Quantum control via geometry: An explicit example
Mile Gu, Andrew C. Doherty, Michael A. Nielsen · Physical Review A · 2008
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 $\mathrm{su}({2}^{n})$ into orthogonal subspaces $\mathfrak{l}$ and $\mathfrak{p}$ such that $[\mathfrak{l},\mathfrak{l}]\ensuremath{\subseteq}\mathfrak{p},[\mathfrak{p},\mathfrak{l}]=\mathfrak{p},[\mathfrak{p},\mathfrak{p}]\ensuremath{\subseteq}\mathfrak{l}$. Motion in the $\mathfrak{l}$ direction is assumed to have negligible cost, where motion in the $\mathfrak{p}$ direction does not. In the special case of two qubits, our results correspond to the minimal interaction cost of a given unitary.