arXiv:2001.06464

Quantum Optimal Control via Magnus Expansion and Non-Commutative Polynomial Optimization

Jakub Mareček, Jiří Vala

Published Jan 17, 2020• 1 citation

Abstract

Quantum optimal control has numerous important applications ranging from pulse shaping in magnetic-resonance imagining to laser control of chemical reactions and quantum computing. Our objective is to address two major challenges that have limited the success of applications of quantum optimal control so far: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. Methodologically, we address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-convexity, we employ non-commutative polynomial optimisation and non-commutative geometry. As a result, we present the first globally convergent methods for quantum optimal control.

DOI: 10.48550/arxiv.2001.06464

Related papers

Quantum Optimal Control via Magnus Expansion: The Non-Commutative Polynomial Optimization Problem

Jakub Mareček, Jiří Vala

Quantum optimal control has important applications in quantum computing. For example in transmon qubits, two-qubit gates are implemented by shaping a microwave pulse. More generally, one can replace the application of an entire quantum circuit with a control signal. Two major challenges have limited the success of applications of quantum optimal control so far: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. We present first globally convergent methods for quantum optimal control. We address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-convexity, we employ non-commutative polynomial optimisation and non-commutative geometry. This makes it possible to revisit much of what is done in quantum computing.

Jan 17, 20201 citation

Lie algebra-assisted quantum simulation and quantum optimal control via high-order Magnus expansions

R. F. dos Santos, S. J. J. M. F. Kokkelmans

The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and non-commutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short time-scales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach that reduces the computational effort to depend only on the degrees of freedom defining the time-dependent control function. We focus specifically on Hamiltonians consisting of a constant drift term and a controllable term. Our method provides a polynomial expression for the Magnus expansion which can be evaluated several orders of magnitude faster than previous techniques, enabling broad applications in the realms of quantum simulation and quantum optimal control. We showcase an application of the method by designing control pulses for the 5-qubit phase gate on a neutral-atom platform utilizing Rydberg atoms.

quant-pharXiv:2512.20357Physical Review ADec 24, 20250 citations

Monotonically convergent optimization in quantum control using Krotov's method

Daniel M. Reich, Mamadou Ndong, Christiane P. Koch

The non-linear optimization method developed by A. Konnov and V. Krotov [Autom. Remote Cont. (Engl. Transl.) 60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the non-linear Schrödinger equation [S. E. Sklarz and D. J. Tannor, Phys. Rev. A 66, 053619 (2002)]. Here we show that based on the Konnov-Krotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to nonlinear equations of motion, control problems that are characterized by non-unitary time evolution, nonlinear dependencies of the Hamiltonian on the control, time-dependent targets, and optimization functionals that depend to higher than second order on the time-evolving states. We furthermore show that the nonlinear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimization functional that is an eighth-degree polynomial in the states. For the "standard" quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, the second-order contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first- and second-order algorithms for two examples.

arXiv:1008.5126The Journal of Chemical PhysicsMar 9, 2012219 citations

Unitary Gate Synthesis via Polynomial Optimization

Llorenç Balada Gaggioli, D.S. Bondar, Jiří Vala +2 more

Quantum optimal control plays a crucial role in the development of quantum technologies, particularly in the design and implementation of fast and accurate gates for quantum computing. Here, we present a method to synthesize gates using the Magnus expansion. In particular, we formulate a polynomial optimization problem that allows us to find the global solution without resorting to approximations of the exponential. The global method we use provides a certificate of globality and lets us do single-shot optimization, which implies it is generally faster than local methods. By optimizing over Hermitian matrices generating the unitaries, instead of the unitaries themselves, we can reduce the size of the polynomial to optimize, leading to fast convergence and scalability. Numerical experiments comparing our results with CRAB and GRAPE show that we maintain high accuracy while providing globality certificates.

arXiv:2508.01356Aug 2, 20250 citations

Quantum-Inspired Optimization for Non-Convex Functions

Jincheng Zhang

Quantum-inspired optimization represents a paradigm shift in the field of optimization, aiming to accelerate the convergence of algorithms for complex, non-convex functions. This paper explores the application of quantum mechanics principles to model objective functions and leverage quantum algorithms to achieve enhanced performance. We present a novel quantum-inspired optimization technique, demonstrating its potential to overcome the limitations of traditional methods in solving challenging optimization problems. The core mechanism hinges on a sophisticated approach to representing the objective function as a quantum system, utilizing quantum algorithms to approximate solutions and thereby accelerate convergence. The research highlights the key advantages of this approach, particularly in tackling non-convex functions, and discusses future directions for research and development.

Sep 3, 20260 citations

Nonlinear Control Theory and Quantum System Modeling: A Combined Approach

Jincheng Zhang

This paper explores the integration of nonlinear control theory with quantum system modeling, presenting a novel approach to designing and optimizing control strategies for quantum systems. The core claim centers on leveraging the optimization techniques within nonlinear control theory to address the inherent complexities of quantum system dynamics. The mechanism involves formulating quantum systems' nonlinear dynamics into equivalent control problems, subsequently employing nonlinear control theory's optimization methods for solution. This combined approach promises enhanced precision and efficiency in quantum system control. The work demonstrates a theoretical framework for tackling this challenge, paving the way for future research and practical applications in areas such as quantum computing and quantum sensing. The key mathematical concepts utilized are presented, including Hamiltonian formalism, Liouville-Perron equation, and nonlinear control theory formulations. The final results offer a pathway to more effective manipulation of quantum states.

Sep 6, 20260 citations