Pith. sign in

REVIEW 3 major objections 5 minor 51 references

Optimizing QUBO on a quantum computer by mimicking imaginary time evolution

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper proposes ITEMC, a hybrid quantum-classical optimizer that mimics imaginary time evolution to solve QUBO problems, reporting approximation ratios above 0.99 up to 150 qubits.

desk verdict Solid incremental algorithm with honest small-scale tests, but the 150-qubit approximation ratios rest on a simulated-annealing denominator and a CVaR tail metric, so the headline performance is unproven. read the letter →

arxiv 2505.22924 v2 pith:3DILHVHG submitted 2025-05-28 quant-ph

Yahui Chai , Alice Di Tucci This is my paper
classification quant-ph
keywords QUBOimaginarytimeevolutionhybridquantum-classicalalgorithmCVaRcombinatorialoptimizationgateorderingapproximationrationear-termquantumcomputing
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

The paper proposes a hybrid quantum-classical algorithm, ITEMC, for Quadratic Unconstrained Binary Optimization (QUBO), built from single-qubit rotations and two-qubit gates whose parameters are tuned to mimic imaginary time evolution. The key cost saving is that parameter selection uses only one- and two-qubit Pauli expectation values, measured once per iteration, instead of the repeated full-Hamiltonian energy evaluations that variational methods such as VQE require. An iterative feedback loop re-initializes the qubits from the lowest-energy tail of the previous measurements, and an adaptive gate-ordering step selects the best arrangement of two-qubit gates. On random instances—complete and intermediate-density graphs up to 22 qubits, and 3-regular graphs up to 150 qubits—the authors report approximation ratios above 0.99 in classical simulation, convergence in about six iterations, and hardware demonstrations at 40, 60, and 80 qubits whose solutions are compatible with simulated annealing. If these results hold, the method is a low-measurement-cost route to approximate QUBO solutions on near-term quantum hardware.

What carries the argument

The ITEMC is a parameterized circuit whose building blocks are single-qubit $R_y$ rotations and two-qubit unitaries $U_{ij}(\theta)=\exp(-i(\theta_{ij,1}\sigma_i^z\sigma_j^y+\theta_{ij,0}\sigma_i^y\sigma_j^z)/2)$. Each unitary is chosen to maximize the overlap with the exact imaginary-time evolved state, and because the QUBO Hamiltonian is diagonal in the computational basis, the overlap cost factorizes into a small set of Pauli expectation values that are measured once and then processed classically. The iterative part uses CVaR, averaging over the lowest $\alpha$ fraction of sampled energies, both to select low-energy bitstrings and to compute the $\sigma^z$ expectations that set the initial $R_y$ angles in the next iteration. The adaptive sorting step tests five gate orderings in the first iteration and fixes the one with the lowest CVaR energy for all later iterations.

What would settle it

Run ITEMC on the same 3-regular random instances at 60 to 150 qubits, but replace the simulated-annealing reference energies with certified lower bounds from an exact or branch-and-bound QUBO solver; if the ratio of ITEMC's CVaR to those certified values drops below 0.99, the paper's central scalability claim is falsified.

Watch

Extended reading notes

Core claim

The central claim is that a fixed-structure circuit can be made to act like imaginary time evolution for QUBO without variational full-energy optimization. Each non-unitary imaginary-time factor $e^{-\tau J_{ij}\sigma_i^z\sigma_j^z}$ is approximated by a two-qubit unitary chosen to maximize its overlap with the exact evolved state, while each single-qubit field term is approximated by an $R_y$ rotation; the overlap cost depends only on a small set of one- and two-qubit Pauli expectation values, which are measured once per iteration and then optimized classically. After each pass, the lowest-energy tail of the sampled bitstrings (CVaR with $\alpha=0.01$) provides both the cost and the qubit $\sigma^z$ expectations that set the initial rotations of the next pass, so the state improves without deepening the circuit. The paper further shows that ordering the two-qubit gates by sorted QUBO coefficients matters, and it uses an adaptive sort that tries five orderings and keeps the best. The headline numerical result is that this procedure reaches approximation ratios above 0.99 for random QUBO instances up to 150 qubits (3-regular graphs in the large-size simulations), and that dense instances are hard for tensor-network simulation because the entanglement entropy grows linearly with system size.

Load-bearing premise

The reported approximation ratios for systems larger than 22 qubits are computed relative to simulated-annealing estimates of the optimal energy that carry no error bars, so a biased reference could make the method look better than it is.

Editorial extensions

If this is right

  • Per-iteration measurement cost is $O(M/\epsilon^2)$ with $M$ the number of Hamiltonian terms, because the gate parameters are fixed by local Pauli expectations rather than by repeated full-energy evaluations.
  • In classical simulation, a 150-qubit 3-regular instance needs only about eleven circuit executions: five to select the gate ordering and roughly six iterative steps, compared with the much larger evaluation counts typical of VQE-style optimization.
  • For graph densities at or above 0.5, the entanglement entropy of the ITEMC state grows linearly with system size, making matrix-product-state simulation expensive and marking dense QUBO instances as the regime where a quantum processor could offer an advantage.
  • On current superconducting hardware, solutions at 40, 60, and 80 qubits match simulated-annealing solutions, with the ground state found at high fidelity for 40 and 60 qubits.

Reading between the lines

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

  • Because the large-scale approximation ratios are referenced to simulated-annealing optima without error bars, an editorial test would be to compare ITEMC against certified optima for moderate sizes (for example, 24 to 40 qubits) before extrapolating the 0.99 figure.
  • The algorithm uses only the fact that the Hamiltonian is diagonal in the computational basis, so the same ansatz and feedback loop should transfer directly to MaxCut and other Ising-type problems beyond QUBO.
  • The adaptive sorting step multiplies the first-iteration shot count by five; a learned or heuristic gate-ordering rule could remove this overhead if stable patterns exist in the optimal orderings.
  • The claimed advantage over VQE depends on the iteration count staying small; the paper demonstrates this on random instances, but adversarial or worst-case QUBO families remain an open test.
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 / 5 minor

Summary. The paper proposes ITEMC, a hybrid quantum-classical algorithm for QUBO optimization that constructs a parameterized circuit whose gates are chosen to mimic imaginary time evolution. The circuit parameters are obtained from local one- and two-qubit expectation values rather than from a full energy evaluation, and an iterative feedback loop updates the initial product state using CVaR-tail expectations from the previous iteration. An adaptive pre-sorting step selects among five gate orderings. The authors report classical simulations with approximation ratios above 0.99 up to 150 qubits (for 3-regular graphs), linear entanglement-entropy scaling for dense graphs, and hardware demonstrations on IBM devices for 40, 60, and 80 qubits. The central claim is that ITEMC achieves high-quality QUBO solutions with substantially lower measurement overhead than VQE.

Significance. The core idea is interesting and potentially useful: replacing full Hamiltonian measurements in the variational loop with local expectation values, combined with an iterative warm-start and a heuristic gate-ordering selection, is a concrete and well-motivated design. The small-system benchmarks are extensive (400 instances per size up to 22 qubits, exact ground state, finite-shot simulations), and the paper explicitly gives resource estimates and compares them with VQE. If the large-scale claims were properly supported, this could be a practical resource-efficient alternative for near-term quantum optimization. However, the headline 150-qubit claim currently rests on simulated-annealing reference energies without error bars, and the hardware results are single-instance demonstrations, so the significance is conditional on fixing those benchmarks.

major comments (3)
  1. [IV.B, Fig. 8, and Abstract] The headline approximation ratios above 0.99 for N>22 are computed with E_opt taken from simulated annealing, and the manuscript does not report the SA schedule, the number of restarts, or any optimality gap. Since SA returns feasible spin configurations, its energy is an upper bound on the true ground-state energy; for the negative energies typical of these spin-glass instances, the ratio CVaR/E_SA is systematically larger than CVaR/E_true and can even exceed 1 if CVaR > E_SA. This affects the central quantitative claim directly. Please provide a comparison of SA with exact optima for N<=22 (e.g., the fraction of instances where SA reaches the exact ground state and the distribution of SA-energy gaps), and for N>22 report a sensitivity analysis or an independent lower bound on E_opt. The MPS truncation error (max bond dimension 100) should also be quantified, since the large-scale simulation results inherit it.
  2. [V, Table II] The hardware experiments consist of one instance per size (40, 60, and 80 qubits) with no repeated runs or error bars. The 'fidelity' for 40 and 60 qubits is 1 only relative to the SA-found best solution, not to a known ground state, and for 80 qubits the fidelity is 2e-5. The conclusion that the hardware results are 'compatible with simulated annealing' is therefore too weak to support the abstract's hardware claim. Please report multiple instances, shot-noise error bars, and exact or independently validated reference energies, or explicitly temper the claim to a single-instance demonstration.
  3. [IV.B, Fig. 7, and footnote [45]] The claim that ITEMC circuits are classically hard to simulate is not supported by the data shown. Fig. 7 extends only to N=22, and the footnote itself states that the data points are insufficient to distinguish sublinear from low-slope linear scaling for 3-regular graphs. The large-scale MPS simulations are restricted to 3-regular graphs, precisely the regime where the entropy scaling is mild, while the linear-entropy regime (densities >= 0.5) is demonstrated only at small sizes where exact simulation is possible. Thus the 'hard-to-simulate' and 'high-performance at large N' regimes do not overlap in the presented evidence. Please provide entropy data for larger N in the dense regime or explicitly state that the hardness extrapolation is speculative.
minor comments (5)
  1. [IV.A] The text says 400 random instances are generated for each problem size and density, but the caption of Fig. 2 says the results are averaged over 100 instances; please reconcile these numbers.
  2. [Table II] The column labels for Table II are ambiguous: the first row reads '40 1 0.759' and the text says the 80-qubit fidelity is 2e-5, but the table row seems to place 0.974 and 2e-5 in the ar and fidelity columns. Please label all columns explicitly and make the order (qubits, ar, fidelity, shots per iteration, iterations, total shots) unambiguous.
  3. [III.C and VI] There are several typos that should be fixed: 'gate ording' in Section III.C, 'simulated annealning' in Section VI, and 'classic numerical results' in the Introduction.
  4. [Eq. (14)] In Eq. (14), E_k are described as 'energy eigenvalues', but in the measurement setting they are sampled bitstring energies; please use a term such as 'sampled energies' to avoid confusion.
  5. [IV (intro)] The sentence 'The hardware runs are demonstrated on IBM's quantum devices for 40, 60, and 80 qubits,' is incomplete as it appears before Section V; either remove it or complete it as a lead-in to Section V.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ITEMC circuit parameters are fitted to imaginary-time evolution, not to the QUBO ground state; the reported approximation ratios are measured against independent references.

full rationale

The central parameter optimization (Eq. 10) maximizes the overlap between the state produced by the parametrized unitary U_ij and the exact imaginary-time-evolved state e^{-τ J_ij σz_i σz_j}|ψ_k⟩. This objective uses only local Pauli expectation values of the current state and does not use the QUBO ground-state energy as a training target. The iterative initialization φ_t^i = arccos(⟨σz_i⟩_{t-1,α}) and the adaptive sorting step do use the CVaR criterion, which is also the final performance metric; however, this is the algorithm's own selection rule, not a hidden fit: the reported CVaR is the value achieved by the constructed circuit, and the gate parameters themselves are not optimized against CVaR. The two-qubit ansatz and the analytical product-state simplification are credited to the authors' prior work [15] rather than rederived, including the O(τ) approximation error; but the paper also tests the approximation numerically (Fig. 5) and the cited claim is parameter-free and externally checkable, so the self-citation is not the sole load-bearing evidence. The use of simulated-annealing energies as E_opt for N>22 is a benchmarking limitation (SA gives an upper bound and no error bars are reported), but E_opt is not an input to the ITEMC optimization; it is an external reference, so this does not make the derivation circular. No equation in the paper reduces a predicted quantity to a fitted input by construction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The algorithm introduces no new physical entities, but it rests on several domain assumptions about the accuracy of the local ITE mimicry, the heuristic feedback update, the CVaR metric, and the reference energies from simulated annealing.

