{"id":"3bd182d4-afeb-4dfa-a2c3-b781c55d9647","arxiv_id":"2505.05411","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Energy-filter based reconstruction of equilibrium response functions from easy-to-prepare states is introduced and demonstrated for the dynamical conductivity of a free-fermion model.","lead":"This paper presents a quantum algorithm for computing dynamical response functions, such as electrical conductivity, of equilibrium states without preparing the equilibrium state itself. The method combines energy filters with classical postprocessing of time-evolution measurements and is demonstrated numerically on a free-fermion model showing energy-dependent localization.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (6) as written is not an identity: Aψ(δ/√2,E) corresponds to a single filter of width δ/2, not δ, so the published MCMC estimator is biased unless the text is corrected to use \\tilde Aψ(δ,E) or Aψ(√2δ,E).","rationale":"The reader identified the stationarity of filtered pure states as the weakest assumption, which is a real but secondary concern because the filter-ensemble route avoids it. My stress-test instead found a concrete algebraic error in the key sampling identity: Eq. (6) mixes filter widths in a way that makes the published estimator biased for generic complete state sets. The correct alternative \\tilde Aψ(δ,E) is present in the text, so the central idea is likely salvageable, but the paper as written is internally inconsistent and Appendix B instructs the wrong quantity. This warrants a conditional verdict: the argument is acceptable only after Eq. (6) and Appendix B are corrected and the numerical figures are confirmed to have used the correct estimator. I do not recommend rejection because the flaw is a specific, fixable typo-level error rather than a failure of the overall approach, and no issues with the circuit constructions or the free-fermion demonstrations were found.","tokens_in":13051,"tokens_out":23334,"duration_ms":244005,"concrete_test":"Run the Appendix B MCMC estimator on a small nonintegrable system (e.g., a 4–6 site XXZ chain) using two versions of the filtered observable in Eq. (6): (i) Aψ(δ/√2,E) as printed, and (ii) \\tilde Aψ(δ,E) or equivalently Aψ(√2δ,E). Compare both to the exact filter-ensemble value Tr[APδ(E)]/Tr[Pδ(E)] computed by exact diagonalization. If version (i) deviates while version (ii) matches, Eq. (6) and Appendix B require correction. A minimal analytical check is the two-level example H=diag(0,ε), E=0, A=|0><0|, basis {|+>,|->}, where Eq. (6) predicts 1/(1+e^{-2ε^2/δ^2}) instead of the exact 1/(1+e^{-ε^2/(2δ^2)}).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The filter-ensemble sampling identity Eq. (6) is mathematically incorrect as written. From Eqs. (3)–(5), P_a(E) ∝ exp[-(H-E)^2/(2a^2)], so P_a^2 ∝ P_{a/√2}. Therefore Aψ(δ/√2,E) = <ψ|P_{δ/√2} A P_{δ/√2}|ψ> / <ψ|P_{δ/√2}^2|ψ> equals \\tilde Aψ(δ/2,E) = <ψ|A P_{δ/2}|ψ> / <ψ|P_{δ/2}|ψ>, not \\tilde Aψ(δ,E). With pψ = <ψ|Pδ(E)|ψ>/Z, the right-hand side of Eq. (6) becomes Σψ pψ <ψ|A P_{δ/2}|ψ>/<ψ|P_{δ/2}|ψ>, which is not generally equal to Tr[APδ(E)]/Tr[Pδ(E)] for a non-eigenbasis. A two-level counterexample with H=diag(0,ε), E=0, A=|0><0|, and complete set {|+>,|->} already fails. The correct identity uses \\tilde Aψ(δ,E) (which the paper mentions as alternative), or equivalently Aψ(√2δ,E), not Aψ(δ/√2,E). Appendix B explicitly instructs averaging Aψ_i(δ/√2,E), so as written the sampling protocol is biased. If the numerics in Fig. 5 were produced with the correct single-filter expression, the text still needs correction; if they used Eq. (6), the numerical validation itself needs rechecking. This is load-bearing because the central claim that easy-to-prepare states can be classically postprocessed to yield equilibrium response functions relies on this averaging identity for the filter-ensemble route, and the stationarity issue flagged by the reader is side-stepped precisely by that route.