Efficient Optimal Control of Open Quantum Systems

TL;DR

This paper introduces quantum-classical hybrid algorithms for efficiently solving optimal control problems in open quantum systems, achieving exponential speedups over classical methods by leveraging quantum simulation and gradient estimation.

Contribution

It presents the first simulation algorithm for time-dependent Lindbladians with an $ ext{l}_1$-norm dependence and integrates quantum and classical techniques for optimal control.

Findings

Algorithms achieve poly-logarithmic complexity in system dimension.

Quantum simulation provides access to gradients for control optimization.

The approach offers exponential speedup over classical algorithms.

Abstract

The optimal control problem for open quantum systems can be formulated as a time-dependent Lindbladian that is parameterized by a number of time-dependent control variables. Given an observable and an initial state, the goal is to tune the control variables so that the expected value of some observable with respect to the final state is maximized. In this paper, we present algorithms for solving this optimal control problem efficiently, i.e., having a poly-logarithmic dependency on the system dimension, which is exponentially faster than best-known classical algorithms. Our algorithms are hybrid, consisting of both quantum and classical components. The quantum procedure simulates time-dependent Lindblad evolution that drives the initial state to the final state, and it also provides access to the gradients of the objective function via quantum gradient estimation. The classical procedure uses the gradient information to update the control variables. At the technical level, we provide the first (to the best of our knowledge) simulation algorithm for time-dependent Lindbladians with an $\ell_1$-norm dependence. As an alternative, we also present a simulation algorithm in the interaction picture to improve the algorithm for the cases where the time-independent component of a Lindbladian dominates the time-dependent part. On the classical side, we heavily adapt the state-of-the-art classical optimization analysis to interface with the quantum part of our algorithms. Both the quantum simulation techniques and the classical optimization analyses might be of independent interest.