Pith. sign in

REVIEW 3 major objections 4 minor 98 references

A fidelity metric for quantum annealing benchmarked by extreme scaling quantum Monte-Carlo simulations

T0 review · 3 major / 4 minor · reviewed 2026-07-14 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Quantum annealers should be judged by equation-of-state accuracy, and classical simulations already beat current Rydberg hardware by orders of magnitude.

desk verdict Solid process-level fidelity metric for QA plus extreme-scale classical benchmarks that already beat estimated Rydberg hardware; the non-cancellation premise is untested but does not sink the numerics. read the letter →

arxiv 2606.26233 v2 pith:P3ODXZLL submitted 2026-06-24 quant-ph cond-mat.str-el

Gabriel Gouraud , Miha Srdinsek , Xavier Waintal This is my paper · ORCID
classification quant-phcond-mat.str-el
keywords quantumannealingfidelitymetricequationofstateRydbergatomsvariationalMonteCarloGreenfunctionsimulatedQUBO
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

Quantum annealers are usually scored only by whether they solve an optimization problem. The authors argue that this is brittle and propose instead a fidelity metric for the annealing process itself: the relative accuracy ε with which the device reconstructs the equation of state E(hx) while the transverse field is ramped. Using two classical quantum Monte-Carlo methods on Rydberg-atom models, they show that a simple thermal-like variational ansatz already matches the quantum ground state to ε ~ 10^{-2}–10^{-3} up to 100 million atoms, while a Green-function method reaches ~10^{-4} up to 100 thousand atoms. These classical benchmarks already outperform present experimental platforms in both size and precision, so any claimed quantum advantage must beat this bar. The same metric also diagnoses whether failures come from noise or from intrinsically vanishing gaps.

What carries the argument

The equation-of-state accuracy ε = δE/E, evaluated along the annealing path by variational Monte-Carlo with a thermal-inspired ansatz (and by Green-function Monte-Carlo as a higher-accuracy reference).

What would settle it

Measure E(hx) on a Rydberg annealer for the same lattice paths used in the paper and check whether the experimental ε matches or beats the classical 10^{-3}–10^{-4} benchmarks; if it does not, the hardware fails the proposed fidelity test even when final QUBO scores look acceptable.

Watch

Extended reading notes

Core claim

The accuracy ε = δE/E of the annealing equation of state E(hx) is a direct fidelity metric for quantum annealing, analogous to gate fidelity. With a thermal-inspired variational Monte-Carlo ansatz, a quantum annealer is indistinguishable from its classical thermal counterpart once ε reaches ~10^{-2}–10^{-3}; Green-function Monte-Carlo tightens this to ~10^{-4}. Both methods scale far beyond current Rydberg hardware and therefore set quantitative performance targets that future devices must clear.

Load-bearing premise

The claim that errors spoiling the equation of state cannot cancel in a way that still leaves the final optimization answer correct.

Editorial extensions

If this is right

  • Hardware groups can report ε alongside success probability, separating noise/decoherence from Landau-Zener failures.
  • Any future claim of quantum advantage on Rydberg annealers must demonstrate an equation of state more accurate than the classical simulations already achieved here.
  • A thermal classical machine that realises the effective Jeff couplings of the variational ansatz would already match ideal quantum annealing to the stated precision.
  • The same ε diagnostic can be applied to superconducting or other annealer platforms without requiring a full QUBO benchmark suite.

Reading between the lines

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

  • If experimental ε stays stuck near a few percent, the 1/r^6 tails of Rydberg interactions may permanently limit how faithfully an industrial QUBO matrix can be embedded.
  • Pushing the thermal ansatz to unrestricted Jeff or multi-body terms could systematically generate new classical heuristics that inherit whatever quantum advantage exists.
  • The Gap-FMC construction itself is a reusable classical tool for locating the hard gap-closing points that any annealer must confront.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a process-oriented fidelity metric for quantum annealing: the relative accuracy ε = δE/E of the equation of state E(h_x) ≡ ⟨Ĥ⟩ reconstructed during the anneal (via Z- and X-basis measurements). It argues this is more robust than end-of-anneal QUBO success rates. Benchmarks are obtained for Rydberg-atom transverse-field Ising models via two classical methods: (i) variational Monte Carlo with a thermal-annealing-inspired ansatz (simple and staggered effective energies), reaching N = 10^8 atoms at ε ∼ 10^{-2}–10^{-3} on a single CPU, and (ii) Green-function Monte Carlo projecting the same ansatz, reaching ε ∼ 10^{-4} up to N = 10^5. The same tools are applied to five N = 50 “fallen-angel” QUBO instances. The authors conclude that, within the quoted ε, a quantum annealer is indistinguishable from its thermal classical counterpart and that the classical results already outperform current Rydberg platforms in size and precision, thereby constraining future hardware.

