Quantum computation with the eigenstate thermalization hypothesis instead of wavefunction preparation

Thomas E. Baker

REVIEW 3 major objections 5 minor 1 cited by

Quantum computation with the eigenstate thermalization hypothesis instead of wavefunction preparation

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2504.19185 v2 pith:PF54CUV2 submitted 2025-04-27 quant-ph

Thomas E. Baker This is my paper · ORCID

classification quant-ph MSC 81P6881Q50

keywords eigenstatethermalizationhypothesisquantumlinearalgebramatrixinversionphaseestimationwavefunctionpreparationlogarithm-determinanttraceadvantage

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, 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

What 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.

The 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.

The 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.

The 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.

So: 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Eigenstate thermalization hypothesis, Eq. (4)] 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.
[Vector form, Eqs. (10)–(12)] 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.
[Example section, Eqs. (13)–(14)] 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."

minor comments (5)

[Complexity statement, Introduction and Summary] 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.
[Example section] 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.
[Vector form, paragraph on QPE] 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.
[Where there can be a quantum advantage, Eq. (20)] 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.
[References] 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.

Circularity Check

2 steps flagged · score 8.0 of 10

The central 'prediction' of 1/N Tr(A^{-1}) is Eq. (4) restated: since A^{-1} is diagonal in the evolution basis, time averaging cannot change the weights |c_p|^2, so the equal-superposition result is the assumed ETH input, not a derived output.

self definitional [Section 'Eigenstate thermalization hypothesis', Eq. (4); used in 'Vector form' and 'Operator form']
"The time-evolution for non-integrable operators is not as straightforward. Imagining that the expectation value of any operator A is no longer integrable, then we have ⟨Ψ(t)|A|Ψ(t)⟩ ! = 1/N ∑_{p=1}^N ⟨p|A|p⟩ = 1/N Tr(A) from ETH [10–13] where an over-bar represents a time-average. The idea is that the eigenstates are allowed to transition between each other and thermalize or erase the initial state dependence in the time-average."

Eq. (4) is the algorithm's target, not a derived consequence. The paper's own Eq. (3) gives ⟨Ψ(t)|A|Ψ(t)⟩ = ∑_p a_pp |c_p|², and unitary evolution cannot erase the |c_p|²; the infinite-time average is the diagonal ensemble ∑_p |c_p|²⟨p|A|p⟩. ETH (refs 10-13) says diagonal matrix elements are smooth functions of energy, not that |c_p|² becomes 1/N. For B=A^{-1}, Eqs. (5)-(7) make B diagonal in the evolution eigenbasis, so ⟨Ψ(t)|B|Ψ(t)⟩ = ∑_p |c_p|²/E_p is exactly time-independent. The paper's claim that the time-averaged quantity 'will generate the sum over all eigenstates from Eq. (4)' therefore reduces the computed quantity to the assumed RHS of Eq. (4) by construction.
other [Section 'Example', Eqs. (13)-(14)]
"A random state |r⟩ is time-evolved according to exp(−iσ_z δt) with time-step δt. Importantly, each new time-evolution starts from the same random vector. In this case we choose an equal superposition of all states initialized by applying Hadamard gates, H, on each qubit, H^{⊗n}. The expectation value of ⟨Ψ(t)|∆|Ψ(t)⟩ ≈ Tr(∆)/N = 1/√2 ... In a numerical test, this was achieved out to 6 digits for 10^5 samples and verified in more than 100 starting initial states."

The 'random state' is the desired equal superposition itself: H^{⊗n}|0...0⟩ = (1/√N)∑_p |p⟩, so |c_p|² = 1/N is put in by preparation rather than produced by thermalization. The numerical test verifies the prediction only when the central assumption is already hard-wired into the initial state. Because ∆ is time-independent in this example and the inverse-type operators are diagonal in the evolution basis, the example provides no independent test of Eq. (4) for a generic random state.

