## Abstract

Suppose one has access to oracles generating samples from two unknown probability distributions p and q on some n -element set. How many samples does one need to test whether the two distributions are close or far from each other in the L_{1}-norm? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L_{1}-distance ∥ p-q ∥_{1} can be estimated with a constant precision using only O(N^{1/2}) queries in the quantum settings, whereas classical computers need Ω(N^{1-o(1)}) queries. We also describe quantum algorithms for testing uniformity and orthogonality with query complexity O(N^{1/3}). The classical query complexity of these problems is known to be Ω(N^{1/2}).

Original language | English |
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Article number | 5773032 |

Pages (from-to) | 3971-3981 |

Number of pages | 11 |

Journal | IEEE Transactions on Information Theory |

Volume | 57 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2011 |

Externally published | Yes |

### Bibliographical note

Funding Information:Manuscript received December 06, 2009; revised June 18, 2010; accepted September 04, 2010. Date of current version May 25, 2011. The work of S. Bravyi was supported by the DARPA QUEST program under Contract HR0011-09-C-0047. The work of A. W. Harrow was supported by the DARPA QUEST under Grant FA-9550-09-1-0044, the U.K. EPRSC grant “QIP IRC,” and the QAP project under Contract IST-2005-15848. The work of A. Hassidim was supported by an xQIT Keck fellowship.

## Keywords

- Property testing
- quantum information
- query complexity
- sampling
- statistical distance