Significance. If the metric is accepted as a faithful diagnostic, the work supplies a concrete, hardware-agnostic figure of merit analogous to gate fidelity and a set of extreme-scale classical reference values that any future Rydberg (or other) annealer must beat. The numerical campaign itself is a clear strength: multiple independent error estimators (GFMC imaginary-time traces, zero-noise extrapolation, V-score, perturbation theory, corner-defect analysis), an original Gap-FMC gap estimator, and reproducible scaling to 10^8 atoms with a lightweight, symmetry-aware ansatz. These results are immediately useful as classical baselines even if the interpretive claims about hardware constraints are moderated.

major comments (3)
  1. [Introduction; Sec. V, Fig. 7] The central interpretive claim—that classical VMC/GFMC already “outperform current Rydberg o platforms o and put severe constraints for future hardware”—rests on the untested premise (Introduction) that errors spoiling the equation of state “are unlikely to compensate and restore the accuracy of the final result.” Section V and Fig. 7 report both δE/E0 and the QUBO residual δF/F for the fallen-angel instances, yet never show the joint distribution or any instance in which δE/E0 remains O(10^{-2}) while δF/F reaches zero. Without that correlation the non-cancellation axiom is unsupported and the hardware-constraint language is not justified by the data.
  2. [Fig. 3; abstract; Sec. VI] The experimental comparison (orange rectangle in Fig. 3, abstract, conclusion) is an informal estimate of “noise levels observed in the cited literature and private discussions,” not a measured experimental ε. Because no published Rydberg data reconstruct E(h_x), the claim of “orders of magnitude” superiority in precision cannot be verified from the manuscript and should be rephrased as a prediction of the precision that future experiments must reach.
  3. [Sec. IV B, Eq. (16), Fig. 5] The staggered thermal ansatz (Eq. 16) is translationally invariant while the open-boundary Rydberg systems are not; the authors themselves note that this shifts the apparent location of the AF–PM transition (Fig. 5). For the metric to be used as a hardware diagnostic, the classical reference must either incorporate the same open boundaries or quantify the systematic bias this approximation introduces into ε near the gap closing.
minor comments (4)
  1. [Figs. 1, 3, 5] Several figures use encoding artifacts (“ans¨ atze”, “δE/E 0”) that should be cleaned for production.
  2. [Appendix G] The definition of the order parameter M (Eq. 15) is given for both lattices, but the triangular-lattice hysteresis (Appendix G) is only briefly mentioned; a short statement of how the first-order character affects the reliability of ε near the jump would help readers.
  3. [Appendix F, Fig. 3] The truncation radius R is chosen so that the 1/R^4 error lies below the reported ε (Appendix F); stating the precise R values used for each N in the main text or a table would improve reproducibility.
  4. [Appendix D] Gap-FMC (Appendix D) is a useful original tool; a one-sentence comparison with existing QMC gap estimators would place it in context.

Circularity Check

1 steps flagged · score 1.0 of 10

No load-bearing circularity: the fidelity metric ε and the VMC/GFMC benchmarks are computed independently of experimental data or final QUBO scores; self-citations supply only implementation details.

  1. self citation load bearing [Sec. III, paragraph after Eq. (13); also Sec. IV references to perceptrain]
    "Our implementations of both VMC and GFMC are described in [69]. … a highly accurate calculation at N=100, h_x=-0.45 using a more sophisticated ansatz. This is a NQS ansatz built with perceptrains using the technique described in [69]."

    The only self-citations supply the Monte-Carlo engine and a small-N reference calculation. They are not required for the definition of ε, the thermal ansatz, the N=10^8 scaling, or the central claim that classical simulations already reach experimental precision; the citations are therefore non-load-bearing and raise the score by at most 1.

full rationale

The paper defines ε = δE/E as the relative accuracy of the annealer equation of state E(h_x) ≡ ⟨H⟩ and obtains numerical values by standard variational Monte Carlo (thermal ansatz of Eqs. 10–16) followed by Green-function Monte Carlo projection. Both procedures sample the transverse-field Ising Hamiltonian from first principles; variational parameters (effective temperatures) are optimized against the variational energy itself, not against experimental spectra or final QUBO bit-strings. The claim that a quantum annealer is indistinguishable from its thermal counterpart inside ε ∼ 10^{-2}–10^{-3} is simply the observed residual of that ansatz relative to GFMC (or zero-noise extrapolation). Self-citations to the authors’ earlier Monte-Carlo framework [69] appear only for code-level details of VMC/GFMC and for a high-accuracy perceptrain reference at N = 100; they do not underwrite the definition of ε, the thermal ansatz, the large-N scaling, or the Gap-FMC gap estimator (presented as original). No uniqueness theorem is imported, no fitted experimental parameter is re-labeled a prediction, and the metric is not defined in terms of the final QUBO residual δF/F. The non-cancellation premise that spoils the equation of state cannot be compensated by a correct final bit-string is an untested modeling assumption, not a circular reduction. Consequently the derivation chain is self-contained against external benchmarks and exhibits only the most minor, non-load-bearing self-citation.

