{"id":"63fe16cf-80c3-4108-959f-6628106e9079","arxiv_id":"2505.06343","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A quasiprobability-sampling algorithm estimates expectation values of imaginary-time evolved states without ancilla qubits, with an 8-qubit simulation and a 2-qubit hardware demonstration.","lead":"This paper proposes a method for computing imaginary-time evolution, a standard tool for finding ground and thermal states, by rewriting each small evolution step as a weighted average of ordinary quantum operations and post-processing the measurement results. The method needs no auxiliary qubits and is designed to work on today's small quantum processors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 2 as printed is biased: the numerator estimator omits the postselection indicators, so the central claim that it estimates the rescaled imaginary-time expectation is false as written.","rationale":"The reader's stated weakest assumption is the applicability of standard additive Trotter-error bounds to non-trace-preserving imaginary-time evolution; that is a legitimate gap, and Section 3.2 itself concedes it. However, the single most load-bearing defect I find is more basic: the pseudocode in Algorithm 1 and Algorithm 2 does not implement the estimator whose unbiasedness is proved in Lemma 1 and Lemma 2. The proof's F and G both contain the postselection-success factors L; the algorithm includes those factors in W (the denominator) but not in M (the numerator). Thus the printed algorithm cannot be correct even before Trotterization or noise are considered. This is a concrete, reproducible correctness failure rather than a conjecture about a bound. It is fixable by multiplying M_j by the product of the I_jk indicators and measuring A only after the full sequence, so it does not invalidate the underlying QPD-ITE idea; the paper's mathematical framework remains plausible. For that reason I would not change the reader's CONDITIONAL verdict, but the revision must fix both pseudocode estimators and ideally add a small numerical check of the kind described above. I also note a second formal issue consistent with this concern: in Appendix E the Trotterized channel is written as P(rho) = P rho P, whereas the operator P is not Hermitian for finite r and the implemented channel is P rho P^dag; the same O(||T-P||) bound goes through with the corrected expression, so I did not make this the headline, but it should be corrected in any revision.","tokens_in":21939,"tokens_out":28204,"duration_ms":310074,"concrete_test":"Use a one-qubit example H = |1><1|, beta = 1, rho = |+><+|, A = |1><1|, with the EBL QPD of T(rho) = e^{-H} rho e^{-H}. Exact rescaled expectation is e^{-2}/(1+e^{-2}). Analytically evaluate (or simulate, with sampling noise removed) Algorithm 2 exactly as printed: because rejected projector branches contribute to M_j but are zeroed in the denominator only, the ratio does not converge to the exact value. Then rerun the same test after replacing M_j by M_j * prod_k I_jk and moving the A measurement after the k-loop; the estimator mean should match the exact value. This isolates the postselection bug from Trotter and sampling errors.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The unbiased estimator proved in Lemma 2 (Appendix D, Eqs. 71-73) is F = gamma^R sgn(q_i1...q_iR) L_iR...L_i1 Y, where each L is the postselection-success indicator. Algorithm 2's line 13 instead sets M_j = gamma^R sgn(q_j1...q_jR) * A_j, dropping the product of indicators I_jk. Line 11 also measures A inside the k-loop (as printed), i.e. after each Trotter layer rather than after the full evolution. Consequently E[M_j] is not tr[AE(rho)]: for every sampled non-trace-preserving basis element, the rejected branch (probability 1 - tr[B_i(rho)]) is still followed by an A measurement whose outcome enters M_j, so the estimator includes contributions of the form tr[A * rho_rejected]. Only the denominator in line 14 uses the product of W_jk. The proof and the implemented algorithm therefore describe different estimators, and the printed Algorithm 2 is biased even in the noiseless, no-Trotter limit. The same omission appears in Algorithm 1, line 12, where M_j should be multiplied by I_j. This is a direct correctness failure of the central algorithm; it is