Classical algorithm inspired by the feedback-based algorithm for quantum optimization and local counterdiabatic driving

Takuya Hatomura

REVIEW 3 major objections 4 minor 26 references

Classical algorithm inspired by the feedback-based algorithm for quantum optimization and local counterdiabatic driving

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A classical spin-dynamics algorithm, CACAO, derived from Lyapunov feedback and local counterdiabatic driving, converges to lower-energy states than quantum annealing, FALQON, and CD-FQA on a 2-SAT-related spin glass, and scales to 10,000…

desk verdict A clean classical spin-dynamics heuristic, but the benchmark always has all-up as the planted optimum and no classical baselines are compared, so the strong performance claim is not yet supported. read the letter →

arxiv 2506.09214 v1 pith:ISAG7O5P submitted 2025-06-10 quant-ph cond-mat.stat-mech

Takuya Hatomura This is my paper · ORCID

classification quant-phcond-mat.stat-mech

keywords combinatorialoptimizationQUBOIsingspinglassquantum-inspiredclassicalalgorithmcounterdiabaticdrivingLyapunovcontrolFALQON2-SAT

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

This paper proposes CACAO, a classical algorithm for combinatorial optimization that takes the feedback structure of a quantum algorithm (FALQON) and combines it with local counterdiabatic driving. The paper's central claim is that this classical spin dynamics converges to lower-energy states of a 2-SAT-related Ising spin glass than quantum annealing, FALQON, and CD-FQA do in a head-to-head numerical benchmark. If correct, the claim matters because it means the useful part of these quantum heuristics is a Lyapunov feedback rule that can run on ordinary hardware with per-timestep cost between $O(NT)$ and $O(N^2T)$, sidestepping the measurement and feedback overhead of the quantum versions. The paper further claims the algorithm works at 10,000 spins with convergence time nearly independent of system size, while also showing that near-degenerate target states slow it down according to $T \propto \Delta E^{-1.08}$.

What carries the argument

The central object is a classical spin dynamics on the Bloch expectation values of a product state, driven by a local Pauli-Y Hamiltonian $\hat H_{\rm CACAO}(\{\alpha_i\})=\sum_i \alpha_i(t)\hat Y_i$. The coefficient on each spin is set by quantum Lyapunov control, $\alpha_i(t)=2(h_i+\sum_{j:(i,j)\in E}J_{ij}m_j^Z)m_i^X$, which is exactly the commutator-feedback rule used in FALQON reduced to a single site. Because the driver is purely local, no entanglement builds up and the Schr\"odinger evolution reduces to the two coupled ODEs $\dot m_i^X=2\alpha_i m_i^Z$ and $\dot m_i^Z=-2\alpha_i m_i^X$; solving these ODEs is the entire algorithm. The Lyapunov structure guarantees $\frac{d}{dt}\langle \hat H_P\rangle = -\sum_i \alpha_i^2 \le 0$, so the energy descends monotonically until the feedback coefficients vanish.

What would settle it

Run the Sec. III B benchmark with an additional control: the same spin ODEs but with $\alpha_i(t)$ replaced by fixed random values of comparable magnitude; if CACAO and the random-coefficient dynamics reach the same final energies on the same instances, the Lyapunov feedback is not the operative mechanism and the paper's central explanation is falsified.

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Extended reading notes

Core claim

On its own terms, the paper establishes that one can take the feedback-based quantum optimization idea, restrict the driving to local $\hat Y_i$ terms with amplitudes set by quantum Lyapunov control, and obtain a closed set of classical equations for the spin expectation values $m_i^X(t)$ and $m_i^Z(t)$: $\alpha_i(t)=2(h_i+\sum_{j:(i,j)\in E} J_{ij} m_j^Z)m_i^X$, $\dot m_i^X=2\alpha_i m_i^Z$, $\dot m_i^Z=-2\alpha_i m_i^X$. The Lyapunov choice makes the problem energy $\langle \hat H_P\rangle$ decrease monotonically. On ten random $L=3$ ($N=9$) instances of a spin glass related to 2-SAT, CACAO achieved the best final energy among QA, FALQON, and CD-FQA across operation times, and on systems up to $N=10{,}000$ the rescaled residual energy stayed at the same level, meaning the algorithm's per-spin performance does not degrade with size. The paper also reports a two-spin convergence-time law $T=\exp(0.5237)\,\Delta E^{-1.08}$, indicating that CACAO, like any gap-sensitive heuristic, becomes slow when the ground state is nearly degenerate.

