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Equating quantum imaginary time evolution, Riemannian gradient flows, and stochastic implementations

Nathan A. McMahon1,2,*, Mahum Pervez3, and Christian Arenz3,†

  • *Contact author: n.a.mcmahon@liacs.leidenuniv.nl
  • †Contact author: carenz1@asu.edu
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Phys. Rev. Research 8, 023024 – Published 7 April, 2026

DOI: https://doi.org/10.1103/ht2m-1j91

Abstract

We identify quantum imaginary time evolution as a Riemannian gradient flow on the unitary group. We develop an upper bound for the error between the two evolutions that can be controlled through the step size of the Riemannian gradient descent that minimizes the energy of the system. We discuss implementations through adaptive quantum algorithms and present a stochastic Riemannian gradient descent algorithm in which each step is efficiently implementable on a quantum computer. We prove that for a sufficiently small step size, the stochastic evolution concentrates around the imaginary time evolution, thereby providing performance guarantees for cooling the system through stochastic Riemannian gradient descent. Finally, we show that Riemannian gradient descent can be understood as an approximation to quantum imaginary time evolution that does not employ classical optimization.

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References (45)

  1. M. Suzuki, Quantum Monte Carlo methods in equilibrium and nonequilibrium systems, in Proceedings of the Ninth Taniguchi International Symposium, Susono, Japan, 1986, Springer Series in Solid-State Sciences Vol. 74 (Springer, Berlin, Heidelberg, 2012).
  2. A. Goldberg and J. L. Schwartz, Integration of the Schrödinger equation in imaginary time, J. Comput. Phys. 1, 433 (1967).
  3. M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. Brandao, and G. K.-L. Chan, Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nat. Phys. 16, 205 (2020).
  4. K. Yeter-Aydeniz, R. C. Pooser, and G. Siopsis, Practical quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and Lanczos algorithms, npj Quantum Inf. 6, 63 (2020).
  5. N. Gomes, F. Zhang, N. F. Berthusen, C.-Z. Wang, K.-M. Ho, P. P. Orth, and Y. Yao, Efficient step-merged quantum imaginary time evolution algorithm for quantum chemistry, J. Chem. Theory Comput. 16, 6256 (2020).
  6. S.-N. Sun, M. Motta, R. N. Tazhigulov, A. T. Tan, G. K.-L. Chan, and A. J. Minnich, Quantum computation of finite-temperature static and dynamical properties of spin systems using quantum imaginary time evolution, PRX Quantum 2, 010317 (2021).
  7. S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Pollmann, Real- and imaginary-time evolution with compressed quantum circuits, PRX Quantum 2, 010342 (2021).
  8. H. Kazuki, T. Kosugi, and Y.-I. Matsushita, Implementation of quantum imaginary-time evolution method on NISQ devices by introducing nonlocal approximation, npj Quantum Inf. 7, 85 (2021).
  9. K. Yeter-Aydeniz, E. Moschandreou, and G. Siopsis, Quantum imaginary-time evolution algorithm for quantum field theories with continuous variables, Phys. Rev. A 105, 012412 (2022).
  10. T. Tsuchimochi, Y. Ryo, S. L. Ten-No, and K. Sasasako, Improved algorithms of quantum imaginary time evolution for ground and excited states of molecular systems, J. Chem. Theory Comput. 19, 503 (2023).
  11. K. Hejazi, M. Motta, and G. K.-L. Chan, Adiabatic quantum imaginary time evolution, Phys. Rev. Res. 6, 033084 (2024).
  12. T. Liu, J.-G. Liu, and H. Fan, Probabilistic nonunitary gate in imaginary time evolution, Quantum Inf. Proc. 20, 204 (2021).
  13. T. Kosugi, Y. Nishiya, H. Nishi, and Y.-I. Matsushita, Imaginary-time evolution using forward and backward real-time evolution with a single ancilla: First-quantized eigensolver algorithm for quantum chemistry, Phys. Rev. Res. 4, 033121 (2022).
  14. J. M. Leamer, D. I. Bondar, and G. McCaul, Quantum dynamical emulation, arXiv:2403.03350.
  15. S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational ansatz-based quantum simulation of imaginary time evolution, npj Quantum Inf. 5, 75 (2019).
  16. T. Jones, S. Endo, S. McArdle, X. Yuan, and S. C. Benjamin, Variational quantum algorithms for discovering Hamiltonian spectra, Phys. Rev. A 99, 062304 (2019).
  17. U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Communications and Control Engineering (Springer, London, 2012).
  18. T. Schulte-Herbrüggen, S. J. Glaser, G. Dirr, and U. Helmke, Gradient flows for optimization in quantum information and quantum dynamics: Foundations and applications, Rev. Math. Phys. 22, 597 (2010).