independent of the Trotter-error gaps in Eqs. (30)-(34) and of the H >= 0 discussion, and it affects both the 2-qubit hardware demonstration and the TPQ simulations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a quasiprobabilistic imaginary-time evolution algorithm for quantum computers. The central idea is to decompose a Trotterized imaginary-time evolution into a probabilistic linear combination of native operations, including non-trace-preserving ones, and to estimate the rescaled expectation value tr[AT(ρ)]/tr[T(ρ)] by separately estimating numerator and denominator with weighted samples. The authors claim an O(γ^{2Lr}/ε^2) circuit complexity, no ancilla qubits, and natural noise resilience when device noise is included in the decomposition. They illustrate the method with classical simulations of thermal pure quantum state preparation for 1D Heisenberg Hamiltonians on up to 8 qubits and with a 2-qubit IBM quantum computer demonstration. The proof appendix gives the intended unbiased estimators.","tokens_in":22262,"tokens_out":6771,"duration_ms":72178,"significance":"If the central claim holds, the work is a useful near-term tool: it extends quasiprobability decomposition techniques from error mitigation to algorithmic simulation of non-unitary evolution, avoids ancilla qubits, and connects the construction to thermal pure quantum states. The paper's strengths are its explicit local QPD formulation, the analytic estimator proofs in Appendix D, the concrete sampling-cost analysis, and the fact that no parameter is fitted to target expectation values in the demonstrations. However, the printed algorithms do not match the proved estimators, and the Trotter-error analysis rests on bounds that the authors themselves flag as potentially invalid for non-unitary evolution. These issues are central to the paper's correctness claims and must be resolved before the results can be relied upon.","major_comments":[{"comment":"The printed estimators omit the postselection indicators. In Algorithm 2, M_j is set to γ^R sgn(q_{j1})...sgn(q_{jR}) A_j, while the proof in Appendix D defines F = γ^R sgn(q_{i1}...q_{iR}) L_{iR}...L_{i1} Y. Because the A measurement is executed even when an intermediate non-TP outcome is rejected, the empirical average of M_j includes contributions from the rejected branches, so E[M_j] is not tr[AE(ρ)]. The same omission appears in Algorithm 1, where M_j should be γ sgn(q_i) I_j A_j. This is a direct correctness failure of the central algorithm as printed, independent of the Trotter-error gaps, and it affects the hardware demonstration and the TPQ simulations if they were run according to the pseudocode. The fix is local: multiply by the indicators (or postselect before measuring A and set A_j=0 on rejection), and Appendix D shows that the intended estimator is then unbiased.","section":"Algorithm 2, line 13 and Algorithm 1, line 12"},{"comment":"The instruction 'Measure A' appears inside the k-loop, so A is measured after each Trotter layer and the variable A_j is overwritten; only the last measurement is retained. The estimator in Lemma 2 is defined for an observable measured after all R applications of T. Unless A commutes with every intermediate map, this procedure yields a different expectation value. The measurement of A should be moved to after the loop and A_j should be defined from that single measurement.","section":"Algorithm 2, line 11"},{"comment":"The Trotter-error claim is made by applying Theorem 6 of Ref. [65], whose hypotheses concern unitary evolution generated by Hermitian terms, to the non-unitary, non-trace-preserving maps e^{-βH_l/r}. Section 3.2 concedes that these bounds 'may not be well-suited for non-trace-preserving imaginary-time evolution.' The choice r = O(β^2L^2/ε) and the total circuit count O(γ^{2Lr}/ε^2) rest on this unverified premise. The authors should either prove a non-unitary version of the Trotter bound or benchmark the operator-norm error numerically for small systems; without this, the stated complexity of the algorithm is not established.","section":"Appendix E, Eqs. (80)-(86); Section 3.2, Eqs. (32)-(34)"},{"comment":"The statement 'We may assume without loss of generality that H≥0' requires knowing the ground-state energy λ0 in order to shift H by αI. Appendix C acknowledges