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arXiv:0811.1545

On interchanging the states of a pair of qudits

Colin Wilmott, Peter Wild

Published Nov 10, 2008• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.0811.1545

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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).
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
Abstract Advances in quantum information science are closely related to our ability to prepare, manipulate, and measure the states of multiple quantum bits (qubits). To this end, quantum gates are the key components by which specific operations over individual and multiple qubits are implemented. In the context of quantum optics, qubits can be encoded in any degree of freedom of a multi-photon state, e.g. polarization, frequency or spatial configuration. The faithful reciprocal exchange of information between two qubits, a so-called SWAP, is one of the fundamental operations that underpin modern quantum architectures. Here, we present a novel method realizing multiple-qubit SWAP operations over several qubits, without relying on any other gates, in an integrated-photonic setting. In principle, our approach allows for the implementation of SWAP operations with unity efficiency regardless of the number of involved qubits. Further, we discuss how this approach can be applied to qudits. Our findings provide a powerful new functional element for the design of compact quantum-photonic circuitry.
In this paper, we exclusively utilize CNOT gates for implementing permutation groups generated by more than two elements. In Lemma 1, we recall that three CNOT gates are both necessary and sufficient to execute a two-qubit swap gate operation. Subsequently, in Lemma 2, we show that the maximum number of CNOT gates needed to carry out an n-qubit substitution operation is 3(n-1). Moving forward, our analysis in Section 3 reveals that utilizing five or fewer CNOT gates is insufficient for implementing a three-qubit swap gate corresponding to the permutation element (123). Hence six CNOT gates are both necessary and sufficient for implementing (123). This is done by employing a graph-theoretic approach to rigorously validate the results in terms of at most five CNOT gates. Using computational tools, we exhaustively explore all valid circuit diagrams containing exactly six CNOT gates to successfully execute the swap gate for (123), by explaining the equivalence classes in Remark 6 and Table 2. We conclude them in Theorem 7.To extend our analysis to the multiqubit scenario, we present the reducible and irreducible permutation elements in Definition 8. We clarify the equivalence between rows in the multi-qubit space and provide an approximate upper bound for multi-qubits to perform the aforementioned operations in Theorem 9. The comprehensive exploration of this paper aims to pave the way for further advancements in understanding quantum circuit optimization via multiple use of a specific two-qubit gate.