01571nas a2200133 4500008004100000245007800041210006900119260001400188520113700202100001701339700002501356700001901381856003701400 2019 eng d00aNumber-Theoretic Characterizations of Some Restricted Clifford+T Circuits0 aNumberTheoretic Characterizations of Some Restricted CliffordT C c8/16/20193 a
Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2×2 unitary matrix V can be exactly represented by a single-qubit Clifford+T circuit if and only if the entries of V belong to the ring Z[1/2–√,i]. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+T circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+T circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/2–√,i]. We focus on the subrings Z[1/2], Z[1/2–√], Z[1/-2−−√], and Z[1/2,i], and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X,CX,CCX} with an analogue of the Hadamard gate and an optional phase gate.
1 aAmy, Matthew1 aGlaudell, Andrew, N.1 aRoss, Neil, J. uhttps://arxiv.org/abs/1908.06076