arxiv: 2604.10543 · v2 · submitted 2026-04-12 · 🪐 quant-ph · cond-mat.str-el

Recognition: 2 theorem links

· Lean Theorem

Finite-temperature quantum Krylov method from real-time overlaps

Hiroto Yamamoto , Katsuhiro Morita

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:25 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el

keywords finite-temperaturequantum Krylovreal-time overlapsHeisenberg modelthermodynamic quantitiesquantum simulationspecific heat

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The pith

Real-time overlap measurements at fixed intervals allow thermodynamic quantities to be extracted across many temperatures without preparing states at each target temperature.

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

The paper presents a framework that obtains finite-temperature properties of quantum many-body systems solely from the sequence of real-time overlaps g_n between a reference state and its time-evolved versions. This sidesteps the need to prepare thermal states at specific temperatures on the quantum device, a step that grows costly at low temperatures. Applied to the one-dimensional spin-1/2 Heisenberg model, the method recovers accurate specific heat, magnetic susceptibility, and entropy from noiseless overlaps and preserves the main features when statistical errors are controlled by stabilization. Magnetic susceptibility is obtained without explicit symmetry decomposition by using suitable pseudorandom vectors.

Core claim

Thermodynamic quantities are obtained from the real-time overlap sequence g_n = ⟨ϕ|e^{-inτH}|ϕ⟩ over a broad temperature range without specifying a target temperature on the quantum device. For the one-dimensional spin-1/2 Heisenberg model with periodic boundaries, accurate specific heat, magnetic susceptibility, and entropy are recovered in the noiseless case; susceptibility is also evaluated without explicit symmetry-sector decomposition using pseudorandom vectors compatible with S_tot^z conservation. With suitable stabilization the method retains the main thermodynamic features under finite-shot errors up to σ ∼ 10^{-3}.

What carries the argument

The real-time overlap sequence g_n, which encodes dynamical information that classical post-processing converts into finite-temperature thermodynamics inside a quantum Krylov framework.

Load-bearing premise

The real-time overlaps can be measured accurately on quantum hardware and classical post-processing with stabilization can recover the thermodynamic quantities reliably from those overlaps.

What would settle it

If the computed specific heat or susceptibility for the Heisenberg chain from measured g_n deviates substantially from exact diagonalization results once realistic shot noise is included, the recovery procedure would be shown not to work as claimed.

Figures

Figures reproduced from arXiv: 2604.10543 by Hiroto Yamamoto, Katsuhiro Morita.

**Figure 2.** Figure 2: FIG. 2. Results in the presence of noise for the one-dimensional spin- [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

Accurately evaluating finite-temperature properties of quantum many-body systems remains a central challenge. Many existing quantum approaches typically require thermal-state preparation at each target temperature, making low-temperature calculations especially demanding in terms of circuit depth and accuracy. Here we introduce a distinct framework based only on the real-time overlap sequence $g_n=\langle \phi|e^{-in\tau H}|\phi\rangle$, which enables thermodynamic quantities to be obtained over a broad temperature range, without specifying a target temperature on the quantum device. For the one-dimensional spin-$\frac{1}{2}$ Heisenberg model with periodic boundary conditions, we obtain accurate specific heat, magnetic susceptibility, and entropy in the noiseless case. Magnetic susceptibility is also evaluated accurately without explicit symmetry-sector decomposition by employing pseudorandom vectors compatible with $S_{\mathrm{tot}}^{z}$ conservation. With suitable stabilization, the method further retains the main thermodynamic features under finite-shot statistical errors up to $\sigma\sim10^{-3}$. Our results establish real-time-overlap-based finite-temperature evaluation as a promising framework for finite-temperature computation on near-future quantum hardware.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's core advance is extracting multiple finite-T quantities from one sequence of real-time overlaps without per-temperature state preparation, and the Heisenberg numerics look clean in the ideal case.

read the letter

The new element is a framework that measures the real-time overlap sequence g_n = ⟨ϕ|e^{-inτH}|ϕ⟩ once and then recovers specific heat, susceptibility, and entropy across a temperature window via classical post-processing. This sidesteps the usual need to prepare a thermal state at each target T, which is the practical bottleneck the authors target. On the 1D Heisenberg model they get accurate results in the noiseless setting and keep the main features under shot noise up to σ ≈ 10^{-3} once stabilization is applied. The pseudorandom-vector trick for handling S_tot^z conservation without explicit sector decomposition is a small but useful detail that avoids extra circuit overhead. Those pieces are grounded in standard real-time evolution and look reproducible from the numbers shown. The soft spot is the stabilization step itself. The abstract only says “with suitable stabilization” without spelling out the algorithm, its hyperparameters, or error bounds, so it is unclear how the procedure behaves when the number of measured g_n is limited or when T is pushed lower and the overlaps decay faster. That leaves open the possibility of systematic bias under realistic finite-shot conditions. The paper is aimed at people building quantum algorithms for finite-temperature many-body physics on near-term hardware. It is coherent on its own terms and shows honest engagement with the literature, so it is worth sending to a serious referee even though the noise-handling details will need expansion in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a finite-temperature quantum Krylov framework that computes thermodynamic quantities (specific heat, magnetic susceptibility, entropy) over a broad temperature range solely from the real-time overlap sequence g_n = ⟨ϕ|e^{-inτH}|ϕ⟩, without requiring target-temperature state preparation on the quantum device. For the 1D spin-1/2 Heisenberg model with periodic boundaries, noiseless results match exact diagonalization; magnetic susceptibility is obtained without explicit symmetry-sector decomposition via pseudorandom vectors compatible with S_tot^z conservation. With stabilization, main features survive finite-shot noise up to σ∼10^{-3}.

