Quantum algorithms for testing properties of distributions

Sergey Bravyi; Aram W. Harrow; Avinatan Hassidim

doi:10.1109/TIT.2011.2134250

Quantum algorithms for testing properties of distributions

Sergey Bravyi, Aram W. Harrow, Avinatan Hassidim

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

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 L₁-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 L₁-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
Article number	5773032
Pages (from-to)	3971-3981
Number of pages	11
Journal	IEEE Transactions on Information Theory
Volume	57
Issue number	6
DOIs	https://doi.org/10.1109/TIT.2011.2134250
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.

Funding

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.

Funders	Funder number
EPRSC	IST-2005-15848
Defense Advanced Research Projects Agency	FA-9550-09-1-0044, HR0011-09-C-0047

Keywords

Property testing
quantum information
query complexity
sampling
statistical distance

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10.1109/TIT.2011.2134250

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title = "Quantum algorithms for testing properties of distributions",

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 L1-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 L1-distance ∥ p-q ∥1 can be estimated with a constant precision using only O(N1/2) queries in the quantum settings, whereas classical computers need Ω(N1-o(1)) queries. We also describe quantum algorithms for testing uniformity and orthogonality with query complexity O(N1/3). The classical query complexity of these problems is known to be Ω(N1/2).",

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AB - 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 L1-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 L1-distance ∥ p-q ∥1 can be estimated with a constant precision using only O(N1/2) queries in the quantum settings, whereas classical computers need Ω(N1-o(1)) queries. We also describe quantum algorithms for testing uniformity and orthogonality with query complexity O(N1/3). The classical query complexity of these problems is known to be Ω(N1/2).

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Quantum algorithms for testing properties of distributions

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