{"id":"a52b9d5b-ad56-4977-be97-a51bf06ee55f","arxiv_id":"2504.19185","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A proposed quantum algorithm uses thermalization under the eigenstate thermalization hypothesis to compute expectation values of inverses and log-determinant gradients without explicit wavefunction preparation.","lead":"This paper proposes a quantum algorithm, ETH-Σ, that uses the eigenstate thermalization hypothesis to replace wavefunction preparation when computing quantities like the inverse of a matrix. The idea is that a time-averaged random state naturally samples all eigenstates, potentially giving poly-logarithmic quantum speedups without costly state initialization.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (4) is not ETH: the time average retains the weights |c_p|^2, so the algorithm computes a diagonal ensemble rather than (1/N)Tr(A), and the example uses a prepared equal superposition.","rationale":"The paper's central claim is that ETH lets a time-averaged random state behave as an equal superposition of eigenstates. The load-bearing condition is Eq. (4): the time-averaged expectation equals the normalized trace. Under the standard ETH cited by the paper, the infinite-time average of an observable is the diagonal ensemble sum_p |c_p|^2 <p|O|p>, not the normalized trace; the initial-state weights survive dephasing. The paper's Eq. (3) already displays these weights, so Eq. (4) is an additional assumption equivalent to full ergodicity in Hilbert space, which unitary dynamics does not provide. The 2x2 example avoids the issue by starting from an equal superposition, and it omits QPE, so it cannot validate the algorithm. The proposed exact-diagonalization test isolates exactly this step: with ideal QPE, the swap-test estimator time-averages to D_r = sum_p |c_p|^2 |<p|Phi>|^2/E_p, and comparing D_r to T = (1/N)sum_p |<p|Phi>|^2/E_p determines whether Eq. (4) holds. This is more specific than the thermalization-time concern: even with infinite averaging time, the discrepancy remains unless |c_p|^2 = 1/N. The paper's own end-matter caveats about the lack of a proof of ETH and unknown thermalization times are honest but do not address this structural problem. I agree with the reader's weakest-assumption analysis, so no change to the REJECT verdict is needed.","tokens_in":8795,"tokens_out":22935,"duration_ms":245119,"concrete_test":"Exact-diagonalize a generic non-integrable local Hamiltonian with no conserved symmetries (e.g., a weakly disordered L=10 spin chain H=sum_i Z_i Z_{i+1} + sum_i(h_i^X X_i + h_i^Z Z_i)). Pick |Phi>=|0...0> and draw 100 Haar-random initial states |r>. For each, compute D_r = sum_p |c_p|^2 |<p|Phi>|^2 / E_p, which is the exact infinite-time average of the Algorithm 1 swap-test estimator with ideal QPE, and compare it to T = (1/N) sum_p |<p|Phi>|^2 / E_p, the value the paper requires. If the typical |D_r - T| exceeds the target precision, the identification of |c_p|^2 with 1/N (Eq. 4) is falsified. The all-Hadamard state used in the 2x2 example is a special case that can pass only when it is already an equal superposition of eigenstates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Idealizing the QPE as perfect, the swap-test estimator in Eqs. (10)-(12) time-averages to D = sum_p |c_p|^2 |<Phi|p>|^2 / E_p. The paper needs this to equal (1/N) sum_p |<Phi|p>|^2 / E_p, i.e., it needs |c_p|^2 = 1/N. That is the content of Eq. (4). But ETH, as stated in Refs. 10-13, gives only the diagonal ensemble: the infinite-time average of <Psi(t)|O|Psi(t)> is sum_p |c_p|^2 <p|O|p>, with the initial-state weights intact. ETH says the diagonal matrix elements are smooth functions of energy, not that the weights become uniform. Unitary evolution is quasiperiodic, so no mechanism in the paper erases the |c_p|^2. The 2x2 example does not test this because its 'random' state is the prepared equal superposition H^{otimes n}|0...0>. For A^{-1}, which is diagonal in the eigenbasis of e^{-iAt}, its expectation is exactly time-independent, sum_p |c_p|^2/E_p, so it cannot thermalize to the trace. Averaging over many random initial states could approximate the trace by Haar concentration, but that is random-state sampling, not ETH, and is not what is claimed. The polylog claim therefore rests on an identification that standard ETH does not provide.