Realization of permutation groups by quantum circuit

TL;DR

This paper investigates the implementation of permutation groups using only CNOT gates in quantum circuits, establishing minimal gate counts for specific permutations and extending the analysis to multi-qubit systems.

Contribution

It provides a detailed analysis of CNOT gate requirements for permutation operations, including necessary and sufficient conditions, and introduces a graph-theoretic approach for validation.

Findings

Six CNOT gates are necessary and sufficient for certain three-qubit swap gates.

An upper bound for multi-qubit permutation implementation is established.

Exhaustive exploration of six-CNOT circuit diagrams supports the theoretical results.

Abstract

In this paper, we exclusively utilize CNOT gates for implementing permutation groups generated by more than two elements. In Lemma 1, we recall that three CNOT gates are both necessary and sufficient to execute a two-qubit swap gate operation. Subsequently, in Lemma 2, we show that the maximum number of CNOT gates needed to carry out an n-qubit substitution operation is 3(n-1). Moving forward, our analysis in Section 3 reveals that utilizing five or fewer CNOT gates is insufficient for implementing a three-qubit swap gate corresponding to the permutation element (123). Hence six CNOT gates are both necessary and sufficient for implementing (123). This is done by employing a graph-theoretic approach to rigorously validate the results in terms of at most five CNOT gates. Using computational tools, we exhaustively explore all valid circuit diagrams containing exactly six CNOT gates to successfully execute the swap gate for (123), by explaining the equivalence classes in Remark 6 and Table 2. We conclude them in Theorem 7.To extend our analysis to the multiqubit scenario, we present the reducible and irreducible permutation elements in Definition 8. We clarify the equivalence between rows in the multi-qubit space and provide an approximate upper bound for multi-qubits to perform the aforementioned operations in Theorem 9. The comprehensive exploration of this paper aims to pave the way for further advancements in understanding quantum circuit optimization via multiple use of a specific two-qubit gate.