Time-optimal control of two-level quantum systems by piecewise constant pulses

TL;DR

This paper derives time-optimal control strategies for two-level quantum systems using piecewise constant pulses, establishing quantum speed limits and analyzing convergence behaviors.

Contribution

It extends the Pontryagin Maximum Principle to quantum control, providing exact speed limits and convergence analysis for piecewise constant pulse controls.

Findings

Exact quantum speed limits as a function of sampling period

Exponential convergence towards minimum time in the continuous limit

Polynomial convergence for linearized quantum systems

Abstract

We apply an extension of the Pontryagin Maximum Principle to derive time-optimal controls of two-level quantum systems by means of piecewise constant pulses. Global optimal solutions are obtained for state-to-state transfer in the cases with one and two controls. Exact quantum speed limits are established as a function of the sampling period. We observe numerically an exponential convergence towards the minimum time in the continuous limit when this period goes to zero. We show that this convergence is only polynomial for a linearized quantum system. We discuss the experimental impact of this result.