Decomposition of unitary matrices and quantum gates

Original · EN

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d-1 classes, where each class is isomorphic to the group of 2× 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2ⁿ⁻¹(2ⁿ-1) fully controlled single-qubit gates chosen from 2ⁿ-1 classes, where the quantum gates in each class share the same n-1 control qubits. Moreover, it is shown that it is easy to adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.