2026-08-14 –, Room 3
We report on recent progress in the numerical optimisation of quantum control systems using OptimalControl.jl. Although the package is designed to optimise general control systems governed by ordinary differential equations, it naturally accommodates quantum problems described by finite-dimensional Schrödinger equations evolving on Lie groups — specifically, bilinear dynamical systems whose state trajectories lie on unitary groups and are expressed compactly in terms of tensor products of complex matrices.
A well-established Julia ecosystem for quantum optimal control already exists, with packages such as QuantumControl.jl, Krotov.jl, and GRAPE.jl providing mature, quantum-tailored implementations of the GRAPE and Krotov algorithms. These methods are effective for a broad class of problems and can accommodate extensions such as free final time or path constraints on controls and states, typically via penalisation of the cost functional. However, penalisation-based approaches offer no rigorous guarantee of constraint satisfaction and can introduce significant ill-conditioning. Our motivation is complementary: to leverage state-of-the-art nonlinear programming solvers that treat such constraints directly, as genuine algebraic equalities and inequalities arising from the transcription of the continuous-time optimal control problem — including additional optimisation variables such as free final time or parameters of the system.
OptimalControl.jl offers a high-level, expressive modelling interface that allows users to specify dynamics, objectives, and constraints in a form close to mathematical notation, with no compromise on performance. Problems are transcribed via direct methods into large-scale sparse nonlinear programmes, which are solved using interior-point methods on both CPU and GPU, exploiting automatic differentiation through ExaModels.jl and MadNLP.jl. Crucially, the framework also supports the combination of direct and indirect methods: direct transcription is used first to identify the qualitative structure of the optimal solution, after which indirect shooting methods — based on the Pontryagin Maximum Principle — can be applied to refine the solution to arbitrary numerical precision.
We present preliminary results on the optimisation of small quantum systems modelling nitrogen-vacancy (NV) centres in diamond. These systems, comprising an electron spin coupled to one or more nuclear spins via hyperfine interaction, are naturally described with bilinear dynamics driven by bounded microwave controls. The combination of hard amplitude constraints, partial controllability (nuclear spins are driven only indirectly through the electron spin), and the need for various costs functionals for gate synthesis makes NV centres a compelling benchmark for our approach. We discuss the formulation of these problems within OptimalControl.jl, and compare the results and computational performance against existing quantum-specific methods.
Nitrogen-vacancy (NV) centres, and more generally colour centres in diamond, are promising physical platforms for quantum sensing and quantum information processing, owing to the long coherence times of the associated spin degrees of freedom, even at room temperature, and to the ability to initialise and read out spin states optically. The results presented in this talk are grounded in real experimental activity at the Institut Carnot de Bourgogne (ICB, Université Bourgogne Europe), where colour centres in diamond are actively investigated. The minimal but physically meaningful model we consider consists of two coupled spin-1/2 particles — one electron spin and one nuclear spin — with microwave control acting exclusively on the electron spin.
After applying the rotating wave approximation, the system reduces to a bilinear control system evolving on SU(4), with a two-dimensional control input constrained to a disc, $u_1^2(t) + u_2^2(t) \leq u^2_{\max}$. A controllability analysis carried out via Lie bracket computations and rank conditions on the Lie algebra generated by the drift and control vector fields reveals that, depending on the hyperfine coupling parameters, not every gate in SU(4) is reachable — a direct consequence of the indirect nature of the nuclear spin control. When the desired target gate lies outside the attainable set, the optimisation is reformulated as fidelity maximisation towards the closest reachable gate. For the evolution times of interest, the dynamics are sufficiently smooth that no specialised exponential integrators are required, and we test several standard numerical integration schemes within the direct transcription framework. The high-level problem modelling, including the specification of dynamics directly in terms of matrix Lie group structure and Kronecker products of complex matrices, is enabled by a new extension to LinearAlgebra within ExaModels.jl developed by the Exanauts team, which retains full compatibility with automatic differentiation.
The resulting nonlinear programmes are solved by interior-point methods from the MadSuite (MadNLP.jl), both on CPU and on GPU via CUDSS.jl. The direct transcription solution additionally serves as a warm start for an indirect shooting method obtained by applying the Pontryagin Maximum Principle to the bilinear system on SU(4), allowing the qualitative structure of the optimal control to be captured first and then refined to arbitrary numerical precision. The talk will cover the formulation within OptimalControl.jl, a comparison of direct and indirect approaches, and GPU performance benchmarks.
The joint project CONV (Control of NV-centres) between Université Côte d'Azur Math lab and Institut Carnot of Université Bourgogne Europe receives financial support from the CNRS through the MITI interdisciplinary program. J.-B. Caillau is also supported by a FACCTS grant of the France-Chicago center.
Professor of applied math at Université Côte d’Azur, CNRS, Inria, LJAD
Scientific interests - Optimisation and control: geometry, algorithms, applications