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Creating subgroups of U(2^w) for quantum-minus computers

Alexis De Vos UGent and Michiel Boes UGent (2011) Journal of Physics Conference Series. 284.
abstract
Classical reversible computers on w bits are isomorphic to the (finite) symmetric group S_{2^w}; quantum computers on w qubits are isomorphic to the (Lie) unitary group U(2^w). Although S_{2^w} is a subgroup of U(2^w), the step from S_{2^w} to U(2^w) is huge. We investigate and classify groups X which are simultaneously supergroup of S_{2^w} and subgroup of U(2^w), such that they represent computers which are intermediate between classical reversible computers and quantum computers. Such intermediate groups X may exist in three flavours: - finite groups of order larger than (2^w)!, - infinite but discrete groups, and - Lie groups of dimension smaller than (2^w)^2. The larger the group, the more powerful the computer may be, but the smaller the group, the easier it can be to build the computer hardware. In the present paper, we investigate the first two flavours only. For our purpose, we start from 1-qubit transformations, represented by 2 * 2 unitary matrices, forming a group which is simultaneously a subgroup of U(2) and a supergroup of the group (isomorphic to S_2) consisting of the 2 * 2 IDENTITY matrix and the 2 * 2 NOT matrix. We call this group the creator X_2. Its members are called gates and act on one qubit. Controlled gates are quantum circuits acting on w qubits, such that the 1-qubit transformation (applied to a particular qubit) depends on the state of the w-1 other qubits. The controlled gates generate a group of 2^w * 2^w matrices, called the creation X. We discuss all creators of order up to 8. Additionally a creator of order 16 and one of order 192 are discussed.
Please use this url to cite or link to this publication:
author
organization
alternative title
Creating subgroups of U(2(w)) for quantum-minus computers
year
type
conference
publication status
published
subject
keyword
group theory, quantum computing
in
Journal of Physics Conference Series
J. Phys. Conf. Ser.
volume
284
issue title
GROUP 28 : Physical and Mathematical Aspects of Symmetry
article_number
012021
pages
6 pages
publisher
IOP Publishing
place of publication
Bristol, UK
conference name
28th International colloquium on Group-Theoretical Methods in Physics
conference location
Newcastle-upon-Tyne, UK
conference start
2010-07-26
conference end
2010-07-30
Web of Science type
Proceedings Paper
Web of Science id
000295845500021
ISSN
1742-6588
DOI
10.1088/1742-6596/284/1/012021
language
English
UGent publication?
yes
classification
P1
copyright statement
I have transferred the copyright for this publication to the publisher
id
1203686
handle
http://hdl.handle.net/1854/LU-1203686
date created
2011-04-06 15:47:43
date last changed
2014-01-02 10:30:58
@inproceedings{1203686,
  abstract     = {Classical reversible computers on w bits are isomorphic to the (finite) symmetric group S\_\{2\^{ }w\}; quantum computers on w qubits are isomorphic to the (Lie) unitary group U(2\^{ }w). Although      S\_\{2\^{ }w\} is a subgroup of U(2\^{ }w), the step from S\_\{2\^{ }w\} to U(2\^{ }w) is huge. We investigate and classify groups X which are simultaneously supergroup of S\_\{2\^{ }w\} and subgroup of U(2\^{ }w), such that they represent computers which are intermediate between classical reversible computers and quantum computers. Such intermediate groups X may exist in three flavours:
- finite groups of order larger than (2\^{ }w)!,
- infinite but discrete groups, and
- Lie groups of dimension smaller than (2\^{ }w)\^{ }2.
The larger the group, the more powerful the computer may be, but the smaller the group, the easier it can be to build the computer hardware.
In the present paper, we investigate the first two flavours only. For our purpose, we start from 1-qubit transformations, represented by 2 * 2 unitary matrices, forming a group which is simultaneously a subgroup of U(2) and a supergroup of the group (isomorphic to S\_2) consisting of the 2 * 2 IDENTITY matrix and the 2 * 2 NOT matrix. We call this group the creator X\_2. Its members are called gates and act on one qubit. Controlled gates are quantum circuits acting on w qubits, such that the 1-qubit transformation (applied to a particular qubit) depends on the state of the w-1 other qubits. 
The controlled gates generate a group of 2\^{ }w * 2\^{ }w matrices, called the creation X.
We discuss all creators of order up to 8. Additionally a creator of order 16 and one of order 192 are discussed.},
  articleno    = {012021},
  author       = {De Vos, Alexis and Boes, Michiel},
  booktitle    = {Journal of Physics Conference Series},
  issn         = {1742-6588},
  keyword      = {group theory,quantum computing},
  language     = {eng},
  location     = {Newcastle-upon-Tyne, UK},
  pages        = {6},
  publisher    = {IOP Publishing},
  title        = {Creating subgroups of U(2\^{ }w) for quantum-minus computers},
  url          = {http://dx.doi.org/10.1088/1742-6596/284/1/012021},
  volume       = {284},
  year         = {2011},
}

Chicago
De Vos, Alexis, and Michiel Boes. 2011. “Creating Subgroups of U(2^w) for Quantum-minus Computers.” In Journal of Physics Conference Series. Vol. 284. Bristol, UK: IOP Publishing.
APA
De Vos, Alexis, & Boes, M. (2011). Creating subgroups of U(2^w) for quantum-minus computers. Journal of Physics Conference Series (Vol. 284). Presented at the 28th International colloquium on Group-Theoretical Methods in Physics, Bristol, UK: IOP Publishing.
Vancouver
1.
De Vos A, Boes M. Creating subgroups of U(2^w) for quantum-minus computers. Journal of Physics Conference Series. Bristol, UK: IOP Publishing; 2011.
MLA
De Vos, Alexis, and Michiel Boes. “Creating Subgroups of U(2^w) for Quantum-minus Computers.” Journal of Physics Conference Series. Vol. 284. Bristol, UK: IOP Publishing, 2011. Print.