Assumptions & free parameters 3 free parameters · 3 assumptions · 2 invented entities

The central claim rests on standard quantum-Monte-Carlo convergence, the definition of ε as relative energy error, a handful of numerical cut-offs chosen for computational convenience, and the modelling assumption that equation-of-state accuracy is a faithful proxy for annealing fidelity. No new physical particles or forces are postulated.

free parameters (3)
  • interaction truncation radius R = R=8.7 or 4.1
    Set to 8.7 (N≤10^4) or 4.1 (N>10^4) so that the O(1/R^4) truncation error stays below the reported ε; chosen by hand for computational cost.
  • number of Monte-Carlo walkers n_w = n_w ~ 10^4 (or 1 for N=10^8 VMC)
    Typically 10^4 for GFMC and smaller sizes; reduced to 1 for the largest VMC runs by self-averaging. Controls statistical error.
  • variational effective temperatures β_eff, β_k = optimized per h_x (see Fig. 8)
    Optimized by gradient descent for each h_x; for N>1600 the N=1600 values are frozen. Directly determine the quality of the thermal ansatz.
assumptions (3)
  • domain assumption The transverse-field Ising Hamiltonian is sign-problem free, so VMC and GFMC converge to the exact ground-state energy in the infinite-sample limit.
    Stated in Sec. III; standard for this model class.
  • ad hoc to paper Relative error in the equation of state E(h_x) is a faithful proxy for the quality of the adiabatic process (errors do not cancel to restore final QUBO accuracy).
    Core methodological claim of the Introduction; not derived from a theorem.
  • standard math Landau–Zener and adiabatic theorems correctly describe the dominant non-adiabatic errors near gap closings.
    Used throughout Sec. II and to interpret the location of maximum ε.
invented entities (2)
  • fidelity metric ε = δE/E for the QA equation of state
    purpose: Provide a process-level figure of merit analogous to gate fidelity, independent of any particular QUBO instance.
    Defined in the abstract and Sec. I; no prior literature uses this exact quantity as the primary QA fidelity.
  • Gap-FMC estimator
    purpose: Extract the spectral gap from the imaginary-time energy decay of a GFMC run.
    Introduced in Appendix D as original; used to locate phase transitions and gap closings.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A fidelity metric for quantum annealing benchmarked by extreme scaling quantum Monte-Carlo simulations." pith.science (2026). https://pith.science/paper/P3ODXZLL

@misc{pith2026260626233,
  author       = {Pith},
  title        = {Pith review of: A fidelity metric for quantum annealing benchmarked by extreme scaling quantum Monte-Carlo simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3ODXZLL}},
  note         = {Machine review of arXiv:2606.26233}
}
abstract

Quantum annealers are supposed to follow adiabatically the ground state of a system as its Hamiltonian slowly interpolates between a trivial phase and a non-trivial one; the non-trivial ground state being the solution to an optimization problem. Overwhelmingly, their performances are measured in terms of how well or fast the optimization problem is solved. While pragmatic, this approach is inherently brittle as it strongly depends on the problem considered and the classical algorithm used as the reference benchmark. Here, we propose a quantity that not only measures the end result but also the quality of the actual quantum annealing process itself. Our metric is the quantum annealing counterpart of the fidelity-per gate of gate-based quantum computers. It takes the form of an accuracy $\epsilon$ for the equation of state of the annealer. We calculate benchmark values of $\epsilon$ using two variants of the simulated quantum annealing technique for Rydberg atoms systems. Our first approach uses variational quantum Monte-Carlo with an ansatz inspired by thermal annealing. It suggests that within $\epsilon \sim 10^{-2}-10^{-3}$, a quantum annealer is indistinguishable from its thermal classical counterpart. Critically, we could reach this precision up to $100,000,000$ atoms on a single CPU. Our second approach (based on Green function quantum Monte-Carlo) reaches accuracies around $\epsilon \sim 10^{-4}$ and we have run it up to $100,000$ atoms. These results outperform current Rydberg atom quantum annealing experimental platforms in both precision and size by orders of magnitude and put severe constraints for future hardware.

Figures

Figures reproduced from arXiv: 2606.26233 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Sketch of the Rydberg atom phase diagrams for [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: ). Perturbation theory in hx for Hˆ = Hˆ z − hxHˆ x gives E0 = F(n˜) − h 2 x X i 1 δi + O(h 3 x ), (C1) with δi defined as δi = Jii + 2(1 − 2˜ni) X j Jijn˜j . (C2) 2. Strong field expansion (hx ≫ Jij ) At large field, the reference state is the ground state of Hˆ x, i.…
Figure 12
Figure 12. Figure 12: shows our gap estimate for the fallen angel QUBO problem of [63]. As advertised, we do observe a gap closing that takes place at vanishing (or at least very small) transverse field hx. All of the 5 individual QUBO instances we had access to had similar gap profiles. A…
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 3
Figure 3. Figure 3: The top panel of Fig. 14 shows the difference in [PITH_FULL_IMAGE:figures/full_fig_p015_3.png]
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: shows a calculation with the staggered thermal ansatz where we have performed a standard annealing (slowly decreasing hx, optimizing the effective temper￾atures at every step) as well as a backward-annealing (slowly increasing hx, also optimizing the effective tem￾per…

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

98 extracted references · 10 linked inside Pith

  1. [1]

    This route is becoming increasingly popular with, e.g., demonstra- tions of up to 2000 artificial spins in photonic systems [34] with all-to-all connectivity

    Thermal annealing The most natural counterpart to QA is to build a clas- sical machine tailored to solve QUBO problems using thermal fluctuations instead of quantum ones. This route is becoming increasingly popular with, e.g., demonstra- tions of up to 2000 artificial spins in photonic systems [34] with all-to-all connectivity. A large number of different...

