Number-Theoretic Characterizations of Some Restricted Clifford

Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2 x 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/root 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/root 2, i]. We focus on the subrings Z[1./2], Z[1/root 2], Z[1/i root 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 Hadarnard gate and an optional phase gate.