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

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.

References in corpus (8)

• Quantum-state preparation with universal gate decompositions
• Efficient Toffoli Gates Using Qudits
• Robust and resource-efficient microwave near-field entangling ${}^9$Be${}^{+}$ gate
• Optimal synthesis of the Fredkin gate in a multilevel system
• How many CNOT gates does it take to generate a three-qubit state ?
• Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates
• All unitaries having operator Schmidt rank 2 are controlled unitaries
• On the Schmidt-rank-three bipartite and multipartite unitary operator

Cited by in corpus (1)

• Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates