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  • Open Access

Efficient Quantum Optimization via Dynamical Simulation

Ahmet Burak Catli1,*, Sophia Simon1,†, and Nathan Wiebe2,3,4,‡

  • *Contact author: catli@cs.toronto.edu
  • †Contact author: sophia.simon@mail.utoronto.ca
  • ‡Contact author: nawiebe@cs.toronto.edu
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PRX Quantum 7, 033026 – Published 12 August, 2026

DOI: https://doi.org/10.1103/t59v-8flk

Abstract

We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. We focus on the setting where the objective function can only be accessed via a phase oracle. Our first two algorithms can find local optima of a differentiable function f:  RN→R by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of O(N2κ2/hx2ϵ) queries to a phase oracle to find an ϵ-approximate local optimum, where κ is the condition number of the Hessian matrix and hx is the discretization spacing. In contrast, we show that methods based on gradient descent require O(N3/2(1/ϵ)κlog(3)/4) queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of f by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggest that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.

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Popular Summary

We have created a quantum algorithm for local optimization problems, which gives favorable results over existing algorithms in situations in which the access to the function is provided through an encoding into the phases of a quantum state, and when the target function changes in one direction way faster than compared to the others. We accomplish this by encoding the target function into a time-dependent Hamiltonian that mimics the dynamics of a time-independent dissipative system. The dissipative dynamics suck energy out of the system, making the system lose energy and converge to a local minima. We give rigorous bounds on convergence time and amount of quantum sources needed to execute the algorithm, in addition to numerical simulations on small systems.

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References (56)