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an algorithm, ETH-Σ, that uses the eigenstate thermalization hypothesis to replace wavefunction preparation in quantum linear algebra. A random initial state is time-evolved under exp(-iAt); invoking Eq. (4), the time average of an expectation value is claimed to equal (1/N)Tr(A). Quantum phase estimation and a diagonal register Υ are then used to weight each eigenstate by 1/E_p, and the paper claims this yields expectation values of A^{-1} and gradients of log-determinants in O(τ T M/ε^3) time, independent of the condition number. A single-qubit example and a discussion of quantum advantage are included.","tokens_in":9119,"tokens_out":6481,"duration_ms":65288,"significance":"The question the paper asks—whether physical thermalization can substitute for explicit state preparation—is timely and could be significant if rigorously established. The QPE-based reweighting construction is a concrete and reusable idea. However, the central identity Eq. (4) is not a consequence of ETH as stated in the cited literature: standard ETH preserves the initial-state weights |c_p|^2 in the diagonal ensemble. The numerical example does not exercise the proposed algorithm. Thus, while the paper is clearly written and honest about the absence of a proof of ETH, the core claim is not supported, and the proposed complexity bound rests on an unjustified identification.","major_comments":[{"comment":"Equation (4) states that the time-averaged expectation value equals (1/N)Tr(A). Standard ETH, as formulated in Refs. [10–13], gives for a nondegenerate spectrum the long-time average ∑_p |c_p|^2 ⟨p|A|p⟩, with the initial-state weights |c_p|^2 intact. ETH concerns the smoothness of diagonal matrix elements, not the uniformity of the weights. A random initial state does not generally satisfy |c_p|^2 = 1/N, and unitary evolution is quasiperiodic, so time averaging cannot erase these weights. For B = A^{-1}, which is diagonal in the time-evolution eigenbasis, ⟨Ψ(t)|B|Ψ(t)⟩ is exactly time-independent, so it cannot thermalize to (1/N)Tr(B) even if A itself satisfies ETH. This invalidates the central assumption underlying Algorithm 1 and the polynomial-time claim.","section":"Eigenstate thermalization hypothesis, Eq. (4)"},{"comment":"The swap-test output, after time averaging, is ∑_p |c_p|^2 |⟨Φ|p⟩|^2 / E_p (assuming perfect QPE and neglecting rapidly oscillating cross terms), not ∑_p |⟨Φ|p⟩|^2 / E_p as claimed via Eq. (4). The manuscript gives no derivation showing that the QPE weighting converts the diagonal-ensemble weights into uniform weights. The end-matter Proposition 2 only re-weights the diagonal elements of ∆ by a function of E_p and asserts that ETH applies \"as normal\"; it does not address the |c_p|^2 dependence. The algorithm therefore computes a weighted diagonal ensemble, not the desired normalized trace, unless an additional, generally false condition is imposed.","section":"Vector form, Eqs. (10)–(12)"},{"comment":"The single-qubit example uses A = σ_z and the initial state H^{⊗n}|0...0⟩, which is a prepared equal superposition of the eigenstates of σ_z, so |c_p|^2 = 1/2 exactly. No thermalization is needed to obtain Eq. (13). A one-qubit unitary evolution is integrable and does not exhibit many-body ETH, and the example omits the QPE, the Υ register, the inverse weighting, and the swap test, so it does not test the proposed pipeline. The notation in Eq. (14) is also inconsistent: ∆ = (1/√2)(I + σ_x) is not the identity matrix, despite the accompanying text \"I being the identity.\"","section":"Example section, Eqs. (13)–(14)"}],"minor_comments":[{"comment":"The definitions of the precision parameters are imprecise: the relation between m, ε, M = 2^m, and the stated O(τ T M/ε^3) complexity should be spelled out, including whether ε is the target additive error for the expectation value or the QPE phase error.","section":"Complexity statement, Introduction and Summary"},{"comment":"The text says \"This was achieved out to 6 digits for 10^5 samples and verified in more than 100 starting initial states,\" but no details of the numerical experiment are given, such as the time-step δt, the averaging time τ, or the precise definition of the random initial states.","section":"Example section"},{"comment":"The statement that the QPE \"discovers all eigenvalues, not to just one as in the standard use case\" is misleading: standard QPE applied to a superposition also processes all eigenvalues in superposition; the distinction is that the present algorithm does not postselect on a specific eigenvalue.","section":"Vector form, paragraph on QPE"},{"comment":"Equation (20) is not fully defined: the right-hand double sum should presumably involve ⟨q|...