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a hybrid quantum-classical algorithm to compute dynamical response functions (e.g., the conductivity) of equilibrium states without preparing the equilibrium state itself. The idea is to apply a Gaussian energy filter to an ensemble of easy-to-prepare states, reconstruct filtered expectation values from time-domain measurements via classically post-processed Fourier transforms, and then average over states sampled from the filter ensemble. Three circuit implementations for the required three-time correlator are given, together with a discretization scheme for the filter. The method is illustrated numerically on a free-fermion Aubry-André model with mobility edges, computing the Drude weight and current fluctuations at different energies.","tokens_in":13469,"tokens_out":16981,"duration_ms":155377,"significance":"The paper addresses an important practical bottleneck in quantum simulation: the preparation of equilibrium (especially low-temperature) states. Its strengths are the explicit circuit constructions, the cost statement in terms of the state overlap rδ(E), and the numerical benchmarking of the filter-ensemble sampler against exact Gaussian-state calculations. The localization-dependent Drude weight computed in Fig. 5 provides a convincing proof-of-principle. At the same time, the central sampling identity contains a mathematical error that must be fixed before the protocol can be used as published. Because the error is local and has straightforward corrections, the approach remains promising.","major_comments":[{"comment":"Equation (6) in the main text and the corresponding estimator in Appendix B are incorrect as written. For the normalized Gaussian filter P_a(E) ∝ exp[-(H-E)^2/(2a^2)], one has P_a(E)^2 ∝ P_{a/√2}(E). Therefore the denominator in Aψ(δ/√2,E) is governed by P_{δ/2}(E), and the right-hand side of Eq. (6) evaluates to Tr[AP_{δ/2}(E)]/Tr[Pδ(E)] rather than Tr[APδ(E)]/Tr[Pδ(E)]. A concrete counterexample is H = diag(0, ε), E = 0, A = |0⟩⟨0|, with the orthonormal set {|+⟩,|-⟩}: the left-hand side is 1/(1+e^{-ε^2/(2δ^2)}) while the right-hand side is 1/(1+e^{-2ε^2/δ^2}). The correct identities are Σψ pψ \\tilde Aψ(δ,E) = Tr[APδ(E)]/Tr[Pδ(E)] with \\tilde Aψ(δ,E) = ⟨ψ|APδ(E)|ψ⟩/⟨ψ|Pδ(E)|ψ⟩, or equivalently Σψ pψ Aψ(√2δ,E) = Tr[APδ(E)]/Tr[Pδ(E)]. Since Appendix B instructs the estimator to average Aψ_i(δ/√2,E), the protocol as written is biased. The agreement in Fig. 5 suggests the numerical implementation may have used a correct expression, but the text and Appendix B need to be amended; if the code used δ/√2, the validation must be repeated. This is load-bearing because the filter-ensemble sampling route is the basis for the claimed equilibrium response computations.","section":"Energy filters, Eq. (6), and Appendix B"}],"minor_comments":[{"comment":"The Metropolis acceptance rule is written as 'accept with probability max{p_{ψn+1}/p_{ψn}, u}'; this is not a valid probability (it can exceed 1). The standard rule is to accept with probability min{1, p_{ψn+1}/p_{ψn}}, i.e., accept if u is less than that value.","section":"Appendix B"},{"comment":"The identity in Eq. (6) relies on Σψ ⟨ψ|O|ψ⟩ = Tr[O], which is only valid for an orthonormal basis (or a tight frame with appropriate weights). The text should specify the nature of the complete set {|ψ⟩} used.","section":"Energy filters, Eq. (6)"},{"comment":"The stationarity assumption for filtered pure states is introduced qualitatively; since the filter-ensemble route is exact, a brief remark emphasizing that the pure-state route is an approximation used for illustration (and controlled by ETH in generic systems) would help avoid overgeneralization.","section":"Quantum simulation of response functions, text after Eq. (6)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript extends Ref. [14] from static observables to dynamical response functions; the new material is the response-function reconstruction and the numerical demonstration. The incorrect factor in Eq. (6) should be caught in revision; the authors should also confirm whether Fig. 5 was generated with the corrected expression. If the code used the published formula, the validation is invalid. In my view, the paper is suitable for publication in a major journal after these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short answer: the reader's take is mostly right about what the paper does well, but they missed an actual error in Eq. (6) that is load-bearing.\n\nWhat's new: the paper extends the energy-filter approach of Lu, Bañuls, and Cirac (PRX Quantum 2021) from static observables to dynamical response functions. Prior response-function algorithms (refs [6-11]) assume the equilibrium state is provided; here they show you can measure time-correlation functions on easy-to-prepare states and classically post-process with a spectral filter. The circuits in Fig. 2 for measuring three-time correlators are a reasonable contribution, and the numerical study on a quasiperiodic free-fermion model is a sensible benchmark: they compare MCMC samples against the exact filter ensemble and find agreement.\n\nThe soft spot: Eq. (6) is not an identity as written. With Pδ(E) a Gaussian filter of width δ, P_{δ/√2} squared is proportional to P_{δ/2}, not Pδ. So Aψ(δ/√2,E) has a denominator ⟨ψ|P_{δ/2}|ψ⟩, and weighting that by pψ ∝ ⟨ψ|Pδ|ψ⟩ does not give Tr[APδ]/Tr[Pδ] unless ⟨ψ|Pδ|ψ⟩ ∝ ⟨ψ|P_{δ/2}|ψ⟩, which is false in general (a two-level example already breaks it). The correct options are either to average \\tilde Aψ(δ,E) = ⟨ψ|APδ|ψ⟩/⟨ψ|Pδ|ψ⟩ with the same pψ, or to use Aψ(√2δ,E). The paper mentions \\tilde Aψ(δ,E) as an alternative, so the fix is close, but Appendix B explicitly instructs the reader to average Aψ_i(δ/√2,E) with pψ from Pδ. That procedure is biased. If Fig. 5 was computed with the \\tilde Aψ route, the text needs to be corrected; if it used Eq. (6), the numerical agreement needs to be re-explained.\n\nThis matters because the filter-ensemble route is exactly the one that bypasses the stationarity assumption on filtered pure states (which the reader flagged). That route is currently broken as written. The stationarity assumption for filtered pure states is also an approximation, but the paper is honest about it and the filter ensemble was supposed to be the rigorous alternative.\n\nBottom line: the idea is good and likely salvageable, but the paper is not sound as written. It should go to peer review with a major-revision request focused on fixing Eq. (6) and either correcting the MCMC protocol or rechecking the numerics. I'd bring it to reading group because the error is instructive, but I wouldn't cite it until the correction lands.","headline":"The reader's accept verdict holds up in spirit, but Eq. (6) is mathematically wrong as written and the MCMC sampling protocol in Appendix B is biased until fixed.","tokens_in":13968,"tokens_out":9428,"would_cite":false,"duration_ms":78737,"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":"This paper claims that dynamical response functions of equilibrium states can be reconstructed from measurements on easy-to-prepare states alone, via classical energy-filter post-processing, and demonstrates it on a quasiperiodic model.","keywords":["dynamical response functions","energy filtering","equilibrium states","quantum simulation","Kubo formula","microcanonical ensemble","dynamical conductivity","localization"],"falsifier":"Take a small non-integrable spin chain, prepare a filtered pure state of width $\\delta$ around a target energy, run the protocol to extract $\\chi_{AB}(\\omega)$, and compare it with the exact microcanonical response function computed by exact diagonalization; if the two disagree or if the extracted response depends on how the three-time correlator is reduced to a time difference, the stationarity assumption has failed. A sharper variant: check whether $\\langle\\psi|P_\\delta(E)[A(t),B(t')]P_\\delta(E)|\\psi\\rangle$ actually depends only on $t-t'$ by varying $t$ and $t'$ while keeping the difference fixed.","tokens_in":12875,"feed_emoji":"⚛️","tokens_out":5249,"duration_ms":52372,"temperature":0.7,"pith_summary":"This paper claims that dynamical response functions of equilibrium states — the frequency-dependent conductivity, response to perturbations, and similar quantities — can be reconstructed from measurements on easy-to-prepare non-equilibrium states, by post-processing time-series data with an energy filter. The method never prepares the equilibrium state itself; it only needs states with substantial weight in a target energy window, and it returns microcanonical (and, via Boltzmann weighting, canonical) response functions. This matters because equilibrium states are difficult to prepare on quantum