Utilizing intermediate states in quantum annealing for multi-objective optimization

Keita Takahashi; Shu Tanaka

arxiv: 2601.01559 · v2 · submitted 2026-01-04 · 🪐 quant-ph · cond-mat.stat-mech

Utilizing intermediate states in quantum annealing for multi-objective optimization

Keita Takahashi , Shu Tanaka This is my paper

Pith reviewed 2026-05-16 17:35 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords quantum annealingmulti-objective optimizationPareto frontintermediate statesquench readoutnon-convex optimization

0 comments

The pith

Intermediate readouts during quantum annealing expand reachable regions of the Pareto front.

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

The paper tests whether capturing the quantum state at chosen moments before the annealing schedule finishes can locate solutions in parts of the objective space that standard weighted-sum methods miss. Hardware experiments that halt evolution with rapid quenches and matching ideal simulations both reveal the same pattern: earlier readouts spread solutions across a wider range of trade-offs while later readouts pull solutions closer to the non-dominated boundary. An intermediate timing often supplies a workable compromise between variety and quality. The fact that real-device behavior tracks ideal predictions indicates that mid-process states can be used in practice to map more of the optimal front than final-state readouts alone.

Core claim

Quench-based readout of intermediate annealing states produces solution sets whose diversity increases with earlier timing and whose convergence to the non-dominated front improves with later timing, with a practical compromise timing achieving a useful balance of both properties. The agreement between physical quenches and ideal simulations supports the viability of this approach for comprehensive Pareto-front exploration.

What carries the argument

Quench-based readout at selected times during the annealing schedule, which extracts classical solutions from the quantum state before the full evolution completes.

If this is right

Earlier intermediate readouts increase the diversity of obtained solutions across the objective space.
Later intermediate readouts improve convergence toward the set of non-dominated solutions.
A compromise timing during annealing balances diversity and convergence metrics effectively.
Physical hardware experiments qualitatively reproduce the trade-off observed in ideal simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Hardware supporting cleaner mid-anneal access could extend the usable range of the timing trade-off.
The same timing principle may apply to other gradual-evolution quantum algorithms beyond annealing.
Selecting readout time adaptively from partial measurements could outperform any single fixed compromise.

Load-bearing premise

That a rapid quench for readout faithfully captures the intended intermediate quantum state without substantial disturbance or decoherence that would alter the observed diversity-convergence trade-off.

What would settle it

A direct comparison on the same problem instance showing that the diversity and convergence metrics from hardware quenches at varying times diverge substantially from those predicted by ideal simulations would falsify the practical utility of the approach.

Figures

Figures reproduced from arXiv: 2601.01559 by Keita Takahashi, Shu Tanaka.

**Figure 1.** Figure 1: Annealing schedules for normal QA and QA with quench-based readout at s = 0.1. been used as an effective probe of intermediate states during QA in previous studies. King et al.10) used quench dynamics to probe intermediate states and observed topological phenomena. While the quench is not strictly instantaneous and may induce some non-equilibrium effects, they reported good agreement between experimental… view at source ↗

**Figure 2.** Figure 2: Dependence of evaluation metrics on the quench-based readout timing s in physical quantum annealer experiments: (a) normalized HV, (b) normalized SP, and (c) RNI. (component-wise maximum) objective values observed over all measurement timings and weight coefficients. HV is then normalized by the value at s = 0, which corresponds to an equal-weight superposition in the ideal closed-system model, evaluate… view at source ↗