Load-bearing premise

The load-bearing premise is that the local, Lyapunov-chosen Pauli-Y nudges guide the continuous spin variables to discrete low-energy configurations rather than stranding them in poor local minima of the relaxed dynamics; the paper tests this only on one family of randomly generated 2-SAT instances.

Editorial extensions

If this is right

CACAO is a deterministic classical heuristic with per-step cost $O(NT)$ to $O(N^2T)$, so the benchmark results transfer directly to ordinary computers; the paper demonstrates this at $N=10{,}000$.
The two-spin convergence law $T=\exp(0.5237)\Delta E^{-1.08}$ predicts that CACAO's runtime grows as the target gap closes, so instances with near-degenerate optima will require long integration times.
Because the construction uses only local fields, the same Lyapunov-feedback scheme extends to higher-order binary optimization (HUBO) without new theoretical machinery, as the paper explicitly notes.
In the reported benchmark, CACAO's final energy is at least as good as the three quantum algorithms at every tested operation time; since the comparison excludes measurement and initialization costs for the quantum algorithms, the practical advantage in wall-clock terms would be larger, not smaller.

Reading between the lines

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

The CACAO equations are a continuous-time, deterministic mean-field descent on the Ising energy; a natural next test, not in the paper, is direct comparison with classical local search and simulated annealing on the same 2-SAT instances.
Because the dynamics is deterministic and cheap, adding random initial tilts $\delta_i$ gives a restart-based heuristic; the paper fixes $\delta_i=0$, leaving the restart-vs-run-time trade-off unexplored.
The residual energy growing extensively with $N$ suggests CACAO is a low-energy-state finder rather than an exact solver; using its converged $m_i^Z$ as a warm start for a discrete local search is a testable extension.
The same Lyapunov rule may transfer to other QUBO classes such as MaxCut or the Sherrington-Kirkpatrick model; this is a direct consequence of the algorithm's derivation, though the paper does not test it.

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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

Colleague,

The punchline: Hatomura derives a classical spin-dynamics heuristic (CACAO) from Lyapunov control with a local Pauli-Y driver, and the math is clean—the energy decreases monotonically and the algorithm is straightforwardly O(NT) to O(N^2 T). But the performance section is much weaker than the abstract suggests. The benchmark family in Eq. (7) is a random 2-SAT-like model where the all-up bitstring is always the planted optimum: every clause is drawn from {(0,1),(1,0),(1,1)}, so the all-up assignment satisfies all clauses and each clause term is nonnegative. The paper actually says this explicitly. That means the problem is a biased ferromagnet-like landscape with a known trivial solution, not a generic spin glass. On top of that, there is no comparison to standard classical heuristics—simulated annealing, local search, or even plain gradient descent on the soft-spin energy. As the stress-test note points out, Eqs. (4)-(5) are exactly a coordinate-scaled continuous-time gradient descent on E(mZ), so the paper is implicitly comparing a particular classical descent against three quantum heuristics on an easy instance family. The metric is also the soft-spin energy EP(T), not a rounded discrete energy or success probability, so even the 'convergence' claim is about the relaxation, not the actual binary objective.

Give credit where it's due. The derivation is correct and clearly presented. The energy-decrease property is guaranteed by Lyapunov control, and the discussion honestly acknowledges that no quantum-vs-classical conclusion should be drawn yet. The two-spin scaling analysis is a nice illustrative result, though it's a fit to one small system with two free constants. The paper is also refreshingly explicit about the costs of feedback in FALQON/CD-FQA that it ignores in the numerics.

The soft spots are real and load-bearing for the central claim. The 'best convergence' headline only holds against three quantum heuristics on a family where all-up is the ground state; it does not establish CACAO as a competitive general QUBO heuristic. The absence of any classical baseline is the biggest gap. A referee should ask for comparisons to simulated annealing or gradient descent, and benchmarks on problems without a planted all-up optimum (e.g., MaxCut, random Ising spin glasses, or weighted Max-2-SAT with random fields). The convergence-time scaling also needs more evidence if it's meant to be a general statement.