free parameters (4)
  • tau (imaginary time step) = 0.3
    Controls the strength of the ITE mimicry; chosen based on performance tuning on 20-qubit instances in Appendix B, not derived from first principles.
  • alpha (CVaR fraction) = 0.01 (also 0.001, 0.0001)
    Selects the low-energy tail used for the cost function and initial-state feedback; chosen by the authors and affects both optimization and evaluation.
  • shots for parameter optimization = 1000 (small systems)
    Number of measurements per expectation value; affects parameter accuracy and is a chosen resource level.
  • shots for CVaR evaluation = 10000 (small systems), up to 100000 (hardware)
    Determines the statistical accuracy of the reported CVaR and expectation values.
assumptions (6)
  • domain assumption The parametrized two-qubit unitary U_ij(theta) can approximate e^{-tau J sigma_z sigma_z} on the current state sufficiently well.
    Used in Section III A; no error bound is provided, only numerical demonstration.
  • standard math The cost functions in Eq. (A1) correctly represent the overlap with the ideal ITE step.
    Adopted from Ref [15]; the derivation is not repeated and the expressions contain complex terms whose maximization criterion is not specified.
  • ad hoc to paper The iterative update phi_i = arccos(<sigma_z_i>_alpha) from the low-energy tail guides the state toward lower energies.
    Introduced in Section III B; it is a heuristic feedback loop without a convergence proof.
  • domain assumption CVaR over the lowest alpha fraction is a suitable proxy for solution quality.
    Used throughout; tail averaging can make the algorithm look better than the mean sample quality.
  • domain assumption Simulated annealing provides a good estimate of the ground-state energy for N > 22.
    Used in Section IV as E_opt; no error bars are given.
  • domain assumption The MPS simulation with bond dimension 100 is accurate for 3-regular graphs up to 150 qubits.
    Used in Section IV B; no convergence check in bond dimension is reported.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimizing QUBO on a quantum computer by mimicking imaginary time evolution." pith.science (2026). https://pith.science/paper/3DILHVHG

@misc{pith2026250522924,
  author       = {Pith},
  title        = {Pith review of: Optimizing QUBO on a quantum computer by mimicking imaginary time evolution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3DILHVHG}},
  note         = {Machine review of arXiv:2505.22924}
}
read the original abstract

We propose a hybrid quantum-classical algorithm for solving QUBO problems using an Imaginary Time Evolution-Mimicking Circuit (ITEMC). The circuit parameters are optimized to closely mimic imaginary time evolution, using only single- and two-qubit expectation values. This significantly reduces the measurement overhead by avoiding full energy evaluation. By updating the initial state based on results from last step iteratively, the algorithm quickly converges to the low-energy solutions. With a pre-sorting step that optimizes quantum gate ordering based on QUBO coefficients, the convergence is further improved. Our classical simulations achieve approximation ratios above 0.99 up to 150 qubits. Furthermore, the linear scaling of entanglement entropy with system size suggests that the circuit is challenging to simulate classically using tensor networks. We also demonstrate hardware runs on IBM's device for 40, 60, and 80 qubits, and obtain solutions compatible with that from simulated annealing.

Figures

Figures reproduced from arXiv: 2505.22924 by the authors.

Figure 1
Figure 1. Schematic illustration of the ITEMC algorithm. (I) Adaptive sorting step: Different gate orderings based on the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Approximation ratio as a function of iteration num [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. The approximation ratio with 95% confidence level [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (7 more)
Figure 2
Figure 2. Figure 2: Log-scale plot for the probability of the best [PITH_FULL_IMAGE:figures/full_fig_p006_2.png]
Figure 6
Figure 6. Figure 6: Evolution of the entanglement entropy along the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 5
Figure 5. Figure 5: Approximation ratio with 95% confidence level for [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 8
Figure 8. Figure 8: Scaling of approximation ratio and iterations for [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Approximation ratio versus the number of iterations [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: Approximation ratio versus the number of itera [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: Performance metrics for 20-qubit ITEMC runs avaraged over 100 instances with varying [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

51 extracted references · 25 canonical work pages

  1. [15]

    M. d. S. Fonseca, C. M. Porto, N. A. C. Carpio, G. S. T. Moraes, N. H. Morgon, R. A. Nome, and C. J. Villas- Boas, An Introduction to Variational Quantum Eigen- solver Applied to Chemistry, arXiv:2505.04768 [quant- ph] (2025)

    arXiv 2025

  2. [1]

    Try different gate order regarding the coefficients in QUBO .Jij

    work page

  3. [2]

    Do the measurements and evaluate the CVaR cost function

    work page

  4. [3]

    Choose the gate order resulting in lowest CVaR as the best gate order. I. Adaptive sorting φti=arccos(⟨σzi⟩t−1α) |+⟩|+⟩|+⟩|+⟩|+⟩|+⟩ ITEMC II. Iterative ITEMC |φt0⟩|φt1⟩|φt2⟩|φt3⟩|φt4⟩|φt5⟩ ⋯⋯ θ0,1 θ4,5 θ1,3 θ3,4

    work page

  5. [4]

