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Trading Inverses for an Irrep in the Solovay-Kitaev Theorem

Authors Adam Bouland , Maris Ozols

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Author Details

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

Funding

  • Bouland, Adam: AB was partially supported by the NSF GRFP under Grant No. 1122374, by the Vannevar Bush Fellowship from the US Department of Defense, by ARO Grant W911NF-12-1-0541, by NSF Grant CCF-1410022, and by the NSF Waterman award under grant number 1249349.
  • Ozols, Maris: Part of this work was done while MO was at the University of Cambridge where he was supported by a Leverhulme Trust Early Career Fellowship (ECF-2015-256). MO also acknowledges hospitality of MIT where this project was initiated.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.TQC.2018.6

Abstract

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
Keywords
  • Solovay-Kitaev theorem
  • quantum gate sets
  • gate set compilation

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