### Abstract

We show that a set of gates that consists of all one-bit quantum gates [U(2)] and the two-bit exclusive-OR gate [that maps Boolean values (x,y) to (x,xy)] is universal in the sense that all unitary operations on arbitrarily many bits n [U(2n)] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical and of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.

Original language | English (US) |
---|---|

Pages (from-to) | 3457-3467 |

Number of pages | 11 |

Journal | Physical Review A |

Volume | 52 |

Issue number | 5 |

DOIs | |

State | Published - 1995 |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Atomic and Molecular Physics, and Optics

### Cite this

*Physical Review A*,

*52*(5), 3457-3467. https://doi.org/10.1103/PhysRevA.52.3457

**Elementary gates for quantum computation.** / Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; Divincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald.

Research output: Contribution to journal › Article

*Physical Review A*, vol. 52, no. 5, pp. 3457-3467. https://doi.org/10.1103/PhysRevA.52.3457

}

TY - JOUR

T1 - Elementary gates for quantum computation

AU - Barenco, Adriano

AU - Bennett, Charles H.

AU - Cleve, Richard

AU - Divincenzo, David P.

AU - Margolus, Norman

AU - Shor, Peter

AU - Sleator, Tycho

AU - Smolin, John A.

AU - Weinfurter, Harald

PY - 1995

Y1 - 1995

N2 - We show that a set of gates that consists of all one-bit quantum gates [U(2)] and the two-bit exclusive-OR gate [that maps Boolean values (x,y) to (x,xy)] is universal in the sense that all unitary operations on arbitrarily many bits n [U(2n)] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical and of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.

AB - We show that a set of gates that consists of all one-bit quantum gates [U(2)] and the two-bit exclusive-OR gate [that maps Boolean values (x,y) to (x,xy)] is universal in the sense that all unitary operations on arbitrarily many bits n [U(2n)] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical and of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.

UR - http://www.scopus.com/inward/record.url?scp=34748841353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34748841353&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.52.3457

DO - 10.1103/PhysRevA.52.3457

M3 - Article

VL - 52

SP - 3457

EP - 3467

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 5

ER -