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arXiv:0811.1879

On Binomial Summations and a Generalised Quantum SWAP Gate

Colin Wilmott, Peter Wild

Published Nov 12, 2008• 0 citations
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Abstract

We give a quantum gate construction - composed entirely from incidents of the CNOT gate - that generalises the qubit SWAP gate to higher dimensions. This new construction is more regular than and is an improvement on the WilNOT quantum gate construction.
DOI
10.48550/arxiv.0811.1879

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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≡3(mod4)d \equiv 3 (mod 4)d≡3(mod4). The task of constructing generalised SWAP gates based on transpositions of qudit states is argued in terms of the signature of a permutation.
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≡3d \equiv 3d≡3 (mod 4).
It is known that SWAP operation for quantum systems of arbitrary dimension can be implemented using various qudit gates. In this paper, equivalence of some important qudit operations is demonstrated. Equivalence of the qudit operations is exploited to minimize the circuit complexity in the previously proposed qudit SWAP gate constructions. Further, constructions of qudit SWAP operation with minimum number of qudit gates are also proposed. Importantly, these circuit constructions of SWAP retain its properties like symmetry and simplicity.