Who is this for? People working on quantum-inspired optimization and on benchmarking quantum heuristics might find CACAO a useful reference point. It's a legitimate proposal and the paper is honest, so it deserves peer review, but the performance claims need substantial strengthening before they should be taken as evidence of a competitive algorithm.

Recommendation: send it to review, with the expectation of major revision. The theory section is fine; the benchmark section needs more baselines and more diverse instances.

Best.

Referee Report

3 major / 4 minor

Summary. The paper proposes CACAO, a classical algorithm for QUBO/Ising optimization that is inspired by quantum Lyapunov control and local counterdiabatic driving. The algorithm evolves soft-spin variables according to Eqs. (4)-(5), with the coefficient alpha_i chosen so that the problem energy decreases monotonically. The paper demonstrates the dynamics on a two-spin example, compares CACAO with QA, FALQON, and CD-FQA on a 2-SAT-related Ising spin-glass model (Sec. III B), and reports results for systems up to 10,000 spins (Sec. III C). The main claim is that CACAO shows the best convergence to low-energy states among the tested algorithms on this benchmark.

Significance. The algorithm is simple, rigorously derived, and has modest classical resource requirements (O(NT) to O(N^2T)). The monotonic energy-decrease property and the absence of divergence are genuine strengths. However, the central performance claim is not established by the evidence presented: the benchmark family is degenerate, the algorithm is not compared with standard classical heuristics, and the reported energy is a continuous soft-spin quantity rather than a rounded discrete solution. If the paper is revised to include proper baselines and discrete metrics, the result could be a useful contribution; as it stands, the significance is mainly as a derivation of a classical spin-dynamics heuristic rather than a demonstrated competitive optimizer.

major comments (3)

[III B, Eq. (7), Fig. 3] The benchmark construction in Eq. (7) is degenerate: each clause is drawn from {(0,1),(1,0),(1,1)}, so the all-zero bitstring (all-up spin state) satisfies every clause and, because each C_ij is nonnegative, is always a global optimum with energy zero. Thus the comparison in Fig. 3 only shows that CACAO beats QA, FALQON, and CD-FQA on a family where the optimal solution is a single fixed bitstring, with only 10 instances at N=9 and no classical baseline such as simulated annealing, simple gradient descent, or even the trivial all-up assignment. This does not support the broader claim that CACAO is a competitive general QUBO heuristic.
[II, Eqs. (4)-(5)] The paper does not identify that the CACAO dynamics is equivalent to a coordinate-scaled continuous-time gradient descent on the soft-spin energy E(mZ)=sum J_ij mZ_i mZ_j + sum h_i mZ_i. Using mX_i^2 + mZ_i^2 = 1, Eq. (5) with Eq. (4) gives dmZ_i/dt = -4(1-mZ_i^2) dE/dmZ_i. Since this is a known classical optimization structure, the paper should either compare CACAO with vanilla gradient descent on the same soft-spin energy or explain precisely what the counterdiabatic-inspired coefficient choice adds over that baseline. Without such a comparison, the claimed advantage over classical methods is not demonstrated.
[III C, Fig. 4] The large-system results report only the continuous soft-spin energy EP(T)/N and do not provide any discrete metric, such as the Hamming distance to the planted all-up optimum, the success probability, or the energy after rounding the final mZ_i to bit values. Since all-up is exactly the ground state for every instance, a trivial algorithm returning all-up would achieve the global optimum, so the statement that CACAO 'worked even for the system with 10,000 spins' is not supported without a comparison to trivial or standard classical baselines. The residual energy per spin being roughly constant with N does not establish solution quality.

minor comments (4)

