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Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets

Forest, Simon and Gosset, David and Kliuchnikov, Vadym and McKinnon, David (2015) Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets. Journal of Mathematical Physics, 56 (8). Art. No. 082201. ISSN 0022-2488. http://resolver.caltech.edu/CaltechAUTHORS:20150121-111212393

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Abstract

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov, and Mosca [Quantum Inf. Comput. 13(7,8), 607–630 (2013)]. Their algorithm takes as input an exactly synthesizable single-qubit unitary—one which can be expressed without error as a product of Clifford and T gates—and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer n, we consider the “Clifford-cyclotomic” gate set consisting of the Clifford group plus a z-rotation by π/n. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of π/n z-rotations. For the Clifford+T case n = 4, the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring Z[e^(i π/n), 1/2]. We prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100%.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1501.04944arXivDiscussion Paper
http://dx.doi.org/10.1063/1.4927100DOIArticle
http://scitation.aip.org/content/aip/journal/jmp/56/8/10.1063/1.4927100PublisherArticle
Additional Information:© 2015 AIP Publishing LLC. Received 13 February 2015; accepted 9 July 2015; published online 5 August 2015. We thank Jean-Francois Biasse and Michele Mosca for helpful discussions. D.G. and D.M. were supported in part by NSERC. D.G. was supported in part by ARO. IQC is supported in part by the Government of Canada and the province of Ontario.
Group:Walter Burke Institute for Theoretical Physics, IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Army Research Office (ARO)UNSPECIFIED
Government of CanadaUNSPECIFIED
Province of OntarioUNSPECIFIED
Non-Subject Keywords:quantum physics
Record Number:CaltechAUTHORS:20150121-111212393
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20150121-111212393
Official Citation:Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets Forest, Simon and Gosset, David and Kliuchnikov, Vadym and McKinnon, David, Journal of Mathematical Physics, 56, 082201 (2015), DOI:http://dx.doi.org/10.1063/1.4927100
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:53941
Collection:CaltechAUTHORS
Deposited By: Joy Painter
Deposited On:21 Jan 2015 20:35
Last Modified:01 Jun 2016 23:01

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