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  • Open Access

Geometric quantum control and the random Schrödinger equation

Rufus Lawrence*, Aleš Wodecki†, and Johannes Aspman‡

Llorenç Balada Gaggioli§

Jakub Mareček∥

  • *Contact author: lawrejam@fel.cvut.cz
  • †Contact author: wodecki.ales@fel.cvut.cz
  • ‡Contact author: johannes.aspman@cvut.cz
  • §Contact author: llorenc.balada.gaggioli@fel.cvut.cz
  • ∥Contact author: jakub.marecek@fel.cvut.cz
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Phys. Rev. Research 8, 013150 – Published 10 February, 2026

DOI: https://doi.org/10.1103/y8nw-n1wb

Abstract

Understanding and mitigating noise in quantum systems is a fundamental challenge in achieving scalable and fault-tolerant quantum computation. Error modeling for quantum systems can be formulated in many ways, some of which are very fundamental, but hard to analyze (evolution by general dynamical map) and others perhaps too simplistic to represent physical reality. In this paper, we present an intermediate approach, introducing the random Schrödinger equation, with a noise term given by a time-varying random Hermitian matrix as a means to model noisy quantum systems. We derive bounds on the error of the synthesized unitary in terms of bounds on the norm of the noise, and show that for certain noise processes these bounds are tight. We then show that in certain situations, minimizing the error is equivalent to finding a geodesic on SU(n) with respect to a Riemannian metric encoding the coupling between the control pulse and the noise process, thus connecting our work to the complexity geometry pioneered by Michael Nielsen.

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References (34)

  1. W. Parvaiz, J. Aspman, A. Wodecki, G. Korpas, and J. Marecek, Identifiability of autonomous and controlled open quantum systems, arXiv:2501.05270.
  2. V. P. Belavkin, A stochastic posterior Schrödinger equation for counting nondemolition measurement, Lett. Math. Phys. 20, 85 (1990).
  3. N. Gisin and I. C. Percival, The quantum-state diffusion model applied to open systems, J. Phys. A: Math. Gen. 25, 5677 (1992).
  4. W. T. Strunz, The Brownian motion stochastic Schrödinger equation, Chem. Phys. 268, 237 (2001).
  5. I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).
  6. A. Chenu, M. Beau, J. Cao, and A. del Campo, Quantum simulation of generic many-body open system dynamics using classical noise, Phys. Rev. Lett. 118, 140403 (2017).
  7. H. Xue and X.-H. Deng, Traversing quantum control robustness landscapes: A new paradigm for quantum gate engineering, arXiv:2412.19473.
  8. R. L. Kosut, G. Bhole, and H. Rabitz, Robust quantum control: Analysis & synthesis via averaging, arXiv:2208.14193.
  9. J. Aspman, V. Kungurtsev, and J. Marecek, Robust quantum gate complexity: Foundations, arXiv:2404.15828.
  10. G. Dirr, U. Helmke, I. Kurniawan, and T. Schulte-Herbrüggen, Lie-semigroup structures for reachability and control of open quantum systems: Kossakowski-Lindblad generators form Lie wedge to Markovian channels, Rep. Math. Phys. 64, 93 (2009).
  11. S. Kallush, M. Khasin, and R. Kosloff, Quantum control with noisy fields: Computational complexity versus sensitivity to noise, New J. Phys. 16, 015008 (2014).
  12. M. M. Müller, S. Gherardini, T. Calarco, S. Montangero, and F. Caruso, Information theoretical limits for quantum optimal control solutions: Error scaling of noisy control channels, Sci. Rep. 12, 21405 (2022).
  13. R. L. Kosut and H. Rabitz, Robust quantum control for set-membership Hamiltonian uncertainty, in 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), Bellevue, WA, USA (IEEE, New York, NY, 2023), Vol. 1, pp. 1304–1307.
  14. C. Altafini, Coherent control of open quantum dynamical systems, Phys. Rev. A 70, 062321 (2004).
  15. S. Kallush, R. Dann, and R. Kosloff, Controlling the uncontrollable: Quantum control of open-system dynamics, Sci. Adv. 8, eadd0828 (2022).
  16. Z. Yang, R. L. Kosut, and K. B. Whaley, Hamiltonian switching control of noisy bipartite qubit systems, New J. Phys. 26, 023051 (2024).
  17. H. N. Nguyen, F. Motzoi, M. Metcalf, K. B. Whaley, M. Bukov, and M. Schmitt, Reinforcement learning pulses for transmon qubit entangling gates, Mach. Learn.: Sci. Technol. 5, 025066 (2024).
  18. N. Funcke and J. Berberich, Robustness of optimal quantum annealing protocols, New J. Phys. 26, 093040 (2024).
  19. M. A. Nielsen, A geometric approach to quantum circuit lower bounds, Quantum Inf. Comput. 6, 213 (2006).
  20. M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Optimal control, geometry, and quantum computing, Phys. Rev. A 73, 062323 (2006).
  21. M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum computation as geometry, Science 311, 1133 (2006).
  22. M. R. Dowling and M. A. Nielsen, The geometry of quantum computation, Quantum Inf. Comput. 8, 861 (2008).
  23. X. Han and P. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution (Springer, 2017), Vol. 85, p. 12.
  24. J. L. Strand, Random ordinary differential equations, J. Diff. Equ. 7, 538 (1970).
  25. N. Lambert, E. Giguère, P. Menczel, B. Li, P. Hopf, G. Suárez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal, A. Galicia, N. Shammah, P. Nation, J. R. Johansson, S. Ahmed, S. Cross, A. Pitchford, and F. Nori, QuTiP 5: The quantum toolbox in Python, arXiv:2412.04705.
  26. P. Martinez-Azcona, A. Kundu, A. Saxena, A. del Campo, and A. Chenu, Quantum dynamics with stochastic non-Hermitian Hamiltonians, Phys. Rev. Lett. 135, 010402 (2025).
  27. E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, 1/f noise: Implications for solid-state quantum information, Rev. Mod. Phys. 86, 361 (2014).
  28. A. R. Brown and L. Susskind, Complexity geometry of a single qubit, Phys. Rev. D 100, 046020 (2019).
  29. E. Meinrenken, Lie groups and Lie algebras, arXiv:2201.09397.
  30. E. Meinrenken, Clifford Algebras and Lie Theory (Springer, 2013).
  31. K. Shizume, T. Nakajima, R. Nakayama, and Y. Takahashi, Quantum computational Riemannian and sub-Riemannian geodesics, Prog. Theor. Phys. 127, 997 (2012).
  32. L. B. Gaggioli and R. Lawrence, Numerical study of the random Schrödinger equation, GitHub (2025), https://github.com/llorebaga/RandomSE.
  33. J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev. 59, 65 (2017).
  34. P. K. Mogensen and A. N. Riseth, Optim: A mathematical optimization package for Julia, J. Open Source Software 3, 615 (2018).

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Phys. Rev. Research 8, 013150– Published 10 February, 2026
  • Received 2 June 2025
  • Accepted 19 December 2025
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