Quantum Control via Geometry: An explicit example

TL;DR

This paper explicitly calculates the optimal control cost for specific geometric quantum control problems based on Cartan decompositions, providing insights into minimal interaction costs in two-qubit systems.

Contribution

It offers a concrete example of computing optimal costs in geometric quantum control using Cartan decompositions, highlighting minimal interaction costs for two-qubit unitaries.

Findings

Explicit computation of optimal control cost for example problems

Connection between Cartan decomposition and control cost

Minimal interaction cost for two-qubit unitaries

Abstract

We explicitly compute the optimal cost for a class of example problems in geometric quantum control. These problems are defined by a Cartan decomposition of $su(2^n)$ into orthogonal subspaces $\mathfrak{l}$ and $\mathfrak{p}$ such that $[\mathfrak{l},\mathfrak{l}] \subseteq \mathfrak{p}, [\mathfrak{p},\mathfrak{l}] = \mathfrak{p}, [\mathfrak{p},\mathfrak{p}] \subseteq \mathfrak{l}$. Motion in the $\mathfrak{l}$ direction are assumed to have negligible cost, where motion in the $\mathfrak{p}$ direction do not. In the special case of two qubits, our results correspond to the minimal interaction cost of a given unitary.