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DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

CHI-KWONG LI

Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA

Corresponding author.

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REBECCA ROBERTS

Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA

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, and

XIAOYAN YIN

Department of Mathematics, Xidian University, Xi'an, Shaanxi, 710071, China

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https://doi.org/10.1142/S0219749913500159Cited by:32 (Source: Crossref)

Abstract

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 2^n-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 one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

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