    2000

  2. [2]

    Simulated thermal annealing The baseline to which many QA articles compare themselves is a simulated version of thermal annealing, often simply referred as simulated annealing (SA). In SA, one uses a Markov Chain to sample a thermal distribu- tionP(n) = (1/Z) exp[−β effF(n)] and then gradually in- crease the inverse temperatureβ eff until one has reached ...

  3. [3]

    Importantly, the QA Hamilto- nian is sign-problemfreeso that these techniques are in principle exact

    Simulated quantum annealing Simulated quantum annealing (SQA) is a generic ter- minology that encompasses various quantum Monte- Carlo techniques that aim at simulating the ground state of the QA problem itself. Importantly, the QA Hamilto- nian is sign-problemfreeso that these techniques are in principle exact. However, SQA potentially suffers from the s...

  4. [4]

    Classical heuristics Many QA applications originate from groups that had a history in solving discrete optimization problems, long before QA became fashionable. While the worst case so- lution of many of these problems is provably NP hard, hence almost certainly exponentially hard, practical in- stances have a lot of extra structure that specialized al- g...

  5. [5]

    As the importance of discrete optimization prob- lems spreads, new classes of classical algorithms are likely to emerge

    Other classical baselines The examples above do not exhaust the classical ap- proaches against which quantum annealing can be com- pared. As the importance of discrete optimization prob- lems spreads, new classes of classical algorithms are likely to emerge. Examples include a combination of belief propagation with a tensor network [49] (Ref. [49] also co...

  6. [6]

    It developed into roughly three stages

    Superconducting circuits The first implementation of QA with superconducting circuits is now already fifteen years old [53]. It developed into roughly three stages. Initially, the focus was to demonstrate that there actu- ally was some quantumness involved in the experiments. Ref. [54] prepared a superposition of the two ferromag- netic states of a small ...

  7. [7]

    EPIQ”, the PEPR “EQUBITFLY

    Rydberg atoms Rydberg atoms QA platforms are more recent than superconducting circuits and the corpus of studied appli- cations is much smaller. Each qubitiis an atom that can be placed at positionr i with optical tweezers resulting in an interaction matrix Jij = (1−δ ij) 2|ri −r j|6 −δ ijhz,(6) whereδ ij is the Kronecker symbol. The off-diagonal part res...

  8. [8]

    It is typically the antiferromagnetic state (possibly with a corner defect, see Fig

    Weak field expansion (h x ≪J ij) Let| ˜n⟩be the ground state of ˆHz. It is typically the antiferromagnetic state (possibly with a corner defect, see Fig. 9). Perturbation theory inh x for ˆH= ˆHz −h x ˆHx gives E0 =F( ˜n)−h 2 x X i 1 δi +O(h 3 x),(C1) withδ i defined as δi =J ii + 2(1−2˜ni) X j Jij ˜nj.(C2)

Show all 98 references
  1. [9]

    Apply perturbation theory to ˆH/hx = ˆHx − 1 hx ˆHz

    Strong field expansion (h x ≫J ij) At large field, the reference state is the ground state of ˆHx, i.e.,|+⟩ ⊗N. Apply perturbation theory to ˆH/hx = ˆHx − 1 hx ˆHz. The perturbation ˆHz contains terms acting on either one site ˆni or two sites ˆniˆnj. This results in the groun...

  2. [10]

    Kochenberger, J.-K

    G. Kochenberger, J.-K. Hao, F. Glover, M. Lewis, Z. L¨ u, H. Wang, and Y. Wang, The unconstrained binary quadratic programming problem: a survey, J. Comb. Op- tim.28, 58 (2014)

    2014

  3. [11]

    Kadowaki and H

    T. Kadowaki and H. Nishimori, Quantum annealing in the transverse Ising model, Phys. Rev. E58, 5355 (1998). 17

    1998

  4. [12]

    Morita and H

    S. Morita and H. Nishimori, Mathematical foundation of quantum annealing, J. Math. Phys.49, 125210 (2008)

    2008

  5. [13]

    Ohzeki and H

    M. Ohzeki and H. Nishimori, Quantum anneal- ing: An introduction and new developments, arXiv 10.48550/arXiv.1006.1696 (2010), 1006.1696

    2010 doi arXiv

  6. [14]

    Phillipson, Quantum computing in logistics and supply chain management an overview (2025), arXiv:2402.17520 [quant-ph]

    F. Phillipson, Quantum computing in logistics and supply chain management an overview (2025), arXiv:2402.17520 [quant-ph]

    2025 arXiv

  7. [15]

    T. Stollenwerket al., Quantum annealing applied to de- conflicting optimal trajectories for air traffic manage- ment, IEEE Transactions on Intelligent Transportation Systems23, 16750 (2021)

    2021

  8. [16]

    Venturelli and A

    D. Venturelli and A. Kondratyev, Reverse quantum an- nealing approach to portfolio optimization problems, Quantum Mach. Intell.1, 17 (2019)

    2019

  9. [17]