|p⟩ matrix elements, and the notation does not make the proposed classical/quantum distinction precise. The argument would benefit from a formal statement of the conditions under which the sum truncates.","section":"Where there can be a quantum advantage, Eq. (20)"},{"comment":"Reference [30] is cited as \"arxiv: 2501.09413\" with incomplete author information; please provide the full citation. Also, the overbar denoting the time average in Eq. (4) is not visible in the rendered text and should be indicated explicitly.","section":"References"}],"recommendation":"reject","confidential_remarks":"The paper's central claim rests on an incorrect reading of ETH: Eq. (4) is not a consequence of the eigenstate thermalization hypothesis, and the numerical example does not test the algorithm. I do not see a repair within the scope of the current manuscript that would preserve the stated polylogarithmic, no-state-preparation claims. The idea of using QPE to weight eigenstates may merit future exploration, but the present form is not suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about this paper: it has a genuinely new idea, and the central step doesn't work as written. Baker proposes using the eigenstate thermalization hypothesis as a resource—let a random state evolve, and the time average of an observable gives the normalized trace, which would replace the expensive state-preparation step in quantum linear algebra. That would be a big deal if it held, and it would be new: I'm not aware of anyone proposing ETH as a computational primitive rather than a limitation.\n\nWhat the paper does well: it identifies a real bottleneck (wavefunction preparation) and suggests a genuinely different way around it. The discussion of where a quantum advantage could survive, and the extension to gradients of log-determinants, is thoughtful. Baker also flags the MBL caveat and admits ETH isn't proven. The writing is clear, and the example is reproducible with a few lines of code.\n\nThe soft spots are load-bearing, though. Equation (4) is the heart of the algorithm, and it is not what ETH gives. The time average of ⟨Ψ(t)|A|Ψ(t)⟩ is always Σ_p |c_p|^2 ⟨p|A|p⟩; unitary evolution cannot erase the initial weights. ETH says the diagonal matrix elements are smooth in energy, but it says nothing about making the weights uniform. For A^{-1}, which is diagonal in the basis of e^{-iAt}, the expectation value is time-independent, so it cannot 'thermalize' to the trace at all. The algorithm is computing a diagonal-ensemble average, not a trace, unless the initial state happens to be an equal superposition.\n\nThe example illustrates the problem rather than solving it: the 'random' state is H^{⊗n}|0...0>, which for σ_z is an exact equal superposition of the eigenstates. That's precisely the kind of prepared state the algorithm claims to avoid. A single qubit also doesn't probe many-body thermalization.\n\nThe complexity claims are also not fully supported. The M = 2^m factor from the QPE register isn't negligible, and the paper never accounts for the precision needed to resolve small eigenvalues, which is where condition-number costs usually hide. Saying the method is independent of condition number needs an argument that isn't there.\n\nSo: the paper is not correct as a working algorithm, and the fix isn't a few added sentences—it would need a different mechanism (e.g., averaging over many random states, which would be a different algorithm with its own cost) or a fundamentally better argument for why the weights vanish. Still, the core idea is original enough that a good referee could help the author see what is needed. I'd send it to peer review rather than desk-reject; it deserves careful scrutiny, and the referee report would be useful to the author. In the meantime, I wouldn't cite it as a method.","headline":"Original idea—use ETH instead of state preparation—but Eq. (4) is not what ETH implies; the algorithm computes a diagonal ensemble, not the trace, and the example quietly prepares the state it claims to avoid.","tokens_in":9632,"tokens_out":4062,"would_cite":false,"duration_ms":38576,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","81Q50"],"pacs":[],"model":"deepseek-v4-flash","headline":"A time-averaged random quantum