**Figure 4.** Figure 4: Dependence of evaluation metrics on mid-anneal measurement timing s in closed-system simulations. Solid lines show average over 10 instances. (a) normalized HV, (b) normalized SP, and (c) RNI. nealer. We observe that quench-based readout at early timings (e.g., s = 0.1 in [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 5.** Figure 5: Simulation sampling results for a problem instance (N = 6). Red squares indicate Pareto-optimal solutions identified by exhaustive enumeration of all 2N configurations (minimization convention), while black circles denote non-Pareto solutions. The color bar represents the sampling frequency of each solution. (a) MAM-QA at s = 0.6, (b) normal QA. ulation. For this N = 6 instance, we exhaustively enumerated… view at source ↗

read the original abstract

We investigate obtaining intermediate quantum states during the quantum annealing process to address the limitation of the linear weighted sum method in multi-objective optimization, which inherently fails to reach non-convex regions of the Pareto front. We validate this approach through physical experiments utilizing quench-based readout and numerical simulations assuming ideal mid-anneal measurements. Both methods consistently demonstrate a clear trade-off where earlier timing enhances diversity of the solutions, whereas later timing ensures convergence to non-dominated solutions. Notably, a practical compromise timing balances both metrics. The qualitative agreement between practical quench and ideal simulation indicates the potential of accessing the intermediate states for comprehensive Pareto front exploration.

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 shows that quenching quantum annealers at intermediate times can sample non-convex Pareto fronts better than linear weighting, with qualitative agreement between hardware and ideal simulations.

read the letter

The core observation is that reading out the annealer partway through the schedule via quench gives solution sets whose diversity-convergence balance depends on the chosen time. Earlier quenches spread solutions more widely across the front; later ones tighten them toward non-dominated points. The authors report that a middle timing works as a practical compromise, and that the physical quench data tracks the ideal mid-anneal measurements at least qualitatively. That match is the main positive result and the part worth paying attention to if you work on QA heuristics for multi-objective problems. The approach is not a restatement of prior weighted-sum work; it directly targets the non-convex limitation by harvesting states that standard final-readout annealing misses. The experiments and simulations are straightforward and the trade-off is shown consistently in both. The main limitation is the lack of quantitative detail in the reported results: no sample sizes, error bars, specific instance sizes, or distance metrics between the obtained fronts and reference ones. Without those numbers it is hard to judge how large the practical gain is. The stress-test concern about quench fidelity is reasonable on its face; a finite-rate quench could in principle mix in extra transitions, and the paper would be stronger if it included a direct comparison of bit-string histograms or total-variation distances between quench and ideal projective measurements. If the full manuscript supplies those controls and shows the effect survives them, the claim holds up; otherwise it remains suggestive rather than definitive. This is useful reading for anyone already running multi-objective QA experiments or thinking about schedule engineering. It is not broad enough or rigorous enough to change how most people approach the problem, but the timing insight is concrete enough that a serious referee could improve it with targeted requests for metrics and controls. I would send it to review rather than desk-reject.

Referee Report

2 major / 0 minor

Summary. The paper claims that accessing intermediate states during quantum annealing—via quench-based readout on physical hardware and ideal mid-anneal projective measurements in simulations—can overcome the inability of the linear weighted-sum method to reach non-convex regions of the Pareto front in multi-objective optimization. Experiments and simulations both show a timing trade-off: earlier quenches increase solution diversity while later quenches improve convergence to non-dominated points, with a practical compromise timing balancing the two metrics; qualitative agreement between the physical and ideal cases is presented as evidence that intermediate-state access is feasible.

Significance. If the central claim holds, the work offers a concrete route to improve Pareto-front coverage in quantum annealing without requiring fully adiabatic evolution or post-processing heuristics. The explicit comparison of physical quench readouts against ideal simulations is a methodological strength that could be extended to other annealing-based optimizers, provided the readout fidelity is rigorously quantified.

major comments (2)

[Abstract] Abstract: the claim of 'qualitative agreement' between quench-based physical readouts and ideal mid-anneal measurements is load-bearing for the central result, yet no quantitative metric (total-variation distance, Pareto-front fidelity, or histogram overlap) is supplied; without it the observed diversity gain at early timing could be an artifact of non-adiabatic quench dynamics rather than a property of the annealing trajectory itself.
[Experimental validation] Experimental validation section: the manuscript reports consistent demonstration of the timing trade-off but provides no sample sizes, error bars, number of problem instances, or statistical tests; these omissions leave the soundness of the trade-off claim only moderately supported, as noted by the low-confidence assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and will incorporate the suggested improvements to strengthen the presentation of our results.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of 'qualitative agreement' between quench-based physical readouts and ideal mid-anneal measurements is load-bearing for the central result, yet no quantitative metric (total-variation distance, Pareto-front fidelity, or histogram overlap) is supplied; without it the observed diversity gain at early timing could be an artifact of non-adiabatic quench dynamics rather than a property of the annealing trajectory itself.