    Benedetti, J

    M. Benedetti, J. Realpe-G´ omez, R. Biswas, and A. Perdomo-Ortiz, Quantum-assisted learning of hardware-embedded probabilistic graphical models, Phys. Rev. X7, 041052 (2017)

    2017

  10. [18]

    Or´ us, S

    R. Or´ us, S. Mugel, and E. Lizaso, Forecasting finan- cial crashes with quantum computing, Phys. Rev. A99, 060301 (2019)

    2019

  11. [19]

    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 Physics85, 104001 (2022)

    2022

  12. [20]

    K. A. Smith-Miles, H. H. Hoos, H. Wang, T. B¨ ack, and T. J. Osborne, The travelling salesperson problem and the challenges of near-term quantum advantage, Quan- tum Science and Technology10, 033001 (2025)

    2025

  13. [21]

    A. D. King, J. Carrasquilla, J. Raymond, I. Ozfidan, E. Andriyash, A. Berkley, M. Reis, T. Lanting, R. Har- ris, F. Altomare, K. Boothby, P. I. Bunyk, C. En- derud, A. Fr´ echette, E. Hoskinson, N. Ladizinsky, T. Oh, G. Poulin-Lamarre, C. Rich, Y. Sato, A. Yu. Smirnov, L. J. S...

    2018

  14. [22]

    A. D. King, J. Raymond, T. Lanting, S. V. Isakov, M. Mohseni, G. Poulin-Lamarre, S. Ejtemaee, W. Bernoudy, I. Ozfidan, A. Yu. Smirnov, M. Reis, F. Altomare, M. Babcock, C. Baron, A. J. Berkley, K. Boothby, P. I. Bunyk, H. Christiani, C. Enderud, B. Evert, R. Harris, E. Hoskins...

    2021

  15. [23]

    A. D. King, S. Suzuki, J. Raymond, A. Zucca, T. Lant- ing, F. Altomare, A. J. Berkley, S. Ejtemaee, E. Hoskin- son, S. Huang, E. Ladizinsky, A. J. R. MacDonald, G. Marsden, T. Oh, G. Poulin-Lamarre, M. Reis, C. Rich, Y. Sato, J. D. Whittaker, J. Yao, R. Harris, D. A. Lidar, H....

    2022

  16. [24]

    Scholl, M

    P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, A. M. L¨ auchli, and A. Browaeys, Quantum simulation of 2D antiferromagnets with hun- dreds of Rydberg atoms, Nature595, 233 (2021)

    2021

  17. [25]

    Ebadi, T

    S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Se- meghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pich- ler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Quantum phases of matter on a 256-atom programmable quantum simulator, Nature 595, 227 (2021)

    2021

  18. [26]

    Ebadi, A

    S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar, X.-Z. Luo, B. Nash, X. Gao, B. Barak, E. Farhi, S. Sachdev, N. Gemelke, L. Zhou, S. Choi, H. Pichler, S.-T. Wang, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Qua...

    2022

  19. [27]

    Scotti, G

    R. Scotti, G. Bettonte, A. Costantini, S. Marzella, D. Ot- taviani, and S. Lodi, A clustering aggregation algo- rithm on neutral-atoms and annealing quantum proces- sors (2024), arXiv:2412.07558 [quant-ph]

    2024 arXiv

  20. [28]

    Zener, Non-adiabatic crossing of energy levels, Proc

    C. Zener, Non-adiabatic crossing of energy levels, Proc. R. Soc. London A137, 696 (1932)

    1932

  21. [29]

    S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

    1992

  22. [30]

    M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Area laws in quantum systems: Mutual infor- mation and correlations, Phys. Rev. Lett.100, 070502 (2008)

    2008

  23. [31]

    Miller and D

    J. Miller and D. A. Huse, Zero-temperature critical be- havior of the infinite-range quantum Ising spin glass, Phys. Rev. Lett.70, 3147 (1993)

    1993

  24. [32]

    van Dam, M

    W. van Dam, M. Mosca, and U. Vazirani, How Powerful is Adiabatic Quantum Computation?, arXiv 10.1109/SFCS.2001.959902 (2002), quant-ph/0206003

    2001 doi

  25. [33]

    B. W. Reichardt, The quantum adiabatic optimization algorithm and local minima, inProceedings of the Thirty- Sixth Annual ACM Symposium on Theory of Computing, STOC ’04 (Association for Computing Machinery, New York, NY, USA, 2004) p. 502–510

    2004

  26. [34]

    J¨ org, F

    T. J¨ org, F. Krzakala, J. Kurchan, and A. C. Maggs, Sim- ple Glass Models and Their Quantum Annealing, Phys. Rev. Lett.101, 147204 (2008)

    2008

  27. [35]

    M. H. S. Amin and V. Choi, First-order quantum phase transition in adiabatic quantum computation, Phys. Rev. A80, 062326 (2009)

    2009

  28. [36]

    Altshuler, H

    B. Altshuler, H. Krovi, and J. Roland, An- derson localization makes adiabatic quan- tum optimization fail, Proceedings of the Na- tional Academy of Sciences107, 12446 (2010), https://www.pnas.org/doi/pdf/10.1073/pnas.1002116107

    2010 doi

  29. [37]