that determining α is challenging in general and that the method's useful behavior is tied to frustration-free Hamiltonians. Because the signal strength tr[P(ρ)] and hence the denominator of the ratio estimator depend on the shift, this should be presented in the main text as an algorithmic condition or a limitation, not as a free assumption.","section":"Section 3, first paragraph; Appendix C, Proposition 1"}],"minor_comments":[{"comment":"There is a typo: 'thermal expecation value' should be 'thermal expectation value'.","section":"Figure 1 caption"},{"comment":"The text says the noiseless simulation implements Algorithm 1, while the Figure 5 caption says the simulation implements Algorithm 2; please make the references consistent.","section":"Section 3.1 and Figure 5 caption"},{"comment":"The expression O(γ^{2Lr}/ε^2) assumes that all local decomposition costs γ_l are equal; the paper should state this assumption explicitly and give the general form O(∏_l γ_l^{2r}/ε^2).","section":"Section 3.2, Eq. (34)"},{"comment":"Reference [58] is a duplicate of reference [25]; consolidate them.","section":"References"},{"comment":"For clarity, the pseudocode should specify that A is measured only if I_j=1, or that A_j is discarded when I_j=0; otherwise the reader might reasonably follow the printed instruction and measure A on every circuit.","section":"Algorithm 1, comment on lines 7-12"}],"recommendation":"major_revision","confidential_remarks":"The mismatch between the pseudocode and the Appendix D estimators is serious but local; the intended estimator is recoverable from the proofs. The Trotter-error issue is deeper and may require additional analysis before the complexity claim can stand. I recommend major revision rather than rejection because the core construction is sound and the fixes are identifiable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the idea is sound and the estimator proofs are right, but the printed pseudocode is wrong. Both Algorithm 1 and Algorithm 2 omit the postselection indicator from the numerator estimator, and Algorithm 2 appears to measure the observable inside the Trotter loop. As written, the algorithms are biased; they do not estimate what the theorems claim. That is the headline.\n\nWhat is genuinely useful: applying the existing QPD/PEC machinery to imaginary-time evolution in a Trotterized, expectation-value form is a natural step that had not been made explicitly. The TPQ application is a good fit, the diamond-norm cost discussion is sensible, and the 8-qubit TPQ simulation plus 2-qubit IBM demonstration give plausible evidence that the protocol works when implemented as intended. Appendix D is careful: the random variables F and G include the indicators, and the Hoeffding analysis is standard and correct. The EBL-vs-Takagi basis comparison is a nice practical touch.\n\nSoft spots, in proportion. The algorithm bug is direct. The fix is mechanical—multiply the numerator by the product of the I_jk’s and measure A after the full evolution—so this is a revision issue rather than a fatal one, but it must be fixed before anyone implements from the paper. Second, the Trotter error bound: the authors explicitly say the standard additive bounds “may not be well-suited” for non-trace-preserving evolution, yet Eqs. (30)–(32) and the overall cost rest on them. That is a real gap, though the authors themselves flag it. Third, the H≥0 claim is less problematic than it first looks—shifting H by a multiple of the identity cancels in the rescaled expectation value—but it does affect the sampling cost, and the paper’s own Appendix C explains the tradeoff. The noise-resilience demonstration is suggestive but limited; the third Trotter step is visibly off, and the authors attribute it to noise drift, which is reasonable but under-supported.\n\nThe citation pattern is fine: they build on Temme, Endo, Takagi, Jiang, and others, with no self-citation problem.