2025-10-02 –, Jean-Baptiste Say Amphitheater
Language: English
Quantum technologies exploit small-scale systems to advance computation, communication, and metrology. In this work we develop theory and algorithms for optimal pulse control using indirect methods based on Pontryagin maximum principle. Simulations and numerical solution are obtained by means of the Julia package OptimalControl.jl to couple direct and indirect methods. We also use Julia to implement efficient algorithms to generate the free Lie algebra in the Hall basis to assess controllability of quantum systems.
Quantum control with Julia: Optimal pulses and Lie algebra methods
D. Tinoco[^1], I. Beschastnyi[^1], J.-B. Caillau[^2]
Abstract
Quantum technologies exploit small-scale systems to advance computation, communication, and metrology. In this work we develop theory and algorithms for optimal pulse control using indirect methods based on Pontryagin maximum principle. Simulations and numerical solution are obtained by means of the Julia package OptimalControl.jl to couple direct and indirect methods. We also use Julia to implement efficient algorithms to generate the free Lie algebra in the Hall basis to assess controllability of quantum systems.
Description
Quantum technologies exploit small-scale physical systems and are expected to revolutionize computing, communication, and metrology fields. However, these systems are highly sensitive and short-lived, making them difficult to control. Our goal is to develop a theoretical and computational framework for efficiently steering quantum systems using pulse controls. This type of control is currently one of the fastest and most practical way to manipulate real-world quantum systems. Our approach is grounded in Pontryagin’s maximum principle, which helps us characterize time-optimal controls. To ensure that we can actually steer a system between desired states, we analyse controllability. This is done by generating the free Lie algebra of the controlled system, using a Hall basis built from Lyndon words. We implement our methods in Julia, both for the controllability analysis and for the numerical resolution with direct transcription and indirect methods (aka shooting) thanks to the package OptimalControl.jl [2]. As a first use case, we successfully reproduce known optimal pulse sequences from the literature [1]. We also verify the controllability of quantum systems generating the associated free Lie algebra. In particular, we leverage some tools from the package PauliStrings.jl [3] that performs fast group operations by efficiently encoding Pauli strings.
- Ansel, Q.; Dionis, E.; Arrouas, F.; Peaudecerf, B.; Guérin, S.; Sugny, D. Introduction to theoretical and experimental aspects of quantum optimal control. J. Phys. B 57 (2024), no. 13, 133001.
- Caillau, J.-B., Cots, O., Gergaud, J., Martinon, P., & Sed, S. OptimalControl.jl: a Julia package to model and solve optimal control problems with ODE's. doi.org/10.5281/zenodo.13336563
- Loizeau, N.; Peacock, C. J.; Sels, D. Quantum many-body simulations with PauliStrings.jl. SciPost Phys. Codebases 54 (2025)
[^1]: Université Côte d'Azur, Inria, CNRS, LJAD
[^2]: Université Côte d'Azur, CNRS, Inria, LJAD
I am currently a PhD student at Inria, supervised by Jean-Baptiste Caillau and Ivan Beschastnyi. My research topic is in quantum control theory, with a focus on nearly optimal pulse control of quantum systems. The aim of this project is to develop both theoretical and numerical methods for the fast and efficient manipulation of quantum systems, which are known to be highly sensitive to noise and decoherence.