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

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

1,* email and 2email
Received: 11 January 2011; in revised form: 24 May 2011 / Accepted: 27 May 2011 / Published: 7 June 2011
(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)
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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
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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.

AMA Style

Vos AD, Baerdemacker SD. Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between. Symmetry. 2011; 3(2):305-324.

Chicago/Turabian Style

Vos, Alexis De; Baerdemacker, Stijn De. 2011. "Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between." Symmetry 3, no. 2: 305-324.


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