Efficient Quantum Algorithms for Quantum Optimal Control
This work addresses a fundamental problem in quantum computing and control theory, with potential applications in quantum technologies and machine learning, though it is incremental as it builds on existing quantum simulation methods.
The paper tackles the quantum optimal control problem, which involves finding control variables to maximize a physical quantity in quantum systems governed by the Schrödinger equation, and presents quantum algorithms that are exponentially faster than classical ones, with comprehensive error analysis.
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 $T$, 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.