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

TL;DR

This paper explicitly constructs three-qubit unitary gates with various Schmidt ranks, analyzes their implementation using CNOT and local gates, and explores the relationship between CNOT count and Schmidt rank.

Contribution

It provides the first explicit construction of three-qubit gates with Schmidt ranks from one to seven and links their implementation complexity to Schmidt rank.

Findings

Toffoli and Fredkin gates have Schmidt ranks two and four.

Three CNOT gates can generate a rank seven gate using Strassen tensor.

Schmidt rank relates to the number of CNOT gates needed for implementation.

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. Our results imply the connection between the number of CNOT gates for implementing multiqubit gates and their Schmidt rank.