    A. P. Young, S. Knysh, and V. N. Smelyanskiy, First- order phase transition in the quantum adiabatic algo- rithm, Phys. Rev. Lett.104, 020502 (2010)

    2010

  30. [38]

    J¨ org, F

    T. J¨ org, F. Krzakala, G. Semerjian, and F. Zamponi, First-order transitions and the performance of quantum algorithms in random optimization problems, Phys. Rev. Lett.104, 207206 (2010)

    2010

  31. [39]

    Farhi, J

    E. Farhi, J. Goldstone, D. Gosset, S. Gutmann, H. B. Meyer, and P. Shor, Quantum Adiabatic Al- gorithms, Small Gaps, and Different Paths, arXiv 10.48550/arXiv.0909.4766 (2009), 0909.4766. 18

    2009 doi arXiv

  32. [40]

    Miyazaki and H

    R. Miyazaki and H. Nishimori, Real-space renormalization-group approach to the random transverse-field Ising model in finite dimensions, Phys. Rev. E87, 032154 (2013)

    2013

  33. [41]

    Bapst, L

    V. Bapst, L. Foini, F. Krzakala, G. Semerjian, and F. Zamponi, The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective, Phys. Rep.523, 127 (2013)

    2013

  34. [42]

    C. R. Laumann, R. Moessner, A. Scardicchio, and S. L. Sondhi, Quantum Adiabatic Algorithm and Scaling of Gaps at First-Order Quantum Phase Transitions, Phys. Rev. Lett.109, 030502 (2012)

    2012

  35. [43]

    Inagaki, Y

    T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Ta- mate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu, O. Tadanaga, H. Takenouchi, K. Aihara, K.- i. Kawarabayashi, K. Inoue, S. Utsunomiya, and H. Take- sue, A coherent Ising machine for 2000-node optimization ...

    2000

  36. [44]

    Mohseni, P

    N. Mohseni, P. L. McMahon, and T. Byrnes, Ising ma- chines as hardware solvers of combinatorial optimization problems, Nature Reviews Physics4, 363 (2022)

    2022

  37. [45]

    Kirkpatrick, C

    S. Kirkpatrick, C. J. Gelatt, and M. Vecchi, Optimization by simulated annealing, Science220, 671 (1983)

    1983

  38. [46]

    ˇCern´ y, Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm, J

    V. ˇCern´ y, Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm, J. Optim. Theory Appl.45, 41 (1985)

    1985

  39. [47]

    Hajek, Cooling schedules for optimal annealing, Math- ematics of Operations Research13, 311 (1988)

    B. Hajek, Cooling schedules for optimal annealing, Math- ematics of Operations Research13, 311 (1988)

    1988

  40. [48]

    A. W. Sandvik, Stochastic series expansion method for quantum ising models with arbitrary interactions, Phys. Rev. E68, 056701 (2003)

    2003

  41. [49]

    D. M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys.67, 279 (1995)

    1995

  42. [50]

    G. E. Santoro, R. Martoˇ n´ ak, E. Tosatti, and R. Car, Theory of Quantum Annealing of an Ising Spin Glass, Science295, 2427 (2002)

    2002

  43. [51]

    B. Heim, T. F. Rønnow, S. V. Isakov, and M. Troyer, Quantum versus classical annealing of Ising spin glasses, Science348, 215 (2015)

    2015

  44. [52]

    Martoˇ n´ ak, G

    R. Martoˇ n´ ak, G. E. Santoro, and E. Tosatti, Quantum annealing of the traveling-salesman problem, Phys. Rev. E70, 057701 (2004)

    2004

  45. [53]

    Seki and H

    Y. Seki and H. Nishimori, Quantum annealing with an- tiferromagnetic fluctuations, Phys. Rev. E85, 051112 (2012)

    2012

  46. [54]

    F. A. Quinton, P. A. S. Myhr, M. Barani, P. Crespo del Granado, and H. Zhang, Quantum annealing applica- tions, challenges and limitations for optimisation prob- lems compared to classical solvers, Scientific Reports15, 12733 (2025)

    2025

  47. [55]

    Carugno, M

    C. Carugno, M. Ferrari Dacrema, and P. Cremonesi, Evaluating the job shop scheduling problem on a D-wave quantum annealer, Sci. Rep.12, 6539 (2022)

    2022

  48. [56]

    M. F. Serret, B. Marchand, and T. Ayral, Solving op- timization problems with rydberg analog quantum com- puters: Realistic requirements for quantum advantage using noisy simulation and classical benchmarks, Phys. Rev. A102, 052617 (2020)

    2020

  49. [57]

    Cazals, A

    P. Cazals, A. Fran¸ cois, L. Henriet, L. Leclerc, M. Marin, Y. Naghmouchi, W. d. S. Coelho, F. Sikora, V. Vitale, R. Watrigant, M. W. Garzillo, and C. Dalyac, Identify- ing hard native instances for the maximum-independent- set problem on neutral-atom quantum processors, Phys....

