Constructing general unitary maps from state preparations

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

Research output: Contribution to journalArticle

10 Citations (Scopus)

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 languageEnglish (US)
Article number023424
JournalPhysical Review A
Volume80
Issue number2
DOIs
StatePublished - Aug 28 2009

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preparation
eigenvectors
Hilbert space
alkali metals
logic
resources
gradients
electronics
atoms

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics

Cite this

Constructing general unitary maps from state preparations. / Merkel, Seth T.; Brennen, Gavin; Jessen, Poul S; Deutsch, Ivan H.

In: Physical Review A, Vol. 80, No. 2, 023424, 28.08.2009.

Research output: Contribution to journalArticle

Merkel, Seth T. ; Brennen, Gavin ; Jessen, Poul S ; Deutsch, Ivan H. / Constructing general unitary maps from state preparations. In: Physical Review A. 2009 ; Vol. 80, No. 2.
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