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Extending the works \\cite{Hastings2017} and \\cite{Campbell2017}, we show that taking probabilistic mixtures of channels to solve fallback \\cite{BRS2015} and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+T<\/mml:mi><\/mml:mrow><\/mml:msqrt><\/mml:math>gate set we achieve an average non-Clifford gate count of0.23<\/mml:mn>log<\/mml:mi>2<\/mml:mn><\/mml:msub>&#x2061;<\/mml:mo>(<\/mml:mo>1<\/mml:mn>\/<\/mml:mo><\/mml:mrow>&#x03B5;<\/mml:mi>)<\/mml:mo>+<\/mml:mo>2.13<\/mml:mn><\/mml:math>and T-count0.56<\/mml:mn>log<\/mml:mi>2<\/mml:mn><\/mml:msub>&#x2061;<\/mml:mo>(<\/mml:mo>1<\/mml:mn>\/<\/mml:mo><\/mml:mrow>&#x03B5;<\/mml:mi>)<\/mml:mo>+<\/mml:mo>5.3<\/mml:mn><\/mml:math>with mixed fallback approximations for diamond norm accuracy&#x03B5;<\/mml:mi><\/mml:math>.This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+T<\/mml:mi><\/mml:mrow><\/mml:msqrt><\/mml:math>). We also provide detailed numerical results for Clifford+T and Clifford+T<\/mml:mi><\/mml:mrow><\/mml:msqrt><\/mml:math>gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices.<\/jats:p>","DOI":"10.22331\/q-2023-12-18-1208","type":"journal-article","created":{"date-parts":[[2023,12,18]],"date-time":"2023-12-18T15:43:05Z","timestamp":1702914185000},"page":"1208","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":25,"title":["Shorter quantum circuits via single-qubit gate approximation"],"prefix":"10.22331","volume":"7","author":[{"given":"Vadym","family":"Kliuchnikov","sequence":"first","affiliation":[{"name":"Microsoft Quantum, Redmond, WA, US"},{"name":"Microsoft Quantum, Toronto, ON, CA"}]},{"given":"Kristin","family":"Lauter","sequence":"additional","affiliation":[{"name":"Facebook AI Research, Seattle, WA, US"}]},{"given":"Romy","family":"Minko","sequence":"additional","affiliation":[{"name":"University of Oxford, Oxford, UK"},{"name":"Heilbronn Institute for Mathematical Research, University of Bristol, Bristol, UK"}]},{"given":"Adam","family":"Paetznick","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Redmond, WA, US"}]},{"given":"Christophe","family":"Petit","sequence":"additional","affiliation":[{"name":"University of Birmingham, Birmingham, UK"},{"name":"Universit\u00e9 Libre de Bruxelles, Brussels, Belgium"}]}],"member":"9598","published-online":{"date-parts":[[2023,12,18]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. 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