@article { ISI:000524978500001,
title = {Number-Theoretic Characterizations of Some Restricted Clifford},
journal = {Quantum},
volume = {4},
year = {2020},
month = {APR 3},
publisher = {VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF},
type = {Article},
abstract = {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.},
issn = {2521-327X},
author = {Amy, Matthew and Glaudell, Andrew N. and Ross, Neil J.}
}
@article {ISI:000473123700004,
title = {Canonical forms for single-qutrit Clifford plus T operators},
journal = {Ann. Phys.},
volume = {406},
year = {2019},
month = {JUL},
pages = {54-70},
publisher = {ACADEMIC PRESS INC ELSEVIER SCIENCE},
type = {Article},
abstract = {We introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. We show that our canonical forms are T-optimal in the sense that among all the single-qutrit Clifford+T circuits implementing a given operator our canonical form uses the least number of T gates. Finally, we provide an algorithm which inputs the description of an operator (as a matrix or a circuit) and constructs the canonical form for this operator. The algorithm runs in time linear in the number of T gates. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits. (C) 2019 Elsevier Inc. All rights reserved.},
keywords = {Quantum circuits, quantum computation, Qutrits, Universal gate sets},
issn = {0003-4916},
doi = {10.1016/j.aop.2019.04.001},
author = {Glaudell, Andrew N. and Ross, Neil J. and Taylor, Jacob M.}
}