### 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) |
---|---|

Article number | 023424 |

Journal | Physical Review A |

Volume | 80 |

Issue number | 2 |

DOIs | |

State | Published - Aug 28 2009 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*Physical Review A*,

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

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

Research output: Contribution to journal › Article

*Physical Review A*, vol. 80, no. 2, 023424. https://doi.org/10.1103/PhysRevA.80.023424

}

TY - JOUR

T1 - Constructing general unitary maps from state preparations

AU - Merkel, Seth T.

AU - Brennen, Gavin

AU - Jessen, Poul S

AU - Deutsch, Ivan H.

PY - 2009/8/28

Y1 - 2009/8/28

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=69449105122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69449105122&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.80.023424

DO - 10.1103/PhysRevA.80.023424

M3 - Article

AN - SCOPUS:69449105122

VL - 80

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 2

M1 - 023424

ER -