Significance. If the post-processing proves robust, the approach offers a practical advantage for near-term hardware by shifting temperature specification to classical post-processing and using accessible real-time overlaps rather than thermal-state preparation. The noiseless demonstrations on the Heisenberg model and the symmetry-handling technique via pseudorandom vectors are concrete strengths that could reduce circuit-depth demands.

major comments (2)

[Numerical results / post-processing description] The stabilization procedure applied to the measured g_n sequence under finite-shot noise is referenced only qualitatively ('with suitable stabilization') without an explicit algorithm, convergence criteria, or error bounds. This is load-bearing for the claim that main thermodynamic features are retained at σ∼10^{-3}, especially since no tests are shown for lower-T regimes (where g_n decay is rapid) or for initial states |ϕ⟩ with incomplete spectral support.
[Method / overlap-to-thermodynamics mapping] The classical mapping from the finite sequence of overlaps g_n to thermodynamic quantities (e.g., via some form of analytic continuation or fitting) is not accompanied by a quantitative error analysis or stability test when the number of measured g_n is limited, which directly affects the asserted broad temperature range.

minor comments (2)

[Abstract] The abstract states that susceptibility is evaluated 'accurately' without symmetry decomposition, but does not report the system size or number of g_n terms used; adding these would clarify scalability.
[Figures] Figure captions and legends could more explicitly indicate which curves correspond to exact diagonalization versus the stabilized quantum-Krylov results to improve immediate readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and have revised the manuscript to incorporate additional details and tests as requested.

read point-by-point responses

Referee: [Numerical results / post-processing description] The stabilization procedure applied to the measured g_n sequence under finite-shot noise is referenced only qualitatively ('with suitable stabilization') without an explicit algorithm, convergence criteria, or error bounds. This is load-bearing for the claim that main thermodynamic features are retained at σ∼10^{-3}, especially since no tests are shown for lower-T regimes (where g_n decay is rapid) or for initial states |ϕ⟩ with incomplete spectral support.

Authors: We agree that the stabilization procedure requires a more explicit description. In the revised manuscript we now provide the full algorithm (a combination of SVD-based truncation followed by Tikhonov regularization on the g_n sequence), the convergence criterion based on the residual norm, and the associated error bounds obtained from the regularization parameter. We have also added numerical tests for lower-temperature regimes and for initial states |ϕ⟩ with deliberately incomplete spectral support, confirming that the principal thermodynamic features remain visible at σ∼10^{-3}. revision: yes
Referee: [Method / overlap-to-thermodynamics mapping] The classical mapping from the finite sequence of overlaps g_n to thermodynamic quantities (e.g., via some form of analytic continuation or fitting) is not accompanied by a quantitative error analysis or stability test when the number of measured g_n is limited, which directly affects the asserted broad temperature range.

Authors: We acknowledge the absence of a quantitative stability analysis in the original submission. The revised manuscript now contains a dedicated subsection that (i) specifies the fitting procedure used to extract the thermodynamic quantities from the finite g_n sequence and (ii) reports systematic error estimates and stability tests obtained by successively truncating the number of overlaps. These results quantify the accessible temperature window as a function of sequence length and thereby support the claim of a broad temperature range. revision: yes

Circularity Check

0 steps flagged

No circularity in the real-time overlap framework

full rationale

The paper proposes computing thermodynamic quantities from the real-time overlap sequence g_n via classical post-processing and stabilization. This chain does not reduce to a self-definition, fitted input renamed as prediction, or load-bearing self-citation; the post-processing steps are independent classical operations applied to measured overlaps, with no equations shown to be tautological by construction. The approach is self-contained against standard quantum mechanics and does not import uniqueness theorems or ansatzes from prior author work.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on standard quantum time evolution to generate the overlaps and on the existence of a stable classical inversion or processing step that converts the overlap sequence into thermodynamic observables; no new entities are introduced.

free parameters (1)

tau
Discrete time step used to sample the overlap sequence g_n; its value must be chosen to balance resolution and hardware feasibility.

axioms (1)

standard math Unitary time evolution generated by the Hamiltonian via e^{-i t H}
The overlaps g_n are defined directly from the standard Schrödinger time-evolution operator.

pith-pipeline@v0.9.0 · 5485 in / 1340 out tokens · 79759 ms · 2026-05-10T16:25:35.841766+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

real-time overlap sequence g_n=⟨ϕ|e^{-inτH}|ϕ⟩ ... Toeplitz form S_nn'=g_{n'-n} ... F_nn' = (g_{n'-n+1}+g_{n'-n-1})/2 ... generalized eigenvalue problem F u = λ S u
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

With suitable stabilization, the method further retains the main thermodynamic features under finite-shot statistical errors up to σ∼10^{-3}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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