[III A, Fig. 2] The scaling fit T = exp(0.5237) Delta E^{-1.08} is presented as a finding, but the fitting procedure, the number of data points, and the goodness of fit are not reported; the abstract's later statement that convergence time 'might scale as O(Delta E^{-1})' should be labeled as a numerical observation, not a derived result.
[III B] The text contains typos such as 'squre lattice' and 'F ALQON' (with a space); the algorithm name should be typeset consistently as FALQON.
[III C] The claim that 'the time for convergence does not depend on the system size' is inferred from Fig. 4, but the figure plots energy versus operation time and does not explicitly define or extract a convergence time; please specify the convergence criterion used.
[IV] The discussion of feedback costs for FALQON and CD-FQA is useful, but it is qualitative; since the numerical comparison already ignores these costs in favor of the quantum algorithms, the conclusion that CACAO is competitive would be strengthened by a quantitative accounting or by a clearly stated caveat.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: CACAO's dynamics is derived from Lyapunov control equations, and its benchmark comparisons are not forced by construction.

full rationale

The derivation chain is self-contained. In Sec. II, the algorithm sets alpha_i(t) via the Lyapunov condition alpha_i = i<[H_P,Y_i]>, which yields Eq. (4), and integrates the classical spin equations (5). These equations are not assumed to contain the low-energy solution that is later reported; they are a well-defined dynamical system whose behavior is then simulated. The benchmark in Sec. III B uses the same energy function as the optimization objective, but that is the intended target of any optimization heuristic, not a prediction that is equivalent to an input by construction. The fitted scaling T = exp(0.5237) Delta E^{-1.08} in Fig. 2 is a post-hoc observation from two-spin runs and is not used as an input to the algorithm or to the benchmark. Self-citations in the paper are contextual or comparative rather than load-bearing: the local Pauli-Y driver is justified by external results on low-order counterdiabatic terms [22,24], and no uniqueness theorem from the authors' prior work is invoked to force the algorithm's choice. The degenerate nature of the Eq. (7) benchmark family, where the all-up state is always a planted ground state, and the absence of classical baselines such as simulated annealing are substantive concerns about benchmark quality and external validity, but they do not make the derivation circular. Therefore the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The algorithm has no fitted internal parameters; the only fitted numbers are in the empirical convergence-time scaling. The main assumptions are the product-state initial condition, the suitability of a local Y-driver, and the sufficiency of monotonic energy decrease for optimization.

free parameters (1)

Convergence-scaling fit constants = A = exp(0.5237), exponent = 1.08
Obtained by fitting the two-spin data in Fig. 2; the paper uses this fit to claim T ~ ΔE^{-1.08}. These constants are not inputs to the CACAO algorithm itself.

assumptions (4)

standard math Schrodinger equation and spin-1/2 commutation relations
Used to derive Eqs. (4) and (5) from the commutator formula for the Lyapunov feedback control.
domain assumption Initial state is a product state with mX_i=1 and mZ_i=0 for all i
Stated in Sec. II and Sec. III B; the product-state property is preserved because the driver is local, but the specific initial condition is a heuristic choice. When all longitudinal fields vanish, a nonzero initial angle delta_i is needed.
domain assumption The local Pauli-Y driver (3) is a suitable approximate counterdiabatic Hamiltonian for optimization
The paper imports the lowest-order local counterdiabatic approximation from [22]. This approximation is not proven optimal for the optimization task, and the algorithm's performance depends on it empirically.
ad hoc to paper Monotonic energy decrease is sufficient to reach good QUBO solutions
The Lyapunov control guarantees only nonincrease of energy, implying convergence to a local minimum of the continuous relaxation. Whether this suffices for global optimization is not proven and is tested only on a narrow benchmark.

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Cite this review

Pith. "Pith review of Classical algorithm inspired by the feedback-based algorithm for quantum optimization and local counterdiabatic driving." pith.science (2026). https://pith.science/paper/ISAG7O5P

@misc{pith2026250609214,
  author       = {Pith},
  title        = {Pith review of: Classical algorithm inspired by the feedback-based algorithm for quantum optimization and local counterdiabatic driving},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISAG7O5P}},
  note         = {Machine review of arXiv:2506.09214}
}

abstract

We propose a quantum-inspired classical algorithm for combinatorial optimization problems, named the counterdiabaticity-assisted classical algorithm for optimization (CACAO). In this algorithm, a solution of a given combinatorial optimization problem is heuristically searched with classical spin dynamics based on quantum Lyapunov control of local counterdiabatic driving. We compare the performance of CACAO with that of quantum time-evolution algorithms, i.e., quantum annealing, the feedback-based algorithm for quantum optimization (known as FALQON), and the counterdiabatic feedback-based quantum algorithm (known as CD-FQA). We also study the performance of CACAO applied to large systems up to $10,000$ spins.

