Constructing general unitary maps from state preparations

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.