On swapping the states of two qudits

TL;DR

This paper investigates whether a quantum circuit made solely of generalized controlled-NOT gates can swap the states of two qudits, showing it is impossible for certain dimensions based on permutation signatures.

Contribution

It proves the impossibility of constructing such swap circuits for qudits of specific dimensions, extending understanding of quantum gate limitations.

Findings

Impossible for dimensions d ≡ 3 (mod 4)

Uses permutation signature analysis

Provides theoretical limitations on quantum circuit design

Abstract

The SWAP gate has become an integral feature of quantum circuit architectures and is designed to permute the states of two qubits through the use of the well-known controlled-NOT gate. We consider the question of whether a two-qudit quantum circuit composed entirely from instances of the generalised controlled-NOT gate can be constructed to permute the states of two qudits. Arguing via the signature of a permutation, we demonstrate the impossibility of such circuits for dimensions $d \equiv 3$ (mod 4).