω − ω′). Here, δl,l′ is the Kronecker delta, and δ(ω − ω′) is the Dirac delta function. The bath operators are...
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Error in the Redfield equation The generator in the Redfield equation reads K(T, J; t)• ≡ X α,β Z ∞ 0 dsCαβ(T, J; s) Aβ(t − s) • Aα(t) − Aα(t)Aβ(t − s) • + H.c., (A9) where Aα(t) = eiHt Aαe−iHt are system operators in the interaction picture. This equation is approximate, and ...
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The set of energy differences is Ω ≡ {Ei − Ej}

Spectral decomposition, rotating-wave approximation and Lindblad equation Let H = P j EjΠj be the spectral decomposition of the system Hamiltonian, where Ej are eigenvalues of H, and Πj is the projection onto the subspace spanned by eigenvectors of Ej. The set of energy differ...
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Let Aα,δ(t, x) = eiHδ(x)tAαe−iHδ(x)t be time-dependent system operators according to the coarse-grained Hamilto- nian

Coarse-grained Hamiltonian The coarse-grained Hamiltonian with parameters ( δ, x) reads Hδ(x) = ∞X j=−∞ (x + 2jδ)Fδ(x + 2jδ) (C1) Then, ∥Hδ(x) − H∥∞ ≤ δ (C2) and ∥e−iHδ(x)t − e−iHt ∥∞ ≤ δ|t|, (C3) where we use |e−iδt − 1| = 2| sin δt 2 | ≤δ|t|. Let Aα,δ(t, x) = eiHδ(x)tAαe−iHδ...
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Spectral decomposition Similarly, let Ω δ ≡ {2jδ|j ∈ Z} and ω, ω′ ∈ Ωδ, follow the spectral decomposition method mentioned in Appendix A 3, we have KR,δ(t, x)• = X ω,ω ′∈Ωδ X α,β ei(ω′−ω)tΓα,β(ω) Aβ,δ (ω, x) • A† α,δ(ω′, x) − A† α,δ(ω′, x)Aβ,δ (ω, x) • + H.c., (C7) where Aα,δ(...
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Dissipation of δ-stationary states We denote the total energy transferred from the system to the bath at the time t according to the first-order contribution of the Redfield equation as DR(t, ρs) ≡ − Z t 0 ds Tr HK(s)ρs (C9) when ρs is the initial state. Substitute Eq. (B6) in...
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To prove this, we perform approximations step by step and calculate the error for each approximation

Proof of Lemma 1 Lemma 1 provides the lower bound on the energy decrease of the system within time t due to the super bath described by the spectral density JS(ω). To prove this, we perform approximations step by step and calculate the error for each approximation. 22 The firs...
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We then demonstrate that the inequality ϵ ≤ P t/2 holds when the parameters t, g, δ, σ, and T are within an appropriate range

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

For all ω, Jsub(ω) is positive semi-definite

[2] [2]

For all ω, ∥Jsub(ω)∥∞ ≤ bJ(ω); (17)

[3] [3]

(19) In this definition, the first condition states that the sub-bath spectral density is a valid spectral density

For all T >0, τR(T, Jsub) ≥ b−1τR(T, J), (18) τB(T, Jsub) ≤ bτB(T, J). (19) In this definition, the first condition states that the sub-bath spectral density is a valid spectral density. The second condition corresponds to assume the existence of a decomposition of the super bath JS into a sub-bath Jsub and a complementary bath JC in terms of an efficienc...

[4] [4]

The spectrum range of the Hamiltonian is h = O(Poly(n))

[5] [5]

(22) is 1/β = O(Poly(n))

The temperature in Eq. (22) is 1/β = O(Poly(n))

[6] [6]

There exists a good sub-bath Jg-sub with a constraint factor b satisfying 1/PK,min(1/β, Jg-sub) and b = 10 O(Poly(n))

[7] [7]

The operator e−iHt can be implemented with an additive error ϵ1 at the time (qubit) cost Poly(n, t,1/ϵ1)

[8] [8]

quantum detailed balance

The superoperator M(H, T, g2JS; t) can be imple- mented with an additive error ϵ2 at the time (qubit) cost Poly(n, 1/T, 1/g, t,1/ϵ2). The time for attaining the ground state depends on the energy dissipation power and the initial energy of the system. Condition 1 introduces a bound on the initial en- ergy of the system. Under condition 2, we can reach a s...