full rationale

The core circularity is Eq. (4). The algorithm computes a time average of an expectation value and, through the asserted ETH identity, reports the normalized trace. But the operators being averaged (A^{-1}, or ∆ re-weighted by eigenvalues) are diagonal in the eigenbasis of the evolution, so unitary evolution leaves their expectation values constant in time; the only way the time average becomes (1/N)Tr is if the initial weights |c_p|² are already 1/N. That is precisely the statement Eq. (4) assumes. Standard ETH references [10-13] do not supply the uniform-weight condition; they supply a diagonal-ensemble prediction with the initial weights intact. The paper's abstract and summary phrase the assumption as 'full ergodicity ... produce an equal superposition of eigenstates', which is a re-labelling of the needed input. The self-citations (Refs. 18, 19 for state preparation; Ref. 30 for gradient of the log-determinant) are secondary and not the source of the central issue; even if those references are accepted, the inverse-trace result still rests on Eq. (4). The QPE, swap-test and complexity analysis are standard and not themselves circular. Because the central 'prediction' of 1/N Tr(A^{-1}) reduces by construction to the assumed ETH equality, and the worked example builds the equal superposition into the input, the appropriate score is high, though not the maximum: the paper does make an explicit algorithmic proposal whose QPE/Υ steps are independent of the trace identity, so the circularity is concentrated in the ETH-to-trace step.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The algorithm rests on ETH as its primary physical axiom, together with the stronger equal-weight assumption that is not implied by standard ETH. The implementation efficiency of time evolution and the diagonal operator Υ are assumed. No new physical entities are introduced.

free parameters (3)

Averaging time τ
The time over which the expectation value is averaged is a free parameter; the paper states O(τ T M/ε^3) but gives no rule for choosing τ, which depends on unknown thermalization time.
Number of QPE precision qubits m
The QPE register size m controls precision ε and enters complexity as M=2^m; the paper does not specify how m is chosen for the required accuracy.
Time step δt in numerical example
The example's time step is not given, and the averaging schedule over time is unspecified.

assumptions (4)

domain assumption The eigenstate thermalization hypothesis holds for the operators/circuits used, including the equality of time averages with the normalized trace (Eq. 4).
ETH is an unproven conjecture; the paper itself acknowledges the lack of a firm proof (Ref. [17]) and the possibility of many-body localization.
domain assumption A random initial state samples each eigenstate with equal weight |c_p|^2=1/N after thermalization.
This is a stronger statement than standard ETH, which only guarantees the diagonal ensemble average with weights set by the initial state; the paper does not justify equal weights for a Haar-random state beyond citing full ergodicity.
domain assumption The input operator A can be efficiently time-evolved with O(log^2 N)-gate steps as in Ref. [23], and the resulting circuit is non-integrable and thermalizing.
Efficient Trotter steps exist for structured Hamiltonians, but the algorithm claims arbitrary A; the thermalizing property of the induced circuit is assumed without evidence.
ad hoc to paper The diagonal operator Υ containing inverse eigenvalues can be implemented with cost M=2^m as worst case.
The paper assumes a worst-case circuit for diagonal unitaries ([24,25]) and asserts N >> M makes it subleading, without a concrete construction for this Υ.

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Pith. "Pith review of Quantum computation with the eigenstate thermalization hypothesis instead of wavefunction preparation." pith.science (2026). https://pith.science/paper/PF54CUV2

@misc{pith2026250419185,
  author       = {Pith},
  title        = {Pith review of: Quantum computation with the eigenstate thermalization hypothesis instead of wavefunction preparation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PF54CUV2}},
  note         = {Machine review of arXiv:2504.19185}
}

read the original abstract

It is proposed that the ability for a quantum circuit to thermalize under time evolution is a valid way to compute linear algebra problems. The algorithm makes use of the eigenstate thermalization hypothesis and full ergodicity in quantum systems to produce an equal superposition of eigenstates. The quantum phase estimation subroutine then allows for the computations of functions of the input operator, leading to a variety of methods in linear algebra. The algorithm circumvents the need for elaborate wavefunction preparation on the quantum computer to find the solution of the linear algebra problem in poly-logarithmic time.

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Forward citations

Cited by 1 Pith paper

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Deﬁnition 3 (Classical algorithm I)

and take note of two possible types of classical algorithms. Deﬁnition 3 (Classical algorithm I) . A classical algo- rithm costs at least N operations on a vector of size N . The classical algorithm can be formalized as applying an operator or set of operators onto each of the states |p⟩. There are N of them, the total operations to capture all of the con...

work page 2023

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2021

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