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OSA Quantum 2.0 Conference

Using Quantum Algorithms to Solve Optimal Control Problems

Tsung‐Wei Huang, Wei‐Chen Chien, Ching‐Ray Chang

Published Jan 1, 2020• 0 citations
Publisher

Abstract

We use quantum algorithms to solve optimal control problems. Compared with the gradient method usually used in classical algorithms, the time complexity down from O(Ni) to O(Ni/2log(N)).
DOI
10.1364/quantum.2020.qw6a.19

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In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical quantity at time TTT, where the system is governed by a time-dependent Schrödinger equation. This type of control problem also has an intricate relation with machine learning. Our algorithms are based on a time-dependent Hamiltonian simulation method and a fast gradient-estimation algorithm. We also provide a comprehensive error analysis to quantify the total error from various steps, such as the finite-dimensional representation of the control function, the discretization of the Schrödinger equation, the numerical quadrature, and optimization. Our quantum algorithms require fault-tolerant quantum computers.
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Optimal control is one of the most popular and efficient control strategies that have been widely used in both classical and quantum systems. Inspired by the recent progresses of quantum algorithms, we find that optimal control protocols can be exponentially accelerated by embedding a quantum controller in the control loop instead of the classical controller. This is the first time to our knowledge that a quantum controller has been shown to exponentially improve the efficiency of control compared with classical controller. Our method opens up new perspectives in various fields, especially those related to the synthesis of large-scale complex systems.
As industrial models and designs grow increasingly complex, the demand for optimal control of large-scale dynamical systems has significantly increased. However, traditional methods for optimal control incur significant overhead as problem dimensions grow. In this paper, we introduce an end-to-end quantum algorithm for linear-quadratic control with provable speedups. Our algorithm, based on a policy gradient method, incorporates a novel quantum subroutine for solving the matrix Lyapunov equation. Specifically, we build a quantum-assisted differentiable simulator for efficient gradient estimation that is more accurate and robust than classical methods relying on stochastic approximation. Compared to the classical approaches, our method achieves a super-quadratic speedup. To the best of our knowledge, this is the first end-to-end quantum application to linear control problems with provable quantum advantage.