



@article {2464,
	title = {Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits},
	year = {2019},
	month = {8/16/2019},
	abstract = {<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>},
	url = {https://arxiv.org/abs/1908.06076},
	author = {Matthew Amy and Andrew N. Glaudell and Neil J. Ross}
}
@article {1925,
	title = {A finite presentation of CNOT-dihedral operators},
	year = {2016},
	month = {2016/12/31},
	abstract = {<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>},
	url = {https://arxiv.org/abs/1701.00140},
	author = {Matthew Amy and Jianxin Chen and Neil J. Ross}
}
