### 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) |
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Pages (from-to) | 3457-3467 |

Number of pages | 11 |

Journal | Physical Review A |

Volume | 52 |

Issue number | 5 |

DOIs | |

State | Published - 1995 |

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

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## Cite this

*Physical Review A*,

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