Constructing general unitary maps from state preparations

Seth T. Merkel, Gavin Brennen, Poul S. Jessen, Ivan H. Deutsch

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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 languageEnglish (US)
Article number023424
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Issue number2
StatePublished - Aug 28 2009

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics


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