Figures

Figures reproduced from arXiv: 2506.09214 by the authors.

**Figure 1.** FIG. 1. Time evolution of the two-spin system with CACAO. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 4.** FIG. 4. The rescaled energy cost for CACAO against the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 21 canonical work pages

[1]

Schrijver, On the history of combinatorial optimiza- tion (till 1960), Handbooks in Operations Research and Management Science 12, 1 (2005)

A. Schrijver, On the history of combinatorial optimiza- tion (till 1960), Handbooks in Operations Research and Management Science 12, 1 (2005)

work page 2005
[2]

Lucas, Ising formulations of many np problems, Fron- tiers in Physics 2, 5 (2014)

A. Lucas, Ising formulations of many np problems, Fron- tiers in Physics 2, 5 (2014)

work page 2014
[3]

Naseri and M

G. Naseri and M. A. Koffas, Application of combinato- rial optimization strategies in synthetic biology, Nature Communications 11, 2446 (2020)

work page 2020
[4]

Yarkoni, E

S. Yarkoni, E. Raponi, T. B¨ ack, and S. Schmitt, Quan- tum annealing for industry applications: introduction and review, Reports on Progress in Physics 85, 104001 (2022)

work page 2022
[5]

Kochenberger, J

G. Kochenberger, J. K. Hao, F. Glover, M. Lewis, Z. L¨ u, H. Wang, and Y. Wang, The unconstrained bi- nary quadratic programming problem: A survey, Journal of Combinatorial Optimization 28, 58 (2014)

work page 2014
[6]

Kadowaki and H

T. Kadowaki and H. Nishimori, Quantum annealing in the transverse ising model, Physical Review E 58, 5355 (1998)

work page 1998
[7]

Kato, On the adiabatic theorem of quantum me- chanics, Journal of the Physical Society of Japan 5, 435 (1950)

T. Kato, On the adiabatic theorem of quantum me- chanics, Journal of the Physical Society of Japan 5, 435 (1950)

1950
[8]

Jansen, M.-B

S. Jansen, M.-B. Ruskai, and R. Seiler, Bounds for the adiabatic approximation with applications to quan- tum computation, Journal of Mathematical Physics 48, 102111 (2007)

work page 2007

Show all 26 references

[9]

A. B. Magann, K. M. Rudinger, M. D. Grace, and M. Sarovar, Feedback-based quantum optimization, Physical Review Letters 129, 250502 (2022)

2022
[10]

A. B. Magann, K. M. Rudinger, M. D. Grace, and M. Sarovar, Lyapunov-control-inspired strategies for quantum combinatorial optimization, Physical Review A 106, 062414 (2022)

2022
[11]

Cong and F

S. Cong and F. Meng, A survey of quantum lyapunov con- trol methods, The Scientific World Journal 2013, 967529 (2013)

2013
[12]

Torrontegui, S

E. Torrontegui, S. Ib´ a˜ nez, S. Mart ´ ınez-Garaot, M. Mod- ugno, A. del Campo, D. Gu´ ery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, Shortcuts to adiabaticity, Ad- vances In Atomic, Molecular, and Optical Physics 62, 117 (2013)

2013
[13]

Gu´ ery-Odelin, A

D. Gu´ ery-Odelin, A. Ruschhaupt, A. Kiely, E. Tor- rontegui, S. Mart ´ ınez-Garaot, and J. G. Muga, Short- cuts to adiabaticity: Concepts, methods, and applica- tions, Reviews of Modern Physics 91, 045001 (2019)

2019
[14]

T. Hatomura, Shortcuts to adiabaticity: theoretical framework, relations between different methods, and ver- satile approximations, Journal of Physics B: Atomic, Molecular and Optical Physics 57, 102001 (2024). 6

2024
[15]

Demirplak and S

M. Demirplak and S. A. Rice, Adiabatic population transfer with control fields, The Journal of Physical Chemistry A 107, 9937 (2003)

2003
[16]

