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On two subgroups of U(n), useful for quantum computing

Alexis De Vos (UGent) and Stijn De Baerdemacker (UGent)
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Abstract
As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit Phi(theta) and the 1-qubit NEGATOR circuit N(theta). Both are roots of the IDENTITY circuit. Indeed: both (NO) and N(0) equal the 2 x 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2(w)); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2(w)). Here, ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n x n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n x n unitary diagonal matrices with first entry equal to 1. Any U(n) matrix can be decomposed into four parts: U = exp(i alpha) Z(1)XZ(2), where both Z(1) and Z(2) are ZU(n) matrices and X is an XU(n) matrix. We give an algorithm to find the decomposition. For n = 2(w) it leads to a four-block synthesis of an arbitrary quantum computer.
Keywords
unitary group, MATRICES, subgroup, quantum computing, reversible computing, COMPUTATION

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Chicago
De Vos, Alexis, and Stijn De Baerdemacker. 2015. “On Two Subgroups of U(n), Useful for Quantum Computing.” In Journal of Physics : Conference Series. Vol. 597. Bristol, UK: IOP.
APA
De Vos, Alexis, & De Baerdemacker, S. (2015). On two subgroups of U(n), useful for quantum computing. Journal of Physics : Conference Series (Vol. 597). Presented at the 30th International colloquium on Group Theoretical Methods in Physics (ICGTMP), Bristol, UK: IOP.
Vancouver
1.
De Vos A, De Baerdemacker S. On two subgroups of U(n), useful for quantum computing. Journal of Physics : Conference Series. Bristol, UK: IOP; 2015.
MLA
De Vos, Alexis, and Stijn De Baerdemacker. “On Two Subgroups of U(n), Useful for Quantum Computing.” Journal of Physics : Conference Series. Vol. 597. Bristol, UK: IOP, 2015. Print.
@inproceedings{5937344,
  abstract     = {As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit Phi(theta) and the 1-qubit NEGATOR circuit N(theta). Both are roots of the IDENTITY circuit. Indeed: both (NO) and N(0) equal the 2 x 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2(w)); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2(w)). Here, ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n x n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n x n unitary diagonal matrices with first entry equal to 1. Any U(n) matrix can be decomposed into four parts: U = exp(i alpha) Z(1)XZ(2), where both Z(1) and Z(2) are ZU(n) matrices and X is an XU(n) matrix. We give an algorithm to find the decomposition. For n = 2(w) it leads to a four-block synthesis of an arbitrary quantum computer.},
  articleno    = {012030},
  author       = {De Vos, Alexis and De Baerdemacker, Stijn},
  booktitle    = {Journal of Physics : Conference Series},
  issn         = {1742-6588},
  keyword      = {unitary group,MATRICES,subgroup,quantum computing,reversible computing,COMPUTATION},
  language     = {eng},
  location     = {Ghent, Belgium},
  pages        = {10},
  publisher    = {IOP},
  title        = {On two subgroups of U(n), useful for quantum computing},
  url          = {http://dx.doi.org/10.1088/1742-6596/597/1/012030},
  volume       = {597},
  year         = {2015},
}

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