hardware, and because classical methods are limited by entanglement growth and the sign problem when computing real-time dynamics. The claim is demonstrated numerically on a quasiperiodic free-fermion model, where the reconstructed conductivity reveals energy-dependent localization, including the mobility edge.","feed_headline":"Response functions without preparing the equilibrium state","feed_subtitle":"A quench-and-postprocess scheme maps easy-to-prepare states onto microcanonical and thermal conductivity data.","key_machinery":"The Gaussian energy filter $P_\\delta(E)=\\frac{1}{\\sqrt{2\\pi\\delta^2}}e^{-(H_0-E)^2/(2\\delta^2)}$, realized through its Fourier representation as a linear combination of time-evolution unitaries $e^{-iH_0 t}$. Its role is to project expectation values onto a narrow energy window of width $\\delta$ around $E$, converting measurements taken on a generic initial state into equilibrium response data; the projection is done entirely in classical post-processing from the measured $f_\\psi(t)$ and $C^{AB}_\\psi(t_1,t_2,t_3)$. The filter can be applied either to individual pure states or to an ensemble, and the ensemble version converges to the microcanonical ensemble as $\\delta\\to 0$.","core_discovery":"The central claim is that the linear response function $\\chi_{AB}(\\omega)$ of an equilibrium state can be inferred by applying a Gaussian energy filter $P_\\delta(E)$ to a readily preparable state $|\\psi\\rangle$ and measuring two time-domain quantities: the Loschmidt echo $f_\\psi(t)=\\langle\\psi|e^{-iH_0 t}|\\psi\\rangle$ and the three-time correlator $C^{AB}_\\psi(t_1,t_2,t_3)=\\langle\\psi|e^{iH_0 t_1}A e^{iH_0 t_2}B e^{iH_0 t_3}|\\psi\\rangle$. Classical post-processing evaluates the filtered expectation values and reconstructs $\\chi_{AB}(\\omega)$; the filter can be narrowed to select a target energy $E$, so the equilibrium response emerges from the dynamics of a state that was never thermalized. The same machinery yields filter-ensemble (microcanonical) expectation values by averaging over states drawn with probability proportional to $\\langle\\psi|P_\\delta(E)|\\psi\\rangle$. For the quasiperiodic Anderson-type model considered, this reproduces the energy-dependent Drude weight and current fluctuations, showing the transition from localized to extended behavior.","pith_inferences":["A practical diagnostic that follows from, but is not explicitly stated in, the paper: the reconstructed $\\chi_{AB}(\\omega)$ could be checked for consistency by extracting it from two different choices of time origin in the three-time correlator; if the results differ, the stationarity assumption on the filtered state has failed.","Editorial extension: the same measured time series could be used to validate eigenstate thermalization in interacting systems by comparing the filtered-pure-state response against the filter-ensemble response at the same energy and filter width.","Editorial extension: for integrable or localized models, individual filtered states can depart strongly from the filter ensemble, so the method would require either ensemble averaging or a carefully chosen family of initial states — a limitation the paper's own numerics illustrate.","Editorial extension: connecting this scheme to phase-estimation-based time-series analysis might reduce the required circuit depth or measurement overhead by reusing the same time-evolution data for multiple energy targets."],"forward_implications":["One can compute response functions such as optical conductivity, magnetic susceptibility, and spin transport at finite energy without preparing ground states or Gibbs states, provided easy initial states with overlap in the target window exist.","Circuit depth scales polynomially in $1/\\delta$ and sampling cost scales with $1/r_\\delta(E)$, the overlap of the initial state with the selected energy window, making the method practical when such states are available.","The method extends to canonical ensembles by Boltzmann-weighted averaging over filter energies, giving finite-temperature response data from the same set of measurements.","For generic non-integrable systems, the eigenstate thermalization hypothesis makes filtered observables state-independent, so a single easy-to-prepare