state, governed by the eigenstate thermalization hypothesis, can replace explicit wavefunction preparation and yield expectation values such as the inverse of an operator in poly-logarithmic time.","keywords":["eigenstate thermalization hypothesis","quantum linear algebra","matrix inversion","quantum phase estimation","wavefunction preparation","logarithm-determinant","trace estimation","quantum advantage"],"falsifier":"Simulate the ETH-Σ protocol on a non-integrable few-qubit system with a genuinely random initial state, and compare the time-averaged estimate of $\\mathrm{Tr}(A^{-1})/N$ with the exact value; if the estimator does not converge within the predicted averaging time and sample count, the central assumption fails. The paper's $\\sigma_z$ example would not settle this because it begins from a prepared equal superposition rather than a thermalizing random state.","tokens_in":8570,"feed_emoji":"⚛️","tokens_out":7466,"duration_ms":71353,"temperature":0.7,"pith_summary":"The paper proposes a quantum algorithm, ETH-Σ, that replaces explicit wavefunction preparation with thermalization. The claim is that if a random state is time-evolved under an operator $A$ and expectation values are time-averaged, the eigenstate thermalization hypothesis makes the average equal the normalized trace $\\frac{1}{N}\\mathrm{Tr}(A)$; applying quantum phase estimation with weights $1/E_p$ then gives $\\frac{1}{N}\\mathrm{Tr}(A^{-1})$. The advertised complexity is poly-logarithmic in the matrix size and independent of the condition number. If this works, it would remove a known bottleneck in quantum linear algebra and would extend to the gradient of a logarithm-determinant. The assumption that thermalization is fast enough for the operators of interest is the load-bearing premise.","feed_headline":"Thermalization replaces wavefunction preparation in quantum computing","feed_subtitle":"A time-averaged random state samples all eigenstates, so matrix-inverse traces can be computed in polylog time.","key_machinery":"The central object is the ETH-Σ estimator: time-averaging an expectation value under $e^{-iAt}$ until Eq. (4) holds, i.e., until the average equals the normalized trace. Quantum phase estimation supplies the diagonal weighting operator $\\Upsilon$, which maps $|E_p\\rangle$ to $f(E_p)|E_p\\rangle$; a swap test (vector form) or an operator $\\Delta$ (operator form) completes the expectation value. The work this does is to convert state preparation into a dynamical sampling problem: instead of building the superposition $\\sum_p |p\\rangle$ by hand, the algorithm lets time evolution generate it and uses quantum phase estimation to attach the spectral function.","core_discovery":"The paper's central claim is that the eigenstate thermalization hypothesis can be used as a computational resource: for non-integrable evolution under $e^{-iAt}$, the time average of $\\langle \\Psi(t)|A|\\Psi(t)\\rangle$ equals $N^{-1}\\sum_p \\langle p|A|p\\rangle$, erasing the initial-state dependence. The ETH-Σ algorithm prepares a random state, evolves it in time, applies quantum phase estimation to read out the eigenvalues, multiplies by a diagonal operator $\\Upsilon$ with entries $1/E_p$ (or $\\sqrt{1/E_p}$ in the vector form), and then time-averages the resulting expectation value, giving $N^{-1}\\mathrm{Tr}(A^{-1})$ without ever constructing the equal superposition explicitly. The same scheme, with $\\Upsilon|E_p\\rangle = f(E_p)|E_p\\rangle$, gives other spectral functions; the paper works out the gradient of the logarithm-determinant as a concrete generalization. The paper also proposes a criterion for when this is genuinely quantum rather than a disguised classical algorithm: poly-logarithmic scaling while all equivalent classical algorithms cost more.","pith_inferences":["A natural stress test is to benchmark ETH-Σ on small non-integrable spin chains starting from genuinely random states, measuring convergence of the estimator to $\\mathrm{Tr}(A^{-1})/N$; the paper's worked example starts from an engineered equal superposition, so it does not exercise the full random-state assumption.","Because Eq. (4) is a trace estimator, ETH-Σ could be combined with stochastic trace-inversion techniques, potentially offering an alternative route to large-scale fermion determinants or lattice observables without full quantum phase estimation.","If some natural operator families thermalize slowly, such as many-body localized