    2026

  50. [58]

    I. A. Luchnikov, E. S. Tiunov, T. Haug, and L. Aolita, Large-scale quantum annealing simulation with tensor networks and belief propagation, arXiv 10.48550/arXiv.2409.12240 (2024), 2409.12240

    2024 doi arXiv

  51. [59]

    T. Qian, G. Qi, Z. Y. Wu, R. Gu, X. Liu, and C. Lyu, Large language model for discrete optimization prob- lems: Evaluation and step-by-step reasoning (2026), arXiv:2603.07733 [cs.AI]

    2026

  52. [60]

    Mauron and G

    L. Mauron and G. Carleo, Challenging the Quan- tum Advantage Frontier with Large-Scale Clas- sical Simulations of Annealing Dynamics, arXiv 10.48550/arXiv.2503.08247 (2025), 2503.08247

    2025 doi arXiv

  53. [61]

    S. W. Shin, G. Smith, J. A. Smolin, and U. Vazi- rani, How ”Quantum” is the D-Wave Machine?, arXiv 10.48550/arXiv.1401.7087 (2014), 1401.7087

    2014 doi arXiv

  54. [62]

    M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lant- ing, 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. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Tr...

    2011

  55. [63]

    Lanting, A

    T. Lanting, A. J. Przybysz, A. Yu. Smirnov, F. M. Spedalieri, M. H. Amin, A. J. Berkley, R. Harris, F. Al- tomare, S. Boixo, P. Bunyk, N. Dickson, C. Enderud, J. P. Hilton, E. Hoskinson, M. W. Johnson, E. Ladizin- sky, N. Ladizinsky, R. Neufeld, T. Oh, I. Perminov, C. Rich, M....

    2014

  56. [64]

    Boixo, T

    S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Evidence for quantum annealing with more than one hun- dred qubits, Nature Physics10, 218 (2014)

    2014

  57. [65]

    A. D. King, J. Raymond, T. Lanting, R. Harris, A. Zucca, F. Altomare, A. J. Berkley, K. Boothby, S. Ejtemaee, C. Enderud, E. Hoskinson, S. Huang, E. Ladizinsky, A. J. R. MacDonald, G. Marsden, R. Molavi, T. Oh, G. Poulin-Lamarre, M. Reis, C. Rich, Y. Sato, N. Tsai, M. Volkmann...

    2023

  58. [66]

    A. D. King, A. Nocera, M. M. Rams, J. Dziarmaga, R. Wiersema, W. Bernoudy, J. Raymond, N. Kaushal, N. Heinsdorf, R. Harris, K. Boothby, F. Altomare, M. Asad, A. J. Berkley, M. Boschnak, K. Chern, H. Christiani, S. Cibere, J. Connor, M. H. Dehn, R. Desh- pande, S. Ejtemaee, P. ...

    2025

  59. [67]

    V. S. Denchev, S. Boixo, S. V. Isakov, N. Ding, R. Bab- bush, V. Smelyanskiy, J. Martinis, and H. Neven, What is the Computational Value of Finite-Range Tunneling?, Phys. Rev. X6, 031015 (2016). 19

    2016

  60. [68]

    Saket, A ptas for the classical ising spin glass problem on the chimera graph structure (2013), arXiv:1306.6943 [cs.DS]

    R. Saket, A ptas for the classical ising spin glass problem on the chimera graph structure (2013), arXiv:1306.6943 [cs.DS]

    2013 arXiv

  61. [69]

    Albash and D

    T. Albash and D. A. Lidar, Demonstration of a scaling advantage for a quantum annealer over simulated anneal- ing, Phys. Rev. X8, 031016 (2018)

    2018

  62. [70]

    M. S. K¨ onz, W. Lechner, H. G. Katzgraber, and M. Troyer, Embedding Overhead Scaling of Optimiza- tion Problems in Quantum Annealing, PRX Quantum2, 040322 (2021)

    2021

  63. [71]

    D. Vert, R. Sirdey, and S. Louise, Benchmarking Quan- tum Annealing Against “Hard” Instances of the Bipartite Matching Problem, SN Comput. Sci.2, 106 (2021)

    2021

  64. [72]

    Leclerc, L

    L. Leclerc, L. Ortiz-Guti´ errez, S. Grijalva, B. Albrecht, J. R. K. Cline, V. E. Elfving, A. Signoles, L. Henriet, G. Del Bimbo, U. A. Sheikh, M. Shah, L. Andrea, F. Ish- tiaq, A. Duarte, S. Mugel, I. C´ aceres, M. Kurek, R. Or´ us, A. Seddik, O. Hammami, H. Isselnane, and D....

    2023

  65. [73]

    Nishimori and Y

    H. Nishimori and Y. Nonomura, Quantum effects in neu- ral networks, Journal of the Physical Society of Japan 65, 3780 (1996), https://doi.org/10.1143/JPSJ.65.3780

    1996 doi

  66. [74]

    Raymond, S

    J. Raymond, S. Yarkoni, and E. Andriyash, Global Warming: Temperature Estimation in Annealers, Front. ICT3, 214379 (2016)

    2016

  67. [75]

    Becca and S

    F. Becca and S. Sorella,Quantum Monte Carlo ap- proaches for correlated systems(Cambridge University Press, 2017)

    2017

  68. [76]

    Sprague and S

    K. Sprague and S. Czischek, Variational Monte Carlo with large patched transformers, Communications Physics7, 90 (2024)

    2024

  69. [77]

