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.

UR - https://arxiv.org/abs/1908.06076 ER - TY - JOUR T1 - A finite presentation of CNOT-dihedral operators Y1 - 2016 A1 - Matthew Amy A1 - Jianxin Chen A1 - Neil J. Ross AB -We give a finite presentation by generators and relations of unitary operators expressible over the {CNOT, T, X} gate set, also known as CNOT-dihedral operators. To this end, we introduce a notion of normal form for CNOT-dihedral circuits and prove that every CNOT-dihedral operator admits a unique normal form. Moreover, we show that in the presence of certain structural rules only finitely many circuit identities are required to reduce an arbitrary CNOT-dihedral circuit to its normal form. By appropriately restricting our relations, we obtain a finite presentation of unitary operators expressible over the {CNOT, T } gate set as a corollary.

UR - https://arxiv.org/abs/1701.00140 ER -