Zermelo Navigation in the Quantum Brachistochrone

TL;DR

This paper explores optimal control strategies for implementing quantum gates under specific constraints, using Zermelo navigation and Randers metrics to identify time-efficient Hamiltonian trajectories in finite-dimensional quantum systems.

Contribution

It introduces a novel geometric approach using Randers metrics to determine time optimal Hamiltonians for quantum gate implementation under norm-based constraints.

Findings

Derived all geodesics for the constrained control problem.

Re-derived results of Carlini et al. using alternative methods.

Formulated a differential equation system for optimal Hamiltonians.

Abstract

We analyse the optimal times for implementing unitary quantum gates in a constrained finite dimensional controlled quantum system. The family of constraints studied is that the permitted set of (time dependent) Hamiltonians is the unit ball of a norm induced by an inner product on su(n). We also consider a generalisation of this to arbitrary norms. We construct a Randers metric, by applying a theorem of Shen on Zermelo navigation, the geodesics of which are the time optimal trajectories compatible with the prescribed constraint. We determine all geodesics and the corresponding time optimal Hamiltonian for a specific constraint on the control i.e. k (Tr(Hc(t)^2) = 1 for any given value of k > 0. Some of the results of Carlini et. al. are re-derived using alternative methods. A first order system of differential equations for the optimal Hamiltonian is obtained and shown to be of the form of the Euler Poincare equations. We illustrate that this method can form a methodology for determining which physical substrates are effective at supporting the implementation of fast quantum computation.