On interchanging the states of a pair of qudits

TL;DR

This paper investigates the possibility of generalizing the SWAP gate to qudits, revealing limitations in quantum circuit architectures based on CNOT gates for certain dimensions.

Contribution

It demonstrates that quantum circuits composed solely of CNOT gates cannot implement qudit transpositions for dimensions where d ≡ 3 (mod 4).

Findings

CNOT-based circuits cannot implement qudit transpositions for certain dimensions

The paper links the problem to permutation signatures

It discusses the construction of generalized SWAP gates for qudits

Abstract

The qubit SWAP gate has been shown to be an integral component of quantum circuitry design. It permutes the states of two qubits and allows for the storage quantum information, teleportation of atomic or ionic states, and is a fundamental element in the circuit implementation of Shor's algorithm. We consider the problem of generalising the SWAP gate beyond the qubit setting. We show that quantum circuit architectures completely described by instances of the CNOT gate can not implement a transposition of a pair of qudits for dimensions $d \equiv 3 (mod 4)$. The task of constructing generalised SWAP gates based on transpositions of qudit states is argued in terms of the signature of a permutation.