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

• Published in: Quantum information processing Vol. 20; no. 3
• Main Authors: Song, Zhiwei, Chen, Lin, Hu, Mengyao
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-03-2021; Springer Nature B.V
• Subjects: Data Structures and Information Theory; Gates; Mathematical Physics; Physics; Physics and Astronomy; Quantum Computing; Quantum Information Technology; Quantum Physics; Qubits (quantum computing); Spintronics; Tensors; CNOT gate; Three-qubit unitary gate; Schmidt rank
• ISSN: 1570-0755; 1573-1332
• DOI: 10.1007/s11128-021-03031-1

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.