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Trading Inverses for an Irrep in the Solovay-Kitaev Theorem

Authors Adam Bouland , Maris Ozols

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Adam Bouland
  • Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA
Maris Ozols
  • QuSoft and University of Amsterdam, Amsterdam, Netherlands

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Adam Bouland and Maris Ozols. Trading Inverses for an Irrep in the Solovay-Kitaev Theorem. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 6:1-6:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error epsilon using merely polylog(1/epsilon) gates from any finite universal quantum gate set G. One drawback to the theorem is that it requires the gate set G to be closed under inversion. Here we show that this restriction can be traded for the assumption that G contains an irreducible representation of any finite group G. This extends recent work of Sardharwalla et al. [Sardharwalla et al., 2016], and applies also to gates from the special linear group. Our work can be seen as partial progress towards the long-standing open problem of proving an inverse-free Solovay-Kitaev theorem [Dawson and Nielsen, 2006; Kuperberg, 2015].

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Solovay-Kitaev theorem
  • quantum gate sets
  • gate set compilation


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