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Symmetry 2011, 3(2), 305-324; doi:10.3390/sym3020305

Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between

1
Department of Electronics and Information Systems, Universiteit Gent, Sint Pietersnieuwstraat 41, B-9000 Gent, Belgium
2
“FWO-Vlaanderen” post-doctoral fellow, Department of Physics and Astronomy, Universiteit Gent, Proeftuinstraat 86, B-9000 Gent, Belgium
*
Author to whom correspondence should be addressed.
Received: 11 January 2011 / Revised: 24 May 2011 / Accepted: 27 May 2011 / Published: 7 June 2011
(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)
Download PDF [202 KB, uploaded 7 June 2011]

Abstract

Whereas quantum computing circuits follow the symmetries of the unitary Lie group, classical reversible computation circuits follow the symmetries of a finite group, i.e., the symmetric group. We confront the decomposition of an arbitrary classical reversible circuit with w bits and the decomposition of an arbitrary quantum circuit with w qubits. Both decompositions use the control gate as building block, i.e., a circuit transforming only one (qu)bit, the transformation being controlled by the other w−1 (qu)bits. We explain why the former circuit can be decomposed into 2w − 1 control gates, whereas the latter circuit needs 2w − 1 control gates. We investigate whether computer circuits, not based on the full unitary group but instead on a subgroup of the unitary group, may be decomposable either into 2w − 1 or into 2w − 1 control gates.
Keywords: reversible computing; quantum computing; group theory reversible computing; quantum computing; group theory
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Vos, A.D.; Baerdemacker, S.D. Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between. Symmetry 2011, 3, 305-324.

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