  1. C. Durr and P. Hoyer, A quantum algorithm for finding the minimum, arXiv:Quant-Ph/9607014.
  2. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  3. N. Wiebe, D. Braun, and S. Lloyd, Quantum algorithm for data fitting, Phys. Rev. Lett. 109, 050505 (2012).
  4. F. G. Brandao and K. M. Svore, Quantum speed-ups for solving semidefinite programs, in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2017), pp. 415–426.
  5. J. Van Apeldoorn, A. Gilyén, S. Gribling, and R. de Wolf, Quantum SDP-solvers: Better upper and lower bounds, in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2017), pp. 403–414.
  6. M. Schuld, A. Bocharov, K. M. Svore, and N. Wiebe, Circuit-centric quantum classifiers, Phys. Rev. A 101, 032308 (2020).
  7. 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 processor, Nat. Commun. 5, 4213 (2014).
  8. M. Kieferová and N. Wiebe, Tomography and generative training with quantum Boltzmann machines, Phys. Rev. A 96, 062327 (2017).
  9. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  10. T. E. O’Brien, M. Streif, N. C. Rubin, R. Santagati, Y. Su, W. J. Huggins, J. J. Goings, N. Moll, E. Kyoseva, M. Degroote et al., Efficient quantum computation of molecular forces and other energy gradients, Phys. Rev. Res. 4, 043210 (2022).
  11. A. Gilyén, S. Arunachalam, and N. Wiebe, Optimizing quantum optimization algorithms via faster quantum gradient computation, in Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (SIAM, 2019), pp. 1425–1444.
  12. J. van Apeldoorn, A. Gilyén, S. Gribling, and R. de Wolf, Convex optimization using quantum oracles, Quantum 4, 220 (2020).
  13. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  14. J. Leng, E. Hickman, J. Li, and X. Wu, Quantum Hamiltonian descent, arXiv:2303.01471.
  15. J. Leng, Y. Zheng, and X. Wu, A quantum-classical performance separation in nonconvex optimization, arXiv:2311.00811.
  16. Z. Chen, Y. Lu, H. Wang, Y. Liu, and T. Li, Quantum Langevin dynamics for optimization, Commun. Math. Phys. 406, 52 (2025).
  17. S. Simon, R. Santagati, M. Degroote, N. Moll, M. Streif, and N. Wiebe, Improved precision scaling for simulating coupled quantum-classical dynamics, PRX Quantum 5, 010343 (2024).
  18. S. Veroni, M. Müller, and G. Giudice, Optimized measurement-free and fault-tolerant quantum error correction for neutral atoms, Phys. Rev. Res. 6, 043253 (2024).
  19. A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 1032009).
  20. A. M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations, Quantum Inf. Comput. 12, 901 (2012).
  21. N. Wiebe, A. Kapoor, and K. M. Svore, Quantum nearest-neighbor algorithms for machine learning, Quantum Inf. Comput. 15, 318 (2015).
  22. C. Gidney, Halving the cost of quantum addition, Quantum 2, 74 (2018).
  23. Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, 1st ed. (Springer Publishing Company, Incorporated, 2014).
  24. S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Mol. Phys. 52, 255 (1984).
  25. S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys. 81, 511 (1984).
  26. N. Linden, S. Popescu, A. J. Short, and A. Winter, Quantum mechanical evolution towards thermal equilibrium, Phys. Rev. E 792009).
  27. A. Haraux and M. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differ. Equations 144, 313 (1998).
  28. P. De Smedt and A. G. López, On the asymptotic behavior of the solutions of the Caldirola-Kanai equation, Lett. Math. Phys. 12, 291 (1986).
  29. G. H. Low and N. Wiebe, Hamiltonian simulation in the interaction picture, arXiv:1805.00675.
  30. M. Kieferová, A. Scherer, and D. W. Berry, Simulating the dynamics of time-dependent hamiltonians with a truncated dyson series, Phys. Rev. A 99, 042314 (2019).
  31. D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Exponential Improvement in Precision for Simulating Sparse Hamiltonians, in Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14 (Association for Computing Machinery, New York, NY, USA, 2014), pp. 283–292.
  32. A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019), pp. 193–204.
  33. A. Wibisono, A. C. Wilson, and M. I. Jordan, A variational perspective on accelerated methods in optimization, Proc. Natl. Acad. Sci. U.S.A. 113, E7351 (2016).
  34. S. Chakrabarti, D. Herman, J. Watkins, E. Fontana, B. Augustino, J. L. Kim, and M. Pistoia, On speedups for convex optimization via quantum dynamics, arXiv:2503.24332.
  35. J. Leng, K. Wu, X. Wu, and Y. Zheng, (sub)exponential quantum speedup for optimization, arXiv:2504.14841.
  36. M. Betancourt, A conceptual introduction to Hamiltonian Monte Carlo, arXiv:1701.02434.
  37. P. H. Hünenberger, Thermostat algorithms for molecular dynamics simulations, in Advanced Computer Simulation: Approaches for Soft Matter Sciences I, edited by C.Holm and K.Kremer (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005), pp. 105–149.
  38. J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, England, 2018).
  39. V. Kliuchnikov, Synthesis of unitaries with clifford+t circuits, arXiv:1306.3200.
  40. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2010).
  41. G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum 3, 163 (2019).
  42. R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear t complexity, Phys Rev X 8, 041015 (2018).
  43. D. Motlagh and N. Wiebe, Generalized quantum signal processing, PRX Quantum 5, 020368 (2024).
  44. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995).
  45. C. Ortiz Marrero, M. Kieferová, and N. Wiebe, Entanglement-induced barren plateaus, PRX Quantum 2, 040316 (2021).
  46. S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, Nat. Commun. 12, 6961 (2021).
  47. S. P. Jordan, Fast quantum algorithm for numerical gradient estimation, Phys. Rev. Lett. 95, 050501 (2005).
  48. E. K. Ryu and S. Boyd, Primer on monotone operator methods, Appl. Comput. Math. 15, 3 (2016).
  49. A. Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9, 543 (1988).
  50. E. R. Anschuetz, A unified theory of quantum neural network loss landscapes, arXiv:2408.11901.
  51. Z. Chen and J. J. Dongarra, Condition numbers of Gaussian random matrices, SIAM J. Matrix Anal. Appl. 27, 603 (2005).
  52. I. D. Kivlichan, N. Wiebe, R. Babbush, and A. Aspuru-Guzik, Bounding the costs of quantum simulation of many-body physics in real space, J. Phys. A 50, 305301 (2017).
  53. G. Wanner and E. Hairer, Solving Ordinary Differential Equations II Springer, Berlin Heidelberg New York, 1996), Vol. 375.
  54. E. Süli and D. F. Mayers, An Introduction to Numerical Analysis (Cambridge University Press, Cambridge, England, 2003).
  55. F. R. Kschischang, The complementary error function (2017).
  56. E. R. Vrscay, Mean value theorems for functions of several variables (2017).

Outline

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PRX Quantum 7, 033026– Published 12 August, 2026
  • Received 24 September 2025
  • Revised 17 March 2026
  • Accepted 14 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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