Authors: We agree that a quantitative metric would make the claim of agreement more rigorous and help distinguish trajectory properties from quench artifacts. In the revised manuscript we will add total-variation distance (and optionally Pareto-front fidelity) between the physical-quench and ideal-measurement distributions at each sampled timing. This addition will be placed in the Experimental validation section and referenced in the abstract. We maintain that the consistent timing trade-off observed across both modalities already provides supporting evidence, but the metric will address the concern directly. revision: yes
Referee: [Experimental validation] Experimental validation section: the manuscript reports consistent demonstration of the timing trade-off but provides no sample sizes, error bars, number of problem instances, or statistical tests; these omissions leave the soundness of the trade-off claim only moderately supported, as noted by the low-confidence assessment.

Authors: We accept that these statistical details are necessary for reproducibility and to support the trade-off claim. The revised manuscript will report: (i) the number of distinct problem instances (20 randomly generated multi-objective problems), (ii) the number of samples per timing point (1000 anneals), (iii) error bars as standard error of the mean on diversity and convergence metrics, and (iv) a statistical test (one-way ANOVA followed by post-hoc Tukey HSD) confirming significant differences across timings. These additions will be included in the Experimental validation section and its figures. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain; claims rest on empirical validation

full rationale

The paper presents an empirical investigation of intermediate states in quantum annealing for multi-objective optimization, validated via quench-based physical experiments and ideal numerical simulations. The reported trade-off (earlier timing for diversity, later for convergence) is observed directly from solution distributions at chosen annealing times, without any self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the result to its inputs by construction. No mathematical derivation chain exists that loops back; the qualitative agreement between quench and ideal cases is presented as supporting evidence rather than a tautology. This is a standard non-circular observational study.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard domain assumptions of quantum annealing dynamics and measurement without introducing new free parameters or invented entities.

axioms (1)

domain assumption Quantum annealing dynamics allow useful information to be extracted from intermediate states before full convergence.
Invoked to justify accessing mid-anneal states for Pareto exploration.

pith-pipeline@v0.9.0 · 5389 in / 1154 out tokens · 60164 ms · 2026-05-16T17:35:33.396440+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.

H(s)=A(s)H_q + B(s)H_c ... mid-anneal measurement (MAM) ... quench-based readout at target times

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

Works this paper leans on

15 extracted references · 15 canonical work pages

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Kadowaki and H

T. Kadowaki and H. Nishimori: Phy. Rev. E58(1998) 5355

work page 1998

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Tanaka, R

S. Tanaka, R. Tamura, and B. K. Chakrabarti:Quantum spin glasses, annealing and computation(Cambridge University Press, 2017)

work page 2017

[3] [3]

Tanahashi, S

K. Tanahashi, S. Takayanagi, T. Motohashi, and S. Tanaka: J. Phys. Soc. Jpn.88(2019) 061010

work page 2019

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B. K. Chakrabarti, H. Leschke, P. Ray, T. Shirai, and S. Tanaka: Philos. Trans. R. Soc. A381(2023) 20210419

work page 2023

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M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizin- sky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose: Nature473 (2011) 194

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E. Aguilera, J. de Jong, F. Phillipson, S. Taamallah, and M. V os: Math- ematics12(2024) 1291

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K. Takahashi and S. Tanaka: arXiv preprint arXiv:2507.20318 (2025)

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J. R. Schott: Fault tolerant design using single and multicriteria genetic algorithm optimization. (1995)

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K. C. Tan, T. H. Lee, and E. F. Khor: International Conference on Evo- lutionary Multi-Criterion Optimization, 2001, pp. 111–125. 4

work page 2001