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

Zhiwei Song, Lin Chen, Mengyao Hu

TL;DR

This work addresses the problem of characterizing three-qubit unitaries by Schmidt rank, extending the known bipartite result to the tripartite setting. It develops a construction framework based on Landsberg's decomposition to produce explicit gates with $\mathrm{sr}(U)$ from 1 to 7, including $\mathrm{sr}=2$ for the Toffoli gate and $\mathrm{sr}=4$ for the Fredkin gate, with a potential $\mathrm{sr}\in\{7,8\}$ for a special gate linked to the Strassen tensor. It then analyzes circuit realizations using CNOT gates and local gates, showing that two CNOTs suffice for $\mathrm{sr}\in\{1,2,4\}$, while at least three CNOTs are needed for $\mathrm{sr}=3,5,6,7$, and that three CNOTs can realize a $\mathrm{sr}=7$ gate via Strassen multiplicative complexity. The results establish a concrete connection between Schmidt rank and gate-synthesis complexity, paving the way for extensions to more qubits and clarifying the role of algebraic decompositions in quantum circuit design.

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.

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

TL;DR

This work addresses the problem of characterizing three-qubit unitaries by Schmidt rank, extending the known bipartite result to the tripartite setting. It develops a construction framework based on Landsberg's decomposition to produce explicit gates with from 1 to 7, including for the Toffoli gate and for the Fredkin gate, with a potential for a special gate linked to the Strassen tensor. It then analyzes circuit realizations using CNOT gates and local gates, showing that two CNOTs suffice for , while at least three CNOTs are needed for , and that three CNOTs can realize a gate via Strassen multiplicative complexity. The results establish a concrete connection between Schmidt rank and gate-synthesis complexity, paving the way for extensions to more qubits and clarifying the role of algebraic decompositions in quantum circuit design.

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.

Paper Structure

This paper contains 4 sections, 7 theorems, 24 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Construction of three-qubit untiary gates
  3. Implementation of three-qubit unitary gates
  4. Conclusions

Key Result

Lemma 1

Suppose $U=\sum^r_{j=1} Q_j \otimes R_j$ is a tripartite matrix where $Q_j$ on ${\cal H}_A \otimes {\cal H}_B$ are linearly independent, and $R_j$ on ${\cal H}_C$ are also linearly independent. Then the Schmidt rank of $U$ is the minimal number of product matrices spanning the space including the sp

Figures (7)

  • Figure 1: The three-qubit Toffoli gate $T_3$ of Schmidt rank two can be realized using two CNOT gates and one CZ gate $\mathop{\rm diag}(1,1,1,-1)$ in the middle. The CZ gate is locally equivalent to the CNOT gate via two Hadamard gates $H$. The local gate $X_A$ flips the qutrits $|0\rangle$ and $|2\rangle$.
  • Figure 2: The three-qubit gate $U_3$ of Schmidt rank three can be implemented using four CNOT gates, local Hadamard gates $H$ and local gate $X_A$ flipping the qutrit $|0\rangle$ and $|2\rangle$.
  • Figure 3: The three-qubit gate $U_4$ of Schmidt rank four consists of two CNOT gates. This is minimum cost of realizing any three-qubit unitary gate of Schmidt rank four.
  • Figure 4: The three-qubit Fredkin gate $F_3$ of Schmidt rank four can be implemented using five CNOT gates and local gates $X_A$ flipping the qutrit $|0\rangle$ and $|2\rangle$.
  • Figure 5: The three-qubit gate $U_5$ of Schmidt rank five can be implemented using six CNOT gates and local gates $X_A$ flipping the qutrit $|0\rangle$ and $|2\rangle$.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Corollary 2
  • Theorem 3
  • Lemma 4
  • Theorem 5
  • Theorem 6
  • Theorem 7