state may suffice to obtain the equilibrium response.","The method is presented as applicable to interacting fermionic systems and spin systems, enabling quantum hardware to probe response functions beyond classically simulable regimes."],"supporting_citations":[{"why":"Supplies the energy-filter formalism and the state-sampling procedure that the response-function protocol builds on.","marker":"[14]"},{"why":"Shows that time-series data can be classically post-processed to extract spectral information, the basis for the paper's classical post-processing step.","marker":"[13]"},{"why":"Defines the Kubo linear response formula that the algorithm is designed to compute.","marker":"[22]"},{"why":"Provides the eigenstate thermalization hypothesis, which the paper relies on for state-independence of filtered observables in generic systems.","marker":"[18–20]"},{"why":"Supplies the Hadamard-test circuit that is generalized to measure the three-time correlator $C^{AB}_\\psi(t_1,t_2,t_3)$.","marker":"[29]"},{"why":"Offers a phase-sensitive measurement scheme without controlled operations, used to measure the Loschmidt echo.","marker":"[26]"},{"why":"Defines the quasiperiodic mosaic lattice with exact mobility edges that serves as the numerical testbed.","marker":"[33]"},{"why":"Establishes the connection between the current-current response function and the Drude weight, which the paper uses to characterize localization.","marker":"[34]"}],"fun_headline_variants":["Energy filter quench recovers response functions from easy states","No equilibrium prep: quench-based response functions","Response functions from non-thermal states via energy filtering","Quench-and-postprocess computes response without thermal states","Quench+filter: response functions without equilibrium state"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reconstruction assumes that the energy-filtered pure state is effectively stationary on the timescales measured, so that the three-time correlator reduces to a function of time differences only; if the filtered state drifts or fails to equilibrate within the measurement window, the inferred response function mixes distinct time arguments and becomes incorrect, and in practice the paper leans on the eigenstate thermalization hypothesis to guarantee this for generic interacting systems.","fun_headline_variants_meta":{"raw":{"variants":["Energy filter quench recovers response functions from easy states","No equilibrium prep: quench-based response functions","Response functions from non-thermal states via energy filtering","Quench-and-postprocess computes response without thermal states","Quench+filter: response functions without equilibrium state"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001545,"raw_usage":{"total_tokens":6178,"prompt_tokens":943,"completion_tokens":5235,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":5159}},"tokens_in":559,"tokens_out":5235,"duration_ms":38116,"temperature":1.0,"reasoning_tokens":5159,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:04:07.159208+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small non-integrable spin chain, prepare a filtered pure state of width $\\delta$ around a target energy, run the protocol to extract $\\chi_{AB}(\\omega)$, and compare it with the exact microcanonical response function computed by exact diagonalization; if the two disagree or if the extracted response depends on how the three-time correlator is reduced to a time difference, the stationarity assumption has failed. A sharper variant: check whether $\\langle\\psi|P_\\delta(E)[A(t),B(t')]P_\\delta(E)|\\psi\\rangle$ actually depends only on $t-t'$ by varying $t$ and $t'$ while keeping the difference fixed.","supporting_citations":[{"cited_title":"Somma, G","cited_arxiv_id":null,"evidence_quote":"Supplies the Hadamard-test circuit that is generalized to measure the three-time correlator $C^{AB}_\\psi(t_1,t_2,t_3)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Offers a phase-sensitive measurement scheme without controlled operations, used to measure the Loschmidt echo."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the quasiperiodic mosaic lattice with exact mobility edges that serves as the numerical testbed."}],"review_version":1}