arxiv: 2605.13082 · v1 · submitted 2026-05-13 · 🪐 quant-ph

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Feedback-based quantum optimization and its classical counterpart: quantum advantage and the power of classical algorithms

Takuya Hatomura, Tomohiro Hattori

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Pith reviewed 2026-05-14 19:03 UTC · model grok-4.3

classification 🪐 quant-ph

keywords feedback-based quantum optimizationFALQONclassical counterpartquantum advantagecombinatorial optimizationspin systemsunconstrained binary optimizationscalability

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The pith

Quantum optimization produces higher-quality solutions than classical counterparts but converges more slowly

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

This paper maps feedback-based quantum optimization to classical algorithms via the quantum-classical correspondence of spin systems. It benchmarks the quantum FALQON approach and its variants against these classical versions on small instances and tests the classical algorithms further on large instances. The results indicate quantum algorithms can deliver better solution quality while classical algorithms converge faster, with one classical method showing strong scalability on higher-order unconstrained binary optimization problems. This comparison clarifies where quantum effects may provide an edge in combinatorial optimization and demonstrates the practical value of classical feedback methods.

Core claim

Using the quantum-classical correspondence of spin systems, the feedback-based algorithm for quantum optimization (FALQON) and its variants are compared directly to newly developed classical counterparts. Benchmark tests on small-scale instances show quantum algorithms can be advantageous in solution quality, while classical algorithms tend to converge faster. One classical algorithm also exhibits significant scalability when applied to large-scale higher-order unconstrained binary optimization problems.

What carries the argument

The quantum-classical correspondence of spin systems, which maps quantum feedback optimization to equivalent classical algorithms for direct performance comparison and enables extension to higher-order classical theory.

Load-bearing premise

The quantum-classical correspondence of spin systems preserves the relative performance ranking between the quantum and classical feedback algorithms on the tested instances.

What would settle it

A new benchmark set where a classical algorithm consistently matches or exceeds the quantum algorithm in final solution quality would undermine the claimed quantum advantage.

Figures

Figures reproduced from arXiv: 2605.13082 by Takuya Hatomura, Tomohiro Hattori.

**Figure 3.** Figure 3: FIG. 3. Energy density comparison as a function of the oper [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Energy density dynamics of classical algorithms ap [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Control strengths of classical algorithms for a 3- [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Energy density of CC-FALQON as a function of the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Standard deviation of the energy density of all the [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Energy density dynamics and control strength of classical algorithms as a function of the operation time. A 3-SAT [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

read the original abstract

Feedback-based quantum optimization is a quantum approach to combinatorial optimization. In this paper, we introduce the classical counterpart of feedback-based quantum optimization by using the quantum-classical correspondence of spin systems to discuss the possibility of quantum advantage. It also enables us to develop higher-order theory of a previously proposed classical approach to feedback-based quantum optimization. First, we compare the feedback-based algorithm for quantum optimization (FALQON) and its variant with their classical counterparts. Then, we perform benchmark tests of various quantum and classical algorithms with small-scale instances, and of classical algorithms with large-scale instances. Main findings are that (i) quantum algorithms can be advantageous to classical algorithms in terms of the quality of solutions, while classical algorithms tend to show faster convergence than quantum ones, and (ii) one of the classical algorithms discussed in this paper shows significant scalability for higher-order unconstrained binary optimization problems. These findings highlight the importance of quantumness and the usefulness of classical approaches.

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 spin-system mapping gives a clean classical counterpart to FALQON plus a higher-order extension, but quantum quality gains appear only on small instances while one classical version scales well.

read the letter

The core contribution is the explicit construction of classical feedback algorithms from FALQON via the established quantum-classical spin correspondence, plus the higher-order generalization of earlier classical feedback ideas. They then benchmark FALQON and a variant against these classical versions on small instances, and run the classical ones alone on larger higher-order binary optimization problems. The main reported pattern is that the quantum versions sometimes reach better solution quality while the classical ones converge faster, with one classical method showing good scalability at larger sizes. That mapping and the higher-order step are genuinely new relative to the cited FALQON work, and the small-scale head-to-head tests give a direct view of the quality-speed trade-off. The limitation is that all quantum-classical comparisons stay on small instances; the large-scale runs are classical only, so there is no evidence whether the quality edge survives at sizes where the correspondence could be simulated classically or where the scalable classical method already dominates. The paper does not report error bars or full implementation details in the abstract, which leaves the performance claims harder to judge. This is useful reading for anyone comparing feedback-based quantum solvers to classical alternatives, especially if they care about the spin mapping as a bridge. It is worth sending to referees because the derivation and the classical scalability result are concrete enough to merit review, even though the quantum-advantage claim will need larger-scale checks.

