Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

Matthew Amy1, Andrew N. Glaudell2,3, and Neil J. Ross1

1Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada
2Institute for Advanced Computer Studies and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA
3Joint Quantum Institute, University of Maryland, College Park, MD, USA

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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 $\mathbb{Z}[1/\sqrt{2},i]$. 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 $\mathbb{Z}[1/\sqrt{2},i]$. We focus on the subrings $\mathbb{Z}[1/2]$, $\mathbb{Z}[1/\sqrt{2}]$, $\mathbb{Z}[1/i\sqrt{2}]$, and $\mathbb{Z}[1/2,i]$, 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.

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Cited by

[1] Yongshan Ding and Frederic T. Chong, "Quantum Computer Systems: Research for Noisy Intermediate-Scale Quantum Computers", Synthesis Lectures on Computer Architecture 15 2, 1 (2020).

[2] Andrew N. Glaudell, Neil J. Ross, and Jacob M. Taylor, "Optimal two-qubit circuits for universal fault-tolerant quantum computation", npj Quantum Information 7 1, 103 (2021).

[3] Xiaoning Bian and Peter Selinger, "Generators and relations for $U_n(\mathbb Z [\frac{1}{2}, i])$", arXiv:2105.14047.

[4] Sarah Meng Li, Neil J. Ross, and Peter Selinger, "Generators and Relations for the Group $\mathrm{O}_n(\mathbb{Z}[1/2])$", arXiv:2106.01175.

The above citations are from Crossref's cited-by service (last updated successfully 2021-08-01 08:20:59) and SAO/NASA ADS (last updated successfully 2021-08-01 08:21:01). The list may be incomplete as not all publishers provide suitable and complete citation data.