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Quantum Information Processing

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01-03-2021

Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates

Authors: Zhiwei Song, Lin Chen, Mengyao Hu

Published in: Quantum Information Processing | Issue 3/2021

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Abstract

It is known that every two-qubit unitary operation has Schmidt rank one, two or four, and the construction of three-qubit unitary gates in terms of Schmidt rank remains an open problem. We explicitly construct the gates of Schmidt rank from one to seven. It turns out that the three-qubit Toffoli and Fredkin gate, respectively, have Schmidt rank two and four. As an application, we implement the gates using quantum circuits of CNOT gates and local Hadamard and flip gates. In particular, the collective use of three CNOT gates can generate a three-qubit unitary gate of Schmidt rank seven in terms of the known Strassen tensor from multiplicative complexity.

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Note that one often uses the notations \(\sigma _+\) and \(\sigma _-\) for \(S_1\) and \(S_2\); here, we do not use them for the consistency of expressions of the formulas in (1).

The notion is equivalent to the tensor rank in matrix multiplication. We denote it as Schmidt rank because a similar use has been proposed in [29].

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Title: Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates
Authors: Zhiwei Song
Lin Chen
Mengyao Hu
Publication date: 01-03-2021
Publisher: Springer US
Published in: Quantum Information Processing / Issue 3/2021
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-021-03031-1

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