### Abstract

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.

Original language | English (US) |
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Article number | 023424 |

Journal | Physical Review A - Atomic, Molecular, and Optical Physics |

Volume | 80 |

Issue number | 2 |

DOIs | |

State | Published - Aug 28 2009 |

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

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## Cite this

*Physical Review A - Atomic, Molecular, and Optical Physics*,

*80*(2), [023424]. https://doi.org/10.1103/PhysRevA.80.023424