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

30th International Colloquium on Group Theoretical Methods in Physics

De Vos, Alexis^1 ; De Baerdemacker, Stijn^2,3

Cmst Vakgroep Elektronika en Informatiesystemen Imec V.z.w., Universiteit Gent, Sint Pietersnieuwstraat 41, Gent

B-9000, Belgium^1

Center for Molecular Modeling, Vakgroep Fysica en Sterrenkunde, Universiteit Gent, Technologiepark 903, Gent

B-9052, Belgium^2

Ghent Quantum Chemistry Group, Vakgroep Anorganische en Fysische Chemie, Universiteit Gent, Krijgslaan 281-S3, Gent

B-9000, Belgium^3

关键词: Basic building block;    Block synthesis;    Diagonal matrices;    Quantum circuit;    Quantum Computing;    Unit matrix;    Unitary group;    Unitary matrix;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012030/pdf
DOI  :  10.1088/1742-6596/597/1/012030

来源: IOP

【 摘 要 】

As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit φ(θ) and the 1-qubit NEGATOR circuit N(θ). Both are roots of the IDENTITY circuit. Indeed: both φ(0) 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(2w); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2w). 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) Z1XZ2, where both Z1and Z2are ZU(n) matrices and X is an XU(n) matrix. We give an algorithm to find the decomposition. For n = 2wit leads to a four-block synthesis of an arbitrary quantum computer.