On two subgroups of U(n), useful for quantum computing
Alexis De Vos, Stijn De Baerdemacker · Journal of Physics Conference Series · 2015
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 × 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 × n unitary matrices with all 2 n 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 × n unitary diagonal matrices with first entry equal to 1. Any U( n ) matrix can be decomposed into four parts: U = exp( iα ) 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.