    Initialize the qubits as |φti⟩=Ry(φt−1i)|+⟩fort>0 |φ0i⟩=|+⟩fort=0

    work page

  6. [5]

    True” in the legend, and random sorting are labeled with “False

    Construct circuit by the two-qubit gate with the best gate order. Each parameter is optimized as Eq. (10) to locally approximate the ITE . Ui,j(θij) θi,j 3.Do the measurement, and calculate the expectation of each qubit with CVaR ⟨σzi⟩t−1α Figure 1. Schematic illustration of the ITEMC algorithm. (I) Adaptive sorting step: Different gate orderings based on...

    work page 2025

  7. [6]

    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. Combin. Optim.28, 58 (2014)

    work page 2014

  8. [7]

    Barahona, On the computational complexity of Ising spin glass models, J

    F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. A: Math.15, 3241 (1982)

    work page 1982

Show all 51 references
  1. [8]

    Abbaset al., Challenges and opportunities in quan- tum optimization, Nature Rev

    A. Abbaset al., Challenges and opportunities in quan- tum optimization, Nature Rev. Phys.6, 718 (2024), arXiv:2312.02279 [quant-ph]

    2024 arXiv

  2. [9]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a photonic quantum pro- 11 0.3 0.4 0.6 0.8 1.0 τ 10−10 10−8 10−6 10−4 10−2 100 fidelity 0.1 0.2 0.3 0.4 0.5 0.6 τ 0.980 0.985 0.990 0....

    2014 arXiv

  3. [10]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio,et al., Variational quantum algorithms, Nat. Rev. Phys.3, 625 (2021)

    2021

  4. [11]

    Farhi, J

    E. Farhi, J. Goldstone, and S. Gutmann, A Quantum Approximate Optimization Algorithm, arXiv:1411.4028 [quant-ph] (2014)

    2014 arXiv

  5. [12]

    Tillyet al., The Variational Quantum Eigensolver: A review of methods and best practices, Phys

    J. Tillyet al., The Variational Quantum Eigensolver: A review of methods and best practices, Phys. Rept.986, 1 (2022), arXiv:2111.05176 [quant-ph]

    2022 arXiv

  6. [13]

    Kandala, A

    A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware- efficient variational quantum eigensolver for small molecules and quantum magnets, Nature549, 242 (2017), arXiv:1704.05018 [quant-ph]

    2017 arXiv

  7. [14]

    H. Lim, D. H. Kang, J. Kim, A. Pellow-Jarman, S. Mc- Farthing, R. Pellow-Jarman, H.-N. Jeon, B. Oh, J.-K. K. Rhee, and K. T. No, Fragment molecular orbital-based variational quantum eigensolver for quantum chemistry in the age of quantum computing, Sci. Rep.14, 2422 (2024)

    2024

  8. [16]

    Y. Chai, L. Funcke, T. Hartung, K. Jansen, S. Kuehn, P. Stornati, and T. Stollenwerk, Optimal Flight-Gate As- signment on a Digital Quantum Computer, Phys. Rev. Applied20, 064025 (2023), arXiv:2302.11595 [quant-ph]

    2023 arXiv

  9. [17]

    Y. Chai, E. Epifanovsky, K. Jansen, A. Kaushik, and S. K¨ uhn, Simulating the flight gate assignment problem on a trapped ion quantum computer, arXiv:2309.09686 [quant-ph] (2023)

    2023 arXiv

  10. [18]

    Amaro, M

    D. Amaro, M. Rosenkranz, N. Fitzpatrick, K. Hirano, and M. Fiorentini, A case study of variational quan- tum algorithms for a job shop scheduling problem, EPJ Quantum Technology9, 10.1140/epjqt/s40507-022- 00123-4 (2022)

    2022 doi

  11. [19]

    Nannicini, Performance of hybrid quantum-classical variational heuristics for combinatorial optimization, Physical Review E99, 10.1103/physreve.99.013304 (2019)

    G. Nannicini, Performance of hybrid quantum-classical variational heuristics for combinatorial optimization, Physical Review E99, 10.1103/physreve.99.013304 (2019)

    2019 doi

  12. [20]

    Y. Chai, K. Jansen, S. K¨ uhn, T. Schw¨ agerl, and T. Stollenwerk, Structure-inspired Ansatz and Warm Start of Variational Quantum Algorithms for Quadratic Unconstrained Binary Optimization Prob- lems, arXiv:2407.02569 [quant-ph] (2024)

