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

Alexis De Vos UGent and Stijn De Baerdemacker UGent (2015) Journal of Physics : Conference Series. 597.
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.
Please use this url to cite or link to this publication:
author
organization
year
type
conference (proceedingsPaper)
publication status
published
subject
keyword
unitary group, MATRICES, subgroup, quantum computing, reversible computing, COMPUTATION
in
Journal of Physics : Conference Series
volume
597
issue title
XXXth International colloquium on group theoretical methods in physics (ICGTMP)
article number
012030
pages
10 pages
publisher
IOP
place of publication
Bristol, UK
conference name
30th International colloquium on Group Theoretical Methods in Physics (ICGTMP)
conference location
Ghent, Belgium
conference start
2014-07-14
conference end
2014-07-18
Web of Science type
Proceedings Paper
Web of Science id
000354929400030
ISSN
1742-6588
DOI
10.1088/1742-6596/597/1/012030
language
English
UGent publication?
yes
classification
P1
copyright statement
I have retained and own the full copyright for this publication
id
5937344
handle
http://hdl.handle.net/1854/LU-5937344
date created
2015-04-20 08:24:22
date last changed
2018-01-29 12:12:26
@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},
}

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.