Constructing general unitary maps from state preparations
Seth Merkel, Gavin K. Brennen, Poul Jessen, Ivan Deutsch · Physical Review A · 2009
We present an efficient algorithm for generating unitary maps on a $d$-dimensional Hilbert space from a time-dependent Hamiltonian through a combination of stochastic searches and geometric construction. The protocol is based on the eigendecomposition of the map. A unitary matrix can be implemented by sequentially mapping each eigenvector to a fiducial state, imprinting the eigenphase on that state, and mapping it back to the eigenvector. This requires the design of only $d$ state-to-state maps generated by control wave forms that are efficiently found by a gradient search with computational resources that scale polynomially in $d$. In contrast, the complexity of a stochastic search for a single wave form that simultaneously acts as desired on all eigenvectors scales exponentially in $d$. We extend this construction to design maps on an $n$-dimensional subspace of the Hilbert space using only $n$ stochastic searches. Additionally, we show how these techniques can be used to control atomic spins in the ground-electronic hyperfine manifold of alkali metal atoms in order to implement general qudit logic gates as well to perform a simple form of error correction on an embedded qubit.