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 threequbit 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.

Synthesis and Upper Bound of Schmidt Rank of Bipartite Controlled‐Unitary Gates

“…are linearly independent local operations acting on respective parties. The smallest possible m is defined as Schmidt rank $$\text{Sch}(\mathcal {U})$$, [ 32 ] which can be used for quantifying the nonlocality of $$\mathcal {U}$$ defined by $$K_{\text{Har}}(\mathcal {U})\equiv \log _2[\text{Sch}(\mathcal {U})]$$, [ 33–35 ] providing a sufficient condition for when $$\mathcal {U}$$ is a controlled‐unitary operation, [ 36–42 ] and optimizing the synthesis of quantum computation [ 43 ] and quantum transistors. [ 30 ] Another common approach, called synthesis or quantum circuit, is to factorize $$\mathcal {U}$$ into fewer achievable simple local and nonlocal operations from a universal library, which may simplify such physical implementations: $$\mathcal {U}= X_1 X_2 \cdots X_m$$.…”