regimes, the algorithm's polylogarithmic claim would fail only for those families; identifying the boundary between fast and slow thermalization for practical operators is an open engineering question.","The paper's classical-versus-quantum criterion suggests a concrete benchmark: compare ETH-Σ against classical trace estimators on dense, ill-conditioned matrices of increasing size; a crossover at large $N$ would be evidence of genuine advantage."],"forward_implications":["If ETH-Σ works as claimed, quantum linear algebra solvers can drop the state-preparation subroutine, and the complexity of computing $A^{-1}$ expectation values becomes $O(\\tau T M/\\varepsilon^3)$ with no dependence on the condition number.","The same circuit pattern gives the gradient of the logarithm-determinant, summing over all eigenstates in a single quantum phase estimation pass, which the paper connects to applications in physics and density-functional theory.","Because the time average samples all eigenstates, the method behaves like a full-configuration-interaction sampler; conserving particle number or other symmetries during thermalization is flagged as a needed extension.","For low-condition-number or mean-field-friendly problems, the algorithm degenerates to a classical algorithm, so its quantum value is confined to problems where classical methods require an exponential number of operations."],"supporting_citations":[{"why":"Supplies the eigenstate thermalization hypothesis: time-averaged expectation values equal normalized traces.","marker":"[10–13]"},{"why":"Shows that state-preparation assumptions can hide exponential speedup, motivating the preparation-free design.","marker":"[2]"},{"why":"Earlier quantum linear-systems solvers that ETH-Σ aims to replace by removing explicit superposition preparation.","marker":"[3–8]"},{"why":"Sets out the hidden costs and condition-number limitations that ETH-Σ claims to avoid.","marker":"[9]"},{"why":"Provides the efficient time-evolution implementation that keeps the per-step cost at $O(\\log^2 N)$.","marker":"[23]"},{"why":"Gives the gradient of the logarithm-determinant that the framework generalizes from the inverse case.","marker":"[30]"}],"fun_headline_variants":["ETH turns thermalization into a quantum algorithm","Quantum linear algebra without state preparation","Thermalization computes matrix inverses in polylog time","Random states plus ETH yield quantum speedup","Skipping state prep: ETH powers quantum circuits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on the assumption that for the operator one wants to invert, a random initial state time-averaged under $e^{-iAt}$ converges to the normalized trace quickly enough and with fluctuations below the target error; the paper does not derive this from any known theorem.","fun_headline_variants_meta":{"raw":{"variants":["ETH turns thermalization into a quantum algorithm","Quantum linear algebra without state preparation","Thermalization computes matrix inverses in polylog time","Random states plus ETH yield quantum speedup","Skipping state prep: ETH powers quantum circuits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00044,"raw_usage":{"total_tokens":2193,"prompt_tokens":865,"completion_tokens":1328,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":1258}},"tokens_in":481,"tokens_out":1328,"duration_ms":9817,"temperature":1.0,"reasoning_tokens":1258,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:59:03.982719+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the ETH-Σ protocol on a non-integrable few-qubit system with a genuinely random initial state, and compare the time-averaged estimate of $\\mathrm{Tr}(A^{-1})/N$ with the exact value; if the estimator does not converge within the predicted averaging time and sample count, the central assumption fails. The paper's $\\sigma_z$ example would not settle this because it begins from a prepared equal superposition rather than a thermalizing random state.","supporting_citations":[{"cited_title":"Buhrman, R","cited_arxiv_id":null,"evidence_quote":"Provides the efficient time-evolution implementation that keeps the per-step cost at $O(\\log^2 N)$."},{"cited_title":"Build your own tensor network library: DMRjulia I. Basic library for the density matrix renormalization group","cited_arxiv_id":"2109.03120","evidence_quote":"Gives the gradient of the logarithm-determinant that the framework generalizes from the inverse case."}],"review_version":1}