\n\nVerdict: this deserves a serious referee. The conceptual contribution is real, the proofs are mostly sound, and the bugs are addressable. I would send it to peer review and ask for a revised version with corrected pseudocode and a more honest treatment of the Trotter error and cost claims. The paper is mainly for people working on near-term quantum simulation and error mitigation; they will want to read it, but they should not code from the current pseudocode.","headline":"Solid QPD-for-ITE idea with correct proofs, but the printed algorithms are biased because they drop the postselection indicators—fixable, but the Trotter cost claims also need tightening.","tokens_in":22736,"tokens_out":3667,"would_cite":false,"duration_ms":37949,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that Trotterized imaginary-time evolution can be simulated \"in expectation\" on a quantum computer by sampling from a quasiprobability decomposition of each local imaginary-time exponential, then classically re-weighting…","keywords":["quasiprobability decomposition","imaginary-time evolution","probabilistic error cancellation","thermal pure quantum state","quantum error mitigation","Trotterization","Heisenberg model","near-term quantum computing"],"falsifier":"For a fixed $k$-local Hamiltonian, say the 2-qubit Heisenberg term, numerically compute the exact operator difference $\\| e^{-\\beta H} - (e^{-\\beta H_1/r} e^{-\\beta H_2/r})^r \\|$ for increasing $r$ at fixed $\\beta$, and compare the decay with the bound $O(\\beta^2 L^2 e^{\\beta L/r}/r)$ from Eq. (81). If the non-trace-preserving composition produces errors that do not shrink as predicted, or if the empirical deviation $|\\operatorname{tr}[A T(\\rho)] - \\operatorname{tr}[A P(\\rho)]|$ saturates or grows with $r$, then the Trotter-step choice and the total circuit count $O(\\gamma^{2Lr}/\\epsilon^2)$ do not follow, and the method's claimed cost would need revision.","tokens_in":21777,"feed_emoji":"🎲","tokens_out":6596,"duration_ms":60949,"temperature":0.7,"pith_summary":"The paper proposes doing imaginary-time evolution on a quantum computer without fault tolerance, by writing each local imaginary-time step $e^{-\\beta H_l / r}$ as a signed, weighted sum of ordinary quantum operations and measurements, sampling those operations, and re-weighting the outcomes classically. The method targets the rescaled expectation value $\\langle A \\rangle_P = \\operatorname{tr}[A P(\\rho)] / \\operatorname{tr}[P(\\rho)]$ for a Trotterized evolution, requires no ancillary qubits, and can absorb a device noise model into the decomposition so that noise does not bias the estimate. The sampling cost is $O(\\gamma^{2Lr}/\\epsilon^2)$ circuits for a Hamiltonian with $L$ local terms and $r$ Trotter steps, where $\\gamma$ is the one-norm of the quasiprobability coefficients. For thermal physics, the authors argue the method is well matched to Clifford-random thermal pure quantum states, whose expectation values converge rapidly to Gibbs-state values as the system grows, and they demonstrate this with an 8-qubit simulation of a 1D Heisenberg chain and a 2-qubit energy estimate on IBM hardware.","feed_headline":"Imaginary-time evolution becomes sampling on quantum chips","feed_subtitle":"A quasiprobability re-weighting estimates ground- and thermal-state energies on noisy hardware with zero ancilla qubits.","key_machinery":"The load-bearing object is the quasiprobability decomposition (QPD) of a $k$-local imaginary-time channel, $T_l(\\rho) = e^{-\\beta H_l/r} \\rho e^{-\\beta H_l/r} = \\sum_i q_i B_i(\\rho)$, with $\\gamma = \\sum_i |q_i|$ and $p_i = |q_i|/\\gamma$, so that $T_l(\\rho) = \\gamma \\sum_i \\operatorname{sgn}(q_i)\\, p_i\\, B_i(\\rho)$. Randomly sampling $i$ from $p_i$, applying $B_i$, and weighting the measurement by $\\gamma \\operatorname{sgn}(q_i)$ turns each circuit into an estimator of the numerator and denominator of the renormalized expectation value. Algorithm 2 repeats this for each of the $Lr$ local factors in a first-order Trotter decomposition, accumulating the product of weights $\\gamma^R \\operatorname{sgn}(q_{i_1})\\cdots \\operatorname{sgn}(q_{i_R})$ with $R = Lr$, so the sampling cost compounds as $\\gamma^{2R}$ while the gate depth stays $O(Lr)$.","core_discovery":"The central claim is that any Hermiticity-preserving, possibly non-trace-preserving linear map that admits a quasiprobability decomposition into implementable basis operations can be simulated in expectation on a quantum computer by sampling those operations and forming weighted measurement estimators. Applied to imaginary-time evolution, Algorithm 2 composes $r$ Trotter steps of $L$ local exponentials $e^{-\\beta H_l/r}$, each decomposed with quasiprobabilities $q_i$ and normalization $\\gamma = \\sum_i |q_i|$, and returns an estimate of the rescaled expectation value $\\langle A \\rangle_P = \\operatorname{tr}[A P(\\rho)] / \\operatorname{tr}[P(\\rho)]$ using $O(\\gamma^{2Lr}/\\epsilon^2)$ circuits and no ancillas, with error bounded by $2\\epsilon / |\\operatorname{tr}[P(\\rho)]|$. Because the same decomposition can fold a device noise model into the basis operations, the estimate is noise-resilient without a separate error-mitigation layer. The authors further claim this construction is naturally suited to thermal pure quantum states, where a Clifford-random state evolved under imaginary time gives Gibbs-state expectation values whose error decays rapidly with qubit number.","pith_inferences":["A corollary the paper does not develop: the same estimators that yield $\\operatorname{tr}[A P(\\rho)]$ and $\\operatorname{tr}[P(\\rho)]$ give direct access to the partition-function-like normalization, so the method could estimate free-energy-type quantities, not just normalized expectation values.","The error bound's denominator $\\operatorname{tr}[P(\\rho)]$ decays as $e^{-2\\beta \\lambda_0}$ for a Hamiltonian with smallest eigenvalue $\\lambda_0$, so even when the quasiprobability cost $\\gamma$ stays flat, the useful precision degrades exponentially in $\\beta$ unless the initial state has good ground-state overlap; this delimits the practical temperature range.","The paper notes its cost analysis relies on standard additive Trotter bounds for non-trace-preserving operators; a concrete testable extension is to verify numerically for a frustrated $k$-local Hamiltonian whether the empirical deviation $|\\operatorname{tr}[A T(\\rho)] - \\operatorname{tr}[A P(\\rho)]|$ obeys the claimed $O(\\beta^2 L^2 e^{\\beta L/r}/r)$ scaling, since the paper only argues signal pr","If mid-circuit measurement with feed-forward becomes available, rejected branches of the post-selected non-trace-preserving basis operations could be terminated early, effectively reducing the per-sample cost; the paper identifies this as a hardware-level optimization but does not quantify the resulting speedup."],"forward_implications":["No ancilla qubits are needed: the method uses only the system qubits, avoiding the multi-qubit-controlled operations and large SWAP networks of ancilla-based Gibbs samplers.","Noise resilience is intrinsic: if the device noise channel is incorporated into the basis operations from calibration data, the estimate tracks the noiseless result, as the 2-qubit hardware demonstration through four Trotter steps indicates.","Thermal expectation values become accessible on near-term devices: Clifford-random thermal pure quantum states combined with Algorithm 2 estimate Gibbs-state expectation values for $k$-local Hamiltonians, with the error in the TPQ expectation value decaying rapidly with the number of qubits.","The machinery applies beyond imaginary-time evolution to any Hermiticity-preserving map that admits a QPD over implementable operations, giving a general tool for estimating rescaled expectation values of non-trace-preserving evolutions.","Trotter step count and sampling cost trade off against each other: $r = O(\\beta^2 L^2 / \\epsilon)$ controls the Trotter error while $\\gamma$ depends on the local exponentials $e^{-\\beta H_l/r}$, so increasing $r$ lowers each local $\\gamma$ but multiplies the number of factors in the cost $O(\\gamma^{2Lr}/\\epsilon^2)$."],"supporting_citations":[{"why":"Supplies the probabilistic error cancellation template: linear-program QPD construction and sampling with sign re-weighting, which the algorithm generalizes to non-trace-preserving maps.","marker":"[39]"},{"why":"Supplies the EBL basis of Clifford unitaries and projectors used as the QPD basis for the demonstrations.","marker":"[41]"},{"why":"Supplies the Takagi basis for 2-qubit CPTP maps, which the paper shows lowers $\\gamma$ for the ITE operator compared to the EBL basis.","marker":"[62]"},{"why":"Supplies the diamond-norm lower bound on $\\gamma$ that the paper uses to gauge the near-optimality of the decompositions.","marker":"[60]"},{"why":"Supplies the Trotter error bounds used to choose the step count $r$ and to bound the difference between