    2024

  13. [21]

    X. Wang, Y. Chai, X. Feng, Y. Guo, K. Jansen, and C. T¨ uys¨ uz, Imaginary Hamiltonian variational Ansatz for combinatorial optimization problems, Phys. Rev. A111, 032612 (2025), arXiv:2408.09083 [quant-ph]

    2025

  14. [22]

    Blekos, D

    K. Blekos, D. Brand, A. Ceschini, C.-H. Chou, R.-H. Li, K. Pandya, and A. Summer, A review on Quantum Approximate Optimization Algorithm and its variants, Phys. Rept.1068, 1 (2024), arXiv:2306.09198 [quant-ph]

    2024 arXiv

  15. [24]

    Z. He, R. Shaydulin, S. Chakrabarti, D. Herman, C. Li, Y. Sun, and M. Pistoia, Alignment between initial state and mixer improves QAOA performance for con- strained optimization, npj Quantum Inf.9, 121 (2023), arXiv:2305.03857 [quant-ph]

    2023 arXiv

  16. [25]

    S. H. Sack and D. J. Egger, Large-scale quantum ap- proximate optimization on nonplanar graphs with ma- chine learning noise mitigation, Phys. Rev. Res.6, 013223 (2024), arXiv:2307.14427 [quant-ph]

    2024 arXiv

  17. [26]

    Herman, R

    D. Herman, R. Shaydulin, Y. Sun, S. Chakrabarti, S. Hu, P. Minssen, A. Rattew, R. Yalovetzky, and M. Pistoia, Constrained optimization via quantum Zeno dynamics, Commun. Phys.6, 219 (2023), arXiv:2209.15024 [quant- ph]

    2023 arXiv

  18. [27]

    Yu, X.-B

    Y. Yu, X.-B. Wang, N. Shannon, and R. Joynt, Warm start adaptive-bias quantum approximate optimization algorithm (2025), arXiv:2503.20048 [quant-ph]. 12

    2025 arXiv

  19. [28]

    Chai, Y.-J

    Y. Chai, Y.-J. Han, Y.-C. Wu, Y. Li, M. Dou, and G.-P. Guo, Shortcuts to the quantum approximate optimiza- tion algorithm, Phys. Rev. A105, 042415 (2022)

    2022

  20. [29]

    Sachdeva, G

    N. Sachdeva, G. S. Hartnett, S. Maity, S. Marsh, Y. Wang, A. Winick, R. Dougherty, D. Canuto, Y. Q. Chong, M. Hush, P. S. Mundada, C. D. B. Bentley, M. J. Biercuk, and Y. Baum, Quantum optimization using a 127-qubit gate-model ibm quantum computer can out- perform quantum anne...

    2024 arXiv

  21. [30]

    Chandarana, N

    P. Chandarana, N. N. Hegade, K. Paul, F. Albarr´ an- Arriagada, E. Solano, A. del Campo, and X. Chen, Digitized-counterdiabatic quantum approximate opti- mization algorithm, Phys. Rev. Res.4, 013141 (2022), arXiv:2107.02789 [quant-ph]

    2022 arXiv

  22. [31]

    Chandarana, N

    P. Chandarana, N. N. Hegade, K. Paul, F. Albarr´ an- Arriagada, E. Solano, A. del Campo, and X. Chen, Digitized-counterdiabatic quantum approximate opti- mization algorithm, Phys. Rev. Res.4, 013141 (2022)

    2022

  23. [32]

    E. R. Anschuetz and B. T. Kiani, Quantum variational algorithms are swamped with traps, Nature Communica- tions13, 7760 (2022)

    2022

  24. [33]

    Larocca, S

    M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Bia- monte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quan- tum computing, Nature Rev. Phys.7, 174 (2025), arXiv:2405.00781 [quant-ph]

    2025 arXiv

  25. [34]

    Schw¨ agerl, Y

    T. Schw¨ agerl, Y. Chai, T. Hartung, K. Jansen, and S. K¨ uhn, Benchmarking Variational Quantum Algo- rithms for Combinatorial Optimization in Practice, arXiv:2408.03073 [quant-ph] (2024)

    2024 arXiv

  26. [35]

