Elementary solution to the time-independent quantum navigation problem

TL;DR

This paper presents a simple, trigonometry-based solution for the time-independent quantum navigation problem in two-dimensional systems, addressing the challenge of finding optimal Hamiltonians under background influences.

Contribution

It introduces an elementary, closed-form solution for the quantum navigation problem in two-dimensional Hilbert spaces, and discusses challenges in extending to higher dimensions.

Findings

Solution based on trigonometric analysis for 2D systems

Discussion on complexities of higher-dimensional generalizations

Foundational approach for quantum control under background Hamiltonians

Abstract

A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realises a required quantum process or task, under the influence of a prevailing `background' Hamiltonian that cannot be manipulated. When the task is to transform one quantum state into another, finding the solution in closed form to the problem is nontrivial even in the case of time-independent Hamiltonians. An elementary solution, based on trigonometric analysis, is found here when the Hilbert space dimension is two. Difficulties arising from generalisations to higher-dimensional systems are discussed.