Demirplak and S

M. Demirplak and S. A. Rice, On the consistency, ex- tremal, and global properties of counterdiabatic fields, The Journal of Chemical Physics 129, 154111 (2008)

2008
[17]

M. V. Berry, Transitionless quantum driving, Journal of Physics A: Mathematical and Theoretical 42, 365303 (2009)

2009
[18]

Kolodrubetz, D

M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Physics Reports 697, 1 (2017)

2017
[19]

Hatomura and K

T. Hatomura and K. Takahashi, Controlling and explor- ing quantum systems by algebraic expression of adiabatic gauge potential, Physical Review A 103, 012220 (2021)

2021
[20]

R. K. Malla, H. Sukeno, H. Yu, T.-C. Wei, A. Weichsel- baum, and R. M. Konik, Feedback-based quantum algo- rithm inspired by counterdiabatic driving, Physical Re- view Research 6, 043068 (2024)

2024
[21]

Chandarana, K

P. Chandarana, K. Paul, K. R. Swain, X. Chen, and A. del Campo, Lyapunov controlled counterdiabatic quantum optimization, arXiv:2409.12525 (2024)

2024 arXiv
[22]

P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Floquet-engineering counterdiabatic protocols in quan- tum many-body systems, Physical Review Letters 123, 090602 (2019)

2019
[23]

M. R. Krom, The decision problem for a class of first- order formulas in which all disjunctions are binary, Math- ematical Logic Quarterly 13, 15 (1967)

1967
[24]

Sels and A

D. Sels and A. Polkovnikov, Minimizing irreversible losses in quantum systems by local counterdiabatic driving., Proceedings of the National Academy of Sciences of the United States of America 114, E3909 (2017)

2017
[25]

Hatomura and T

T. Hatomura and T. Mori, Shortcuts to adiabatic classi- cal spin dynamics mimicking quantum annealing, Physi- cal Review E 98, 032136 (2018)

2018
[26]

Hatomura, Iterative classical superadiabatic algorithm for combinatorial optimization, Journal of Physics A: Mathematical and Theoretical 53, 205302 (2020)

T. Hatomura, Iterative classical superadiabatic algorithm for combinatorial optimization, Journal of Physics A: Mathematical and Theoretical 53, 205302 (2020)

2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Schrijver, On the history of combinatorial optimiza- tion (till 1960), Handbooks in Operations Research and Management Science 12, 1 (2005)

A. Schrijver, On the history of combinatorial optimiza- tion (till 1960), Handbooks in Operations Research and Management Science 12, 1 (2005)

work page 2005

[2] [2]

Lucas, Ising formulations of many np problems, Fron- tiers in Physics 2, 5 (2014)

A. Lucas, Ising formulations of many np problems, Fron- tiers in Physics 2, 5 (2014)

work page 2014

[3] [3]

Naseri and M

G. Naseri and M. A. Koffas, Application of combinato- rial optimization strategies in synthetic biology, Nature Communications 11, 2446 (2020)

work page 2020

[4] [4]

Yarkoni, E

S. Yarkoni, E. Raponi, T. B¨ ack, and S. Schmitt, Quan- tum annealing for industry applications: introduction and review, Reports on Progress in Physics 85, 104001 (2022)

work page 2022

[5] [5]

Kochenberger, J

G. Kochenberger, J. K. Hao, F. Glover, M. Lewis, Z. L¨ u, H. Wang, and Y. Wang, The unconstrained bi- nary quadratic programming problem: A survey, Journal of Combinatorial Optimization 28, 58 (2014)

work page 2014

[6] [6]

Kadowaki and H

T. Kadowaki and H. Nishimori, Quantum annealing in the transverse ising model, Physical Review E 58, 5355 (1998)

work page 1998

[7] [7]

Kato, On the adiabatic theorem of quantum me- chanics, Journal of the Physical Society of Japan 5, 435 (1950)

T. Kato, On the adiabatic theorem of quantum me- chanics, Journal of the Physical Society of Japan 5, 435 (1950)

1950

[8] [8]

Jansen, M.-B

S. Jansen, M.-B. Ruskai, and R. Seiler, Bounds for the adiabatic approximation with applications to quan- tum computation, Journal of Mathematical Physics 48, 102111 (2007)

work page 2007