



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<p>Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2&times;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&ndash;&radic;,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&ndash;&radic;,i]. We focus on the subrings Z[1/2], Z[1/2&ndash;&radic;], Z[1/-2&minus;&minus;&radic;], 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.</p>1 aAmy, Matthew1 aGlaudell, Andrew, N.1 aRoss, Neil, J.  uhttps://arxiv.org/abs/1908.0607601044nas a2200133   4500008004100000245005300041210005000094260001500144520066000159100001700819700001800836700001900854856003700873       2016                        eng d00aA finite presentation of CNOT-dihedral operators0 afinite presentation of CNOTdihedral operators  c2016/12/313 a<p>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.</p>1 aAmy, Matthew1 aChen, Jianxin1 aRoss, Neil, J.  uhttps://arxiv.org/abs/1701.00140