Referee Report

2 major / 2 minor

Summary. The paper introduces classical counterparts to feedback-based quantum optimization algorithms such as FALQON and its variants by invoking the quantum-classical correspondence of spin systems. This enables both a direct performance comparison and the development of higher-order theory for classical feedback methods. On small-scale instances the authors report that the quantum algorithms achieve superior solution quality while the classical counterparts converge faster; on large-scale instances one classical algorithm demonstrates strong scalability for higher-order unconstrained binary optimization problems.

Significance. If the reported quality advantage of the quantum algorithms survives at larger scales, the work would usefully clarify the contribution of quantum coherence versus classical feedback dynamics in combinatorial optimization and would strengthen the case for using spin-system correspondences to derive scalable classical algorithms. The explicit large-scale classical benchmarks already constitute a concrete contribution to the classical side of the literature.

major comments (2)

[Benchmark tests] Benchmark section: quantum advantage in solution quality is demonstrated exclusively on small-scale instances, while all large-scale tests are performed only with the classical algorithms. No scaling extrapolation, intermediate-size verification, or discussion of how the observed ranking behaves when the classical scalable method becomes applicable is provided, leaving the central quantum-advantage claim unsupported beyond the tested regime.
[Quantum-classical correspondence] Quantum-classical correspondence section: the paper derives the classical algorithms from the quantum ones via the established spin-system mapping and uses the same mapping to motivate higher-order extensions, yet supplies no numerical check that the relative performance ordering between quantum and classical feedback is preserved at sizes where the correspondence could be used to simulate the quantum dynamics classically.

minor comments (2)

Figure captions and legends should explicitly state the instance sizes and the number of independent runs used to generate each curve so that readers can assess statistical reliability without consulting the main text.
The definition of the higher-order classical update rule (derived from the correspondence) would benefit from an explicit equation number and a short derivation sketch in the main text rather than relegation to an appendix.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments, which identify key limitations in the scope of our benchmarks and the validation of the quantum-classical mapping. We will revise the manuscript to add discussion and intermediate-size checks, while acknowledging the inherent constraints on large-scale quantum simulations.

read point-by-point responses

Referee: Benchmark section: quantum advantage in solution quality is demonstrated exclusively on small-scale instances, while all large-scale tests are performed only with the classical algorithms. No scaling extrapolation, intermediate-size verification, or discussion of how the observed ranking behaves when the classical scalable method becomes applicable is provided, leaving the central quantum-advantage claim unsupported beyond the tested regime.

Authors: We agree that quantum simulations are restricted to small instances due to exponential resource requirements, which is why large-scale tests use only the classical algorithms. The quality advantage is reported specifically for the tested small-scale regime. We will add a dedicated discussion subsection on scaling implications, incorporating qualitative extrapolation from the observed trends and the spin-system correspondence, along with caveats that direct quantum verification at large scales is currently infeasible. This will clarify that the classical scalability result stands as an independent contribution. revision: partial
Referee: Quantum-classical correspondence section: the paper derives the classical algorithms from the quantum ones via the established spin-system mapping and uses the same mapping to motivate higher-order extensions, yet supplies no numerical check that the relative performance ordering between quantum and classical feedback is preserved at sizes where the correspondence could be used to simulate the quantum dynamics classically.

Authors: We will add numerical comparisons at intermediate sizes (e.g., 10-20 spins) where classical simulation of the quantum feedback dynamics remains tractable. These checks will confirm whether the performance ordering observed at small scales is preserved in the regime accessible to both approaches, thereby supporting the validity of the mapping for deriving and extending the classical algorithms. revision: yes

standing simulated objections not resolved

Direct quantum benchmarks on large-scale instances to test whether the solution-quality advantage persists where classical methods demonstrate scalability.

