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European Journal of Pure and Applied Mathematics

A Note on Quantum Gates SWAP and iSWAP in Higher Dimensions

Arash Pourkia

Published Jul 30, 2023• 0 citations
Publisher

Abstract

We present explicit descriptions for the swap gate and the iswap gate in any arbitrary dimension $d \geq 2$, in terms of permutation matrices. Moreover, we unify these gates by introducing a more general gate xSWAP which includes SWAP and iSWAP for $x=1$ and $x=i$ (i.e. $\sqrt{-1}$), respectively. The higher dimensional xSWAP e.g., the swap and iswap gates for $d > 2$ serve as quantum logic gates that operate on two $d$-level qudits. For $d=2$, it is well known that iSWAP unlike SWAP is universal for quantum computing. We will prove this fact for xSWAP in any dimension $d$, when $x \neq \pm 1$. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.

DOI
10.29020/nybg.ejpam.v16i3.4824

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We present explicit descriptions for the swap gate and the iswap gate in any arbitrary dimension d≥2d \geq 2d≥2, in terms of permutation matrices. Moreover, we unify these gates by introducing a more general gate xSWAP which includes SWAP and iSWAP for x=1x=1x=1 and x=ix=ix=i (i.e. −1\sqrt{-1}−1​), respectively. The higher dimensional xSWAP e.g., the swap and iswap gates for d>2d > 2d>2 serve as quantum logic gates that operate on two ddd-level qudits. For d=2d=2d=2, it is well known that iSWAP unlike SWAP is universal for quantum computing. We will prove this fact for xSWAP in any dimension ddd, when x≠±1x \neq \pm 1x=±1. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.
We introduce an infinite family of universal quantum logic gates that includes not only higher-dimensional versions of the usual SWAP and i\mathbf{i}iSWAP gates, but also their previously known extensions. This family consists of permutation-like matrices with nonzero entries of the form eiαie^{\mathbf{i}\alpha_i}eiαi​, where the αi\alpha_iαi​ are arbitrary real numbers. Moreover, we show that these gates, which we refer to as αSWAP\alpha SWAPαSWAP, provide unitary solutions to the constant Yang--Baxter equation in all dimensions.
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.
The concern of quantum computation is the computation of\nquantum phenomena as observed in Nature. A prerequisite for the\nattainment of such computation is a set of unitary\ntransformations that describe the operational process within\nthe quantum system. Since operational transformations\ninevitably interact with elements outside of the quantum\nsystem, we therefore have quantum gate evolutions determined\nwith less than absolute precision. Consequently, the theory of\nquantum error-correction has developed to meet this difficulty.\nFurther, it is increasingly evident that much effort is being\nmade into finding efficient quantum circuits in the sense that\nfor a library of realisable quantum gates there is no smaller\ncircuit that achieves the same task with the same library of\ngates. A reason for this concerted effort is primarily due to\nthe principle of decoherence. I give the construction for the\nset of unitary transformations that describe an error model\nthat acts on a d-dimensional quantum system. I also give an\noverview of the theoretical framework associated with such\nunitary transformations and generalise results to cater for\nd-dimensional quantum states. I introduce two quantum gate\nconstructions that generalise the qubit SWAP gate to higher\ndimensions. The first of these constructions is the WilNOT gate\nand the second is an efficient design also based on binomial\nsummations. Both of these constructions yield a quantum qudit\nSWAP gate determined only in the CNOT gate. Furthermore, the\ntask of constructing generalised SWAP gates based on\ntranspositions of qudit states is argued in terms of the\nsignature of a permutation. Based on this argument, we show\nthat circuit architectures completely described by instances of\nthe CNOT gate cannot implement a transposition of a pair of\nqudits over dimensions d = 3 mod 4. Consequently, our quantum\ncircuits are of interest because it is not possible to\nimplement a SWAP of qutrits by a sequence of transpositions of\nqutrits if only CNOT gates are used. I also give bounds on\nnumbers of quantum codes predicated on d-dimensional quantum\nsystems, and generalise the encoding and decoding architectures\nfor qudit codes.\n