Quantum Control via Geometry: An explicit example

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.

6 pages, 2 figures. accepted into PRA