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arXiv:1101.4159

On swapping the states of two qudits

Colin Wilmott

Published Jan 21, 2011• 0 citations
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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).

DOI
10.48550/arxiv.1101.4159

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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
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.
Quantum algorithms can be realized in the form of a quantum circuit. To map a quantum circuit for a specific quantum algorithm to quantum hardware, qubit mapping is an imperative technique based on the qubit topology. Due to the neighborhood constraint of qubit topology, the implementation of the quantum algorithm rightly, is essential for moving information around in a quantum computer. Swapping of qubits using a swap gate moves the quantum state between two qubits and solves the neighborhood constraint of qubit topology. Although, one needs to decompose the swap gate into three controlled-not gates to implement the swap gate efficiently, but unwillingly quantum cost with respect to the gate count and depth increases. In this paper, a formalism of moving quantum states without using swap operation is introduced. Moving quantum states through qubits have been attained with the adoption of temporary intermediate qudit states. This introduction of intermediate qudit states has exhibited a three times reduction in quantum cost with respect to the gate count and approximately two times reduction with respect to circuit depth compared to the state-of-the-art approach of the swap gate insertion. We also exhibit that the adoption of the intermediate qudit makes the approach sublimer than the existing works by obtaining a better fidelity. Furthermore, the proposed approach is generalized to any finite-dimensional quantum system.