{"id":215209,"date":"2018-11-06T17:20:18","date_gmt":"2018-11-07T01:20:18","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/optimal-ancilla-free-pauliv-circuits-for-axial-rotations\/"},"modified":"2018-11-06T17:20:18","modified_gmt":"2018-11-07T01:20:18","slug":"optimal-ancilla-free-pauliv-circuits-for-axial-rotations","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/optimal-ancilla-free-pauliv-circuits-for-axial-rotations\/","title":{"rendered":"Optimal Ancilla-free Pauli+V Circuits for Axial Rotations"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We address the problem of optimal representation of single-qubit rotations in a certain unitary basis consisting of the so-called V gates and Pauli matrices. The V matrices were proposed by Lubotsky, Philips, and Sarnak [Commun. Pure Appl. Math. 40, 401\u2013420 (1987)] as a purely geometric construct in 1987 and recently found applications in quantum computation. They allow for exceptionally simple quantum circuit synthesis algorithms based on quaternionic factorization. We adapt the deterministic-search technique initially proposed by Ross and Selinger to synthesize approximating Pauli+V circuits of optimal depth for single-qubit axial rotations. Our synthesis procedure based on simple <i>SL<\/i><sub>2<\/sub>(\u2124) geometry is almost elementary.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We address the problem of optimal representation of single-qubit rotations in a certain unitary basis consisting of the so-called V gates and Pauli matrices. The V matrices were proposed by Lubotsky, Philips, and Sarnak [Commun. Pure Appl. Math. 40, 401\u2013420 (1987)] as a purely geometric construct in 1987 and recently found applications in quantum computation. 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