    Motta, C

    M. Motta, C. Sun, A. T. K. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. S. L. Brand˜ ao, and G. K.-L. Chan, Determining eigenstates and thermal states on a quan- tum computer using quantum imaginary time evolution, Nature Physics16, 205–210 (2019)

    2019

  27. [36]

    McArdle, T

    S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational ansatz-based quantum simula- tion of imaginary time evolution, npj Quantum Informa- tion5, (2019)

    2019

  28. [37]

    S.-N. Sun, M. Motta, R. N. Tazhigulov, A. T. Tan, G. K.-L. Chan, and A. J. Minnich, Quantum computa- tion of finite-temperature static and dynamical proper- ties of spin systems using quantum imaginary time evo- lution, PRX Quantum2, 010317 (2021)

    2021

  29. [38]

    X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Theory of variational quantum simulation, Quantum3, 191 (2019)

    2019

  30. [39]

    Amaro, C

    D. Amaro, C. Modica, M. Rosenkranz, M. Fioren- tini, M. Benedetti, and M. Lubasch, Filtering vari- ational quantum algorithms for combinatorial opti- mization, Quantum Sci. Technol.7, 015021 (2022), arXiv:2106.10055 [quant-ph]

    2022 arXiv

  31. [40]

    Benedetti, M

    M. Benedetti, M. Fiorentini, and M. Lubasch, Hardware- efficient variational quantum algorithms for time evolu- tion, Phys. Rev. Res.3, 033083 (2021)

    2021

  32. [41]

    T. D. Morris, A. Kaushik, M. Roetteler, and P. C. Lot- shaw, Performant near-term quantum combinatorial op- timization, arXiv:2404.16135 [quant-ph] (2024)

    2024 arXiv

  33. [42]

    Ushijima-Mwesigwa, C

    H. Ushijima-Mwesigwa, C. F. A. Negre, and S. M. Mniszewski, Graph partitioning using quantum anneal- ing on the d-wave system (2017), arXiv:1705.03082 [quant-ph]

    2017 arXiv

  34. [43]

    Jain, Solving the traveling salesman problem on the d- wave quantum computer, Frontiers in Physics9, 760783 (2021)

    S. Jain, Solving the traveling salesman problem on the d- wave quantum computer, Frontiers in Physics9, 760783 (2021)

    2021

  35. [44]

    Hadfield, On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Comput- ing, ACM Transactions on Quantum Computing2, 18:1 (2021)

    S. Hadfield, On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Comput- ing, ACM Transactions on Quantum Computing2, 18:1 (2021)

    2021

  36. [45]

    R. J. Banks, D. E. Browne, and P. Warburton, Rapid quantum approaches for combinatorial optimisation in- spired by optimal state-transfer, Quantum8, 1253 (2024)

    2024

  37. [46]

    N. N. Hegade, K. Paul, Y. Ding, M. Sanz, F. Albarr´ an- Arriagada, E. Solano, and X. Chen, Shortcuts to adia- baticity in digitized adiabatic quantum computing, Phys. Rev. Appl.15, 024038 (2021)

    2021

  38. [47]

    A. G. Cadavid, A. Dalal, A. Simen, E. Solano, and N. N. Hegade, Bias-field digitized counterdiabatic quantum op- timization, Phys. Rev. Res.7, L022010 (2025)

    2025

  39. [48]

    Acerbi and D

    C. Acerbi and D. Tasche, On the coherence of expected shortfall, Journal of Banking & Finance26, 1487 (2002)

    2002

  40. [49]

    D ´ ıez-Valle, D

    P. D ´ ıez-Valle, D. Porras, and J. J. Garc ´ ıa-Ripoll, Quan- tum variational optimization: The role of entanglement and problem hardness, Phys. Rev. A104, 062426 (2021)

    2021

  41. [50]

    7 are insufficient to definitively distinguish between sublinear (e.g., logarithmic) and low- slope linear scaling

    The data points in Fig. 7 are insufficient to definitively distinguish between sublinear (e.g., logarithmic) and low- slope linear scaling. A more detailed analysis would be needed to clarify this, but it is beyond the scope of the present work

  42. [51]

    D.-W. S. Inc., neal: Simulated annealing sam- pler for d-wave ocean software,https://github.com/ dwavesystems/dwave-neal(2023), accessed: 2024-04-11

    2023

  43. [52]

    J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A94, 052325 (2016)

    2016

Pith tools

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