Circularity Check

0 steps flagged

No significant circularity; derivation grounded in external correspondence and explicit benchmarks

full rationale

The paper derives classical counterparts from the quantum-classical correspondence of spin systems, an established physical mapping independent of the present work. Performance claims rest on explicit numerical benchmarks for small instances (both quantum and classical) and large instances (classical only), with no fitted parameters renamed as predictions and no load-bearing self-citation chains. The central findings on solution quality and convergence are directly supported by the reported test results rather than reducing to definitions or prior self-references by construction. No steps meet the criteria for circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central comparison rests on the quantum-classical spin correspondence (standard in the field) and on the assumption that small-instance benchmarks extrapolate to the claimed advantage and scalability.

axioms (1)

domain assumption Quantum-classical correspondence of spin systems preserves relative algorithmic performance
Invoked to justify direct comparison of FALQON and its classical twin.

pith-pipeline@v0.9.0 · 5465 in / 1110 out tokens · 30304 ms · 2026-05-14T19:03:14.521273+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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The single parameterβ X(t) is given by βX(t) = i⟨Ψ(t)| " ˆHP, NX i=1 ˆXi # |Ψ(t)⟩,(6) which guarantees the reduction of the cost function (4)

FALQON In the original proposal of FALQON [48, 49], the fol- lowing Hamiltonian ˆH(t) = ˆHP +β X(t) NX i=1 ˆXi,(5) was considered. The single parameterβ X(t) is given by βX(t) = i⟨Ψ(t)| " ˆHP, NX i=1 ˆXi # |Ψ(t)⟩,(6) which guarantees the reduction of the cost function (4)
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[49], we can introduce multiple parameters

iFALQON As summarized above and mentioned in Refs. [49], we can introduce multiple parameters. For example, we can consider the following Hamiltonian ˆH(t) = ˆHP + NX i=1 βX i (t) ˆXi,(7) with the parameters βX i (t) =i⟨Ψ(t)|[ ˆHP, ˆXi]|Ψ(t)⟩. (8) This algorithm is inspired by inhomogeneous driving for quantum annealing [75, 76], and thus we call it inhom...
[3]

CC-FALQON We introduce the classical counterpart of FALQON (CC-FALQON). In CC-FALQON, we consider the fol- lowing Hamiltonian Ht =H P +β X(t) NX i=1 mX i , (12) 4 with the parameter βX(t) =−2 NX i=1 mY i ∂HP ∂mZ i .(13) This is the exact classical counterpart of FALQON
[4]

CC-iFALQON We also introduce the classical counterpart of iFALQON (CC-iFALQON). In CC-iFALQON, we con- sider the following Hamiltonian Ht =H P + NX i=1 βX i (t)mX i , (14) with the parameters βX i (t) =−2m Y i ∂HP ∂mZ i .(15) This is the exact classical counterpart of iFALQON
[5]

CACAO In CACAO [63], which was derived from the theory of FALQON, we consider the following Hamiltonian Ht = NX i=1 βY i (t)mY i , (16) with the parameters βY i (t) = 2mX i ∂HP ∂mZ i .(17) In the previous study [63], faster convergence than that of FALQON was confirmed, whereas the quality of solutions was worse than that of FALQON
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III A enables to develop the higher-order theory of CACAO (HOT- CACAO)

HOT-CACAO The general idea introduced in Sec. III A enables to develop the higher-order theory of CACAO (HOT- CACAO). For example, HOT-CACAO with the second- order counterdiabatic term uses the following Hamilto- nian Ht = NX i=1 βY i (t)mY i + NX i,j=1 (i̸=j) βij(t)(mY i mZ j +m Z i mY j ),(18) with the parameters βY i (t) = 2mX i ∂HP ∂mZ i , βij(t) = 2m...
[7]

HOT-CACAO+ Finally, we introduce HOT-CACAO+, which includes all the terms discussed in this paper. The Hamiltonian is given by Ht =HP + X W={X,Y} NX i=1 βW i (t)mW i + NX i,j=1 (i̸=j) βij(t)(mY i mZ j +m Z i mY j ), (20) with the parameters βX i (t) =−2m Y i ∂HP ∂mZ i , βY i (t) = 2mX i ∂HP ∂mZ i , βij(t) = 2mX i mZ j ∂HP ∂mZ i . (21) This algorithm has t...
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