  19. E. Malvetti, C. Arenz, G. Dirr, and T. Schulte-Herbrüggen, Randomized gradient descents on Riemannian manifolds: Almost sure convergence to global minima in and beyond quantum optimization, arXiv:2405.12039.
  20. R. Wiersema and N. Killoran, Optimizing quantum circuits with Riemannian gradient flow, Phys. Rev. A 107, 062421 (2023).
  21. A. B. Magann, S. E. Economou, and C. Arenz, Randomized adaptive quantum state preparation, Phys. Rev. Res. 5, 033227 (2023).
  22. H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall, An adaptive variational algorithm for exact molecular simulations on a quantum computer, Nat. Commun. 10, 3007 (2019).
  23. H. L. Tang, V. O. Shkolnikov, G. S. Barron, H. R. Grimsley, N. J. Mayhall, E. Barnes, and S. E. Economou, Qubit-adapt-vqe: An adaptive algorithm for constructing hardware-efficient ansätze on a quantum processor, PRX Quantum 2, 020310 (2021).
  24. A. B. Magann, K. M. Rudinger, M. D. Grace, and M. Sarovar, Feedback-based quantum optimization, Phys. Rev. Lett. 129, 250502 (2022).
  25. J. B. Larsen, M. D. Grace, A. D. Baczewski, and A. B. Magann, Feedback-based quantum algorithms for ground state preparation, Phys. Rev. Res. 6, 033336 (2024).
  26. A. B. Magann, K. M. Rudinger, M. D. Grace, and M. Sarovar, Lyapunov-control-inspired strategies for quantum combinatorial optimization, Phys. Rev. A 106, 062414 (2022).
  27. H. L. Tang, Y. Chen, P. Biswas, A. B. Magann, C. Arenz, and S. E. Economou, Nonvariational ADAPT algorithm for quantum simulations, Phys. Rev. Res. 7, 023275 (2025).
  28. J. Stokes, B. Chen, and S. Veerapaneni, Numerical and geometrical aspects of flow-based variational quantum Monte Carlo, Mach. Learn.: Sci. Technol. 4, 021001 (2023).
  29. J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020).
  30. M. Gluza, J. Son, B. H. Tiang, Y. Suzuki, Z. Holmes, and N. H. Ng, Double-bracket quantum algorithms for quantum imaginary-time evolution, Phys. Rev. Lett. 136, 020601 (2026).
  31. R. Zander, R. Seidel, L. Xiaoyue, and M. Gluza, Role of Riemannian geometry in double-bracket quantum imaginary-time evolution, in Geometric Science of Information: GSI 2025, edited by F. Nielsen and F. Barbaresco, Lecture Notes in Computer Science Vol. 16035 (Springer, Cham, 2026).
  32. M. Gluza, Double-bracket quantum algorithms for diagonalization, Quantum 8, 1316 (2024).
  33. R. W. Brockett, Least squares matching problems, Lin. Alg. Appl. 122-124, 761 (1989).
  34. R. W. Brockett, Differential geometry and the design of gradient algorithms, in Proceedings of Symposia in Pure Mathematics, edited by R. Greene and S. T. Yau (American Mathematical Society, Providence, Rhode Island, 1993), Vol. 54.1, pp. 69–92.
  35. C. M. Dawson, J. Eisert, and T. J. Osborne, Unifying variational methods for simulating quantum many-body systems, Phys. Rev. Lett. 100, 130501 (2008).
  36. M. B. Hastings, On Lieb-Robinson bounds for the double bracket flow, in The Physics and Mathematics of Elliott Lieb, edited by R. L. Frank, A. Laptev, M. Lewin, and R. Seiringer (EMS Press, Berlin, 2022), p. 515.
  37. 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).
  38. With a slight abuse of notation, from now on we denote the Riemannian gradient by gradJ=[H,|ϕ〉〈ϕ|], noting that due to the invariance of the Hilber-Schmidt inner product 〈X,Y〉=Tr{X†Y} with respect to U, the inner product between tangent space elements X,Y∈TUSU(d) and the inner product between elements X,Y∈su(d) is the same.
  39. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ht2m-1j91 for detailed proofs of Lemma t1, Theorem  t2, and Theorem t3 as well as details of simulations for projected RGD.
  40. S. R. White, Minimally entangled typical quantum states at finite temperature, Phys. Rev. Lett. 102, 190601 (2009).
  41. K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  42. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  43. D. H. Gutman and N. Ho-Nguyen, Coordinate descent without coordinates: Tangent subspace descent on Riemannian manifolds, Math. Oper. Res. 48, 127 (2023).
  44. J. Kempe, A. Kitaev, and O. Regev, The complexity of the local Hamiltonian problem, SIAM J. Comput. 35, 1070 (2006).
  45. N. McMahon, ProjRGD_ITE, Code for paper Equating quantum imaginary time evolution, Riemannian gradient flows, and stochastic implementations (2025), https://github.com/qnla/ProjRGD_ITE.

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Phys. Rev. Research 8, 023024– Published 7 April, 2026
  • Received 25 April 2025
  • Revised 19 November 2025
  • Accepted 25 February 2026
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