Constructing general unitary maps from state preparations

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

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)
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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
Article number023424
Pages (from-to)1-8
Number of pages8
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Issue number2
Publication statusPublished - 28 Aug 2009

Bibliographical note

Merkel ST, Brennen G, Jessen PS, Deutsch IH, Physical Review A. Atomic, Molecular, and Optical Physics, Vol 80, Issue 2, 023424, 2009. Copyright 2009 by the American Physical Society. The original article can be found at


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