On extrema of the objective functional for short-time generation of single qubit quantum gates

TL;DR

This paper investigates the presence of local extrema, or traps, in the optimization landscape for short-time quantum gate generation in two-level systems, proving their absence for most cases and providing bounds for others.

Contribution

It extends previous results by showing traps are absent for almost all target unitaries at arbitrarily small control durations and establishes new bounds for the remaining cases.

Findings

Traps are absent for almost all target unitaries at very short control durations.

A new lower bound for control duration guarantees trap-free optimization.

Most quantum gate generation problems are free of local extrema in the short-time regime.

Abstract

In the present work the extrema of the objective functional for the problem of generation of quantum gates (logical elements for quantum computations) for two--level systems are investigated for short duration of the control. The problem of existence of local but not global extrema, the so called traps, is considered. In prior works the absence of traps was proved for a sufficiently long control duration. In this paper we prove that for almost all target unitary operators and system Hamiltonians traps are absent for an arbitrarily small control duration. For the remainder target unitary operators and Hamiltonians we obtain a new estimate for the lower boundary of the control duration which guarantees the absence of traps.