Creating subgroups of U(2^w) for quantumminus computers
 Author
 Alexis De Vos (UGent) and Michiel Boes (UGent)
 Organization
 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 1qubit 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 1qubit transformation (applied to a particular qubit) depends on the state of the w1 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.
 Keywords
 group theory, quantum computing
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU1203686
 Chicago
 De Vos, Alexis, and Michiel Boes. 2011. “Creating Subgroups of U(2^w) for Quantumminus 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 quantumminus computers. Journal of Physics Conference Series (Vol. 284). Presented at the 28th International colloquium on GroupTheoretical Methods in Physics, Bristol, UK: IOP Publishing.
 Vancouver
 1.De Vos A, Boes M. Creating subgroups of U(2^w) for quantumminus 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 Quantumminus Computers.” Journal of Physics Conference Series. Vol. 284. Bristol, UK: IOP Publishing, 2011. Print.
@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 1qubit 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 1qubit transformation (applied to a particular qubit) depends on the state of the w1 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 = {17426588}, keywords = {group theory,quantum computing}, language = {eng}, location = {NewcastleuponTyne, UK}, pages = {6}, publisher = {IOP Publishing}, title = {Creating subgroups of U(2^w) for quantumminus computers}, url = {http://dx.doi.org/10.1088/17426596/284/1/012021}, volume = {284}, year = {2011}, }
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