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

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.

Automated summary

This study demonstrates the construction of three-qubit unitary gates at different levels of Schmidt rank, showing the Schmidt rank of Toffoli and Fredkin gates. The gates can be implemented using CNOT gates, local Hadamard gates, and flip gates. Additionally, a three-qubit unitary gate of Schmidt rank seven can be achieved using the known Strassen tensor.