Algebraic synthesis of time-optimal unitaries in SU(2) with alternating controls

TL;DR

This paper develops an algebraic method to determine the shortest possible control sequences for implementing any unitary in SU(2) with two non-parallel control axes, improving efficiency in quantum control.

Contribution

It introduces a novel algebraic framework that simplifies the analysis of time-optimal unitaries in SU(2) with restricted controls, providing new bounds and structural insights.

Findings

Finite time-optimal sequences are at most five concatenations with bidirectional controls.

Strict bounds on sequence length and structure improve numerical search for optimal controls.

Results differ significantly depending on whether rotations are unidirectional or bidirectional.

Abstract

We present an algebraic framework to study the time-optimal synthesis of arbitrary unitaries in SU(2), when the control set is restricted to rotations around two non-parallel axes in the Bloch sphere. Our method bypasses commonly used control-theoretical techniques, and easily imposes necessary conditions on time-optimal sequences. In a straightforward fashion, we prove that time-optimal sequences are solely parametrized by three rotation angles and derive general bounds on those angles as a function of the relative rotation speed of each control and the angle between the axes. Results are substantially different whether both clockwise and counterclockwise rotations about the given axes are allowed, or only clockwise rotations. In the first case, we prove that any finite time-optimal sequence is composed at most of five control concatenations, while for the more restrictive case, we present scaling laws on the maximum length of any finite time-optimal sequence. The bounds we find for both cases are stricter than previously published ones and severely constrain the structure of time-optimal sequences, allowing for an efficient numerical search of the time-optimal solution. Our results can be used to find the time-optimal evolution of qubit systems under the action of the considered control set, and thus potentially increase the number of realizable unitaries before decoherence.