Quantum Algorithms for Quantum Optimal Control Problems with Precise Cost Estimates

Quantum properties are largely responsible for many recent developments in material science and chemical engineering. Such properties are best utilized with external controls. Quantum optimal control (QOC) algorithms, which use first principle-based computer simulations to identify the desired control variables, have been an important route in this direction.

QOC problems have also emerged in recent quantum computing and quantum information technology. For example, the paradigm of QOC has remarkable connections to variational quantum algorithms (VQA) and quantum machine learning. Yes, another interesting observation is that quantum control has the potential to suppress decoherence. It is increasingly clear that an efficient algorithm for quantum optimal control problems can have an impact far beyond the domains from which it originated.



Recent development in quantum algorithms has pointed in a promising direction. However, due to the various source of errors, including discretization, gradient estimation, and optimization errors, and the non-convexity of the objective function, optimizing the complexity using state-of-the-art quantum algorithms is an outstanding challenge. We present two quantum algorithms for solving the quantum optimal control problem: one is based on simulating time-dependent Hamiltonian evolution and quantum gradient estimation, and the other is based on solving linear systems and phase estimation. We quantify the error from various steps in the approximation, including the finite-dimensional representation of the wave function, the discretization of the Schrodinger equation, the numerical quadrature, and optimization. The error quantification leads to explicit complexity estimates.