ideal and Trotterized imaginary-time evolution.","marker":"[65]"},{"why":"Supplies the Clifford-random thermal pure quantum state construction (Eq. 39) that Algorithm 2 is applied to for estimating Gibbs-state expectation values.","marker":"[72]"},{"why":"Supplies the canonical thermal pure quantum state concept and its concentration property that justifies estimating thermal expectation values from pure states.","marker":"[67]"},{"why":"Supplies the complexity baseline showing thermal equilibrium estimation is hard in general, motivating the heuristic near-term approach.","marker":"[14]"}],"fun_headline_variants":["Sampling imaginary time: error-mitigated ground states","Quasiprobability trick simulates imaginary-time evolution","No ancillas: probabilistic imaginary-time on noisy chips","Imaginary-time evolution via quasiprobabilistic sampling","Thermal states from imaginary-time sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Trotter error bounds used to choose the step count $r = O(\\beta^2 L^2/\\epsilon)$ are standard additive bounds derived for unitary evolution, and the paper concedes these may not be well-suited to the non-trace-preserving imaginary-time operators, so the claimed step count and total cost $O(\\gamma^{2Lr}/\\epsilon^2)$ rest on an unverified premise.","fun_headline_variants_meta":{"raw":{"variants":["Sampling imaginary time: error-mitigated ground states","Quasiprobability trick simulates imaginary-time evolution","No ancillas: probabilistic imaginary-time on noisy chips","Imaginary-time evolution via quasiprobabilistic sampling","Thermal states from imaginary-time sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1362,"prompt_tokens":1032,"completion_tokens":330,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":648,"completion_tokens_details":{"reasoning_tokens":253}},"tokens_in":648,"tokens_out":330,"duration_ms":3368,"temperature":1.0,"reasoning_tokens":253,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:46:38.452860+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed $k$-local Hamiltonian, say the 2-qubit Heisenberg term, numerically compute the exact operator difference $\\| e^{-\\beta H} - (e^{-\\beta H_1/r} e^{-\\beta H_2/r})^r \\|$ for increasing $r$ at fixed $\\beta$, and compare the decay with the bound $O(\\beta^2 L^2 e^{\\beta L/r}/r)$ from Eq. (81). If the non-trace-preserving composition produces errors that do not shrink as predicted, or if the empirical deviation $|\\operatorname{tr}[A T(\\rho)] - \\operatorname{tr}[A P(\\rho)]|$ saturates or grows with $r$, then the Trotter-step choice and the total circuit count $O(\\gamma^{2Lr}/\\epsilon^2)$ do not follow, and the method's claimed cost would need revision.","supporting_citations":[{"cited_title":"Error Mitigation for Short-Depth Quan- tum Circuits","cited_arxiv_id":null,"evidence_quote":"Supplies the probabilistic error cancellation template: linear-program QPD construction and sampling with sign re-weighting, which the algorithm generalizes to non-trace-preserving maps."},{"cited_title":"Practical Quantum Error Mitigation for Near- Future Applications","cited_arxiv_id":null,"evidence_quote":"Supplies the EBL basis of Clifford unitaries and projectors used as the QPD basis for the demonstrations."},{"cited_title":"Optimal resource cost for error mitigation","cited_arxiv_id":null,"evidence_quote":"Supplies the Takagi basis for 2-qubit CPTP maps, which the paper shows lowers $\\gamma$ for the ITE operator compared to the EBL basis."},{"cited_title":"Theory of trotter error with commutator scaling","cited_arxiv_id":null,"evidence_quote":"Supplies the Trotter error bounds used to choose the step count $r$ and to bound the difference between ideal and Trotterized imaginary-time evolution."},{"cited_title":"Predicting Gibbs-state expectation values with pure thermal shadows","cited_arxiv_id":null,"evidence_quote":"Supplies the Clifford-random thermal pure quantum state construction (Eq. 39) that Algorithm 2 is applied to for estimating Gibbs-state expectation values."},{"cited_title":"Quantum Hamiltonian complexity in thermal equilibrium","cited_arxiv_id":null,"evidence_quote":"Supplies the complexity baseline showing thermal equilibrium estimation is hard in general, motivating the heuristic near-term approach."}],"review_version":1}