On two subgroups of U(n), useful for quantum computing
 Author
 Alexis De Vos (UGent) and Stijn De Baerdemacker (UGent)
 Organization
 Abstract
 As two basic building blocks for any quantum circuit, we consider the 1qubit PHASOR circuit Phi(theta) and the 1qubit 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 fourblock synthesis of an arbitrary quantum computer.
 Keywords
 unitary group, MATRICES, subgroup, quantum computing, reversible computing, COMPUTATION
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU5937344
 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 1qubit PHASOR circuit Phi(theta) and the 1qubit 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 fourblock synthesis of an arbitrary quantum computer.}, articleno = {012030}, author = {De Vos, Alexis and De Baerdemacker, Stijn}, booktitle = {Journal of Physics : Conference Series}, issn = {17426588}, 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/10.1088/17426596/597/1/012030}, volume = {597}, year = {2015}, }
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