    Naumann, E

    J. Naumann, E. L. Weerda, M. Rizzi, J. Eisert, and P. Schmoll, An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation, SciPost Phys. Lect. Notes , 86 (2024)

    2024

  70. [78]

    Srdinˇ sek and X

    M. Srdinˇ sek and X. Waintal, Hybrid between biologically and quantum-inspired many-body states, Phys. Rev. B 113, 125107 (2026)

    2026

  71. [79]

    Louvet, T

    T. Louvet, T. Ayral, and X. Waintal, Feasibility of per- forming quantum chemistry calculations on quantum computers, Phys. Rev. B113, 125112 (2026)

    2026

  72. [80]

    Ardonne, P

    E. Ardonne, P. Fendley, and E. Fradkin, Topological or- der and conformal quantum critical points, Annals of Physics310, 493 (2004)

    2004

  73. [81]

    C. L. Henley, From classical to quantum dynamics at Rokhsar–Kivelson points, J. Phys.: Condens. Matter16, S891 (2004)

    2004

  74. [82]

    Stochastic Matrix Form

    C. Castelnovo, C. Chamon, C. Mudry, and P. Pujol, From quantum mechanics to classical statistical physics: Generalized Rokhsar–Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition, Ann. Phys. 318, 316 (2005)

    2005

  75. [83]

    Verstraete, M

    F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States, Phys. Rev. Lett.96, 220601 (2006)

    2006

  76. [84]

    P. S. Tarabunga and C. Castelnovo, Magic in general- ized Rokhsar-Kivelson wavefunctions, Quantum8, 1347 (2024), 2311.08463v2

    2024 arXiv

  77. [85]

    Kalinowski, R

    M. Kalinowski, R. Samajdar, R. G. Melko, M. D. Lukin, S. Sachdev, and S. Choi, Bulk and Boundary Quan- tum Phase Transitions in a Square Rydberg Atom Array, Physical Review B105, 174417 (2022), arXiv:2112.10790 [quant-ph]

    2022 arXiv

  78. [86]

    Moessner, S

    R. Moessner, S. L. Sondhi, and P. Chandra, Two- Dimensional Periodic Frustrated Ising Models in a Trans- verse Field, Phys. Rev. Lett.84, 4457 (2000)

    2000

  79. [87]

    Moessner and S

    R. Moessner and S. L. Sondhi, Ising models of quantum frustration, Phys. Rev. B63, 224401 (2001)

    2001

  80. [88]

    S. V. Isakov and R. Moessner, Interplay of quantum and thermal fluctuations in a frustrated magnet, Phys. Rev. B68, 104409 (2003)

    2003

  81. [89]

    Koziol, S

    J. Koziol, S. Fey, S. C. Kapfer, and K. P. Schmidt, Quantum criticality of the transverse-field Ising model with long-range interactions on triangular-lattice cylin- ders, Phys. Rev. B100, 144411 (2019)

    2019

  82. [90]

    S. Fey, S. C. Kapfer, and K. P. Schmidt, Quantum Criti- cality of Two-Dimensional Quantum Magnets with Long- Range Interactions, Phys. Rev. Lett.122, 017203 (2019)

    2019

  83. [91]

    Villain, R

    J. Villain, R. Bidaux, J.-P. Carton, and R. Conte, Order as an effect of disorder, J. Phys.41, 1263 (1980)

    1980

  84. [92]

    S. Guo, J. Huang, J. Hu, and Z.-X. Li, Order by disorder and an emergent Kosterlitz-Thouless phase in a triangu- lar Rydberg array, Phys. Rev. A108, 053314 (2023)

    2023

  85. [93]

    Bombieri, T

    L. Bombieri, T. V. Zache, G. Calliari, M. D. Lukin, H. Pichler, and D. Gonz´ alez-Cuadra, Deconfined quan- tum criticality on a triangular rydberg array, Phys. Rev. Lett.135, 233602 (2025)

    2025

  86. [94]

    Kashima and M

    T. Kashima and M. Imada, Path-Integral Renormaliza- tion Group Method for Numerical Study on Ground States of Strongly Correlated Electronic Systems, J. Phys. Soc. Jpn.70, 2287 (2001)

    2001

  87. [95]

    Janke and R

    W. Janke and R. Villanova, Three-dimensional 3-state potts model revisited with new techniques, Nuclear Physics B489, 679 (1997)

    1997

  88. [96]

    Mora and X

    C. Mora and X. Waintal, Variational Wave Functions and Their Overlap with the Ground State, Phys. Rev. Lett. 99, 030403 (2007)

    2007

  89. [97]

    D. Wu, R. Rossi, F. Vicentini, N. Astrakhantsev, F. Becca, X. Cao, J. Carrasquilla, F. Ferrari, A. Georges, M. Hibat-Allah, M. Imada, A. M. L¨ auchli, G. Mazzola, A. Mezzacapo, A. Millis, J. R. Moreno, T. Neupert, Y. Nomura, J. Nys, O. Parcollet, R. Pohle, I. Romero, M. Schmid...

    2024

  90. [98]

    P. M. Fenwick, A new data structure for cumulative fre- quency tables, Softw.: Pract. Exper.24, 327 (1994)

    1994

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.