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Symmetry groups for the decomposition of reversible computers, quantum computers, and computers in between

Alexis De Vos (UGent) and Stijn De Baerdemacker (UGent)
(2011) SYMMETRY-BASEL. 3(2). p.305-324
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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 2^w-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 2^w-1 control gates.
Keywords
reversible computing, group theory, quantum computing

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Citation

Please use this url to cite or link to this publication:

MLA
De Vos, Alexis, and Stijn De Baerdemacker. “Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in Between.” SYMMETRY-BASEL, vol. 3, no. 2, 2011, pp. 305–24, doi:10.3390/sym3020305.
APA
De Vos, A., & De Baerdemacker, S. (2011). Symmetry groups for the decomposition of reversible computers, quantum computers, and computers in between. SYMMETRY-BASEL, 3(2), 305–324. https://doi.org/10.3390/sym3020305
Chicago author-date
De Vos, Alexis, and Stijn De Baerdemacker. 2011. “Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in Between.” SYMMETRY-BASEL 3 (2): 305–24. https://doi.org/10.3390/sym3020305.
Chicago author-date (all authors)
De Vos, Alexis, and Stijn De Baerdemacker. 2011. “Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in Between.” SYMMETRY-BASEL 3 (2): 305–324. doi:10.3390/sym3020305.
Vancouver
1.
De Vos A, De Baerdemacker S. Symmetry groups for the decomposition of reversible computers, quantum computers, and computers in between. SYMMETRY-BASEL. 2011;3(2):305–24.
IEEE
[1]
A. De Vos and S. De Baerdemacker, “Symmetry groups for the decomposition of reversible computers, quantum computers, and computers in between,” SYMMETRY-BASEL, vol. 3, no. 2, pp. 305–324, 2011.
@article{1258500,
  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 2^w-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 2^w-1 control gates.}},
  author       = {{De Vos, Alexis and De Baerdemacker, Stijn}},
  issn         = {{2073-8994}},
  journal      = {{SYMMETRY-BASEL}},
  keywords     = {{reversible computing,group theory,quantum computing}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{305--324}},
  title        = {{Symmetry groups for the decomposition of reversible computers, quantum computers, and computers in between}},
  url          = {{http://doi.org/10.3390/sym3020305}},
  volume       = {{3}},
  year         = {{2011}},
}

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