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 fundamental unit of quantum information processing is qubit which can be realized by two-level quantum mechanical systems. The two-dimensional Hilbert space representing a single qubit can be extended to higher dimensions of

Two-qubit operations (also known as two-qubit gates) are fundamental in the realization of universal quantum circuits [

Extension of two-qubit operation to qudit operation is not always straightforward. In particular, retaining the features of two-qubit operation in the generalization to qudit operation is not easy. One such instance is the implementation of SWAP operation in the qudit setting. A few generalizations of CNOT operation to the qudit level are proposed with the aim to implement qudit SWAP operation. However, the number of gates utilized in the previously proposed circuit constructions of SWAP gate is not less than four [

SWAP operation has been employed in circuit designs of many quantum operations (e.g., [

First of all, the existence of equivalence between some of the qudit gates is shown. Then the equivalence of gates is exploited to minimize the number of gates used in the previously proposed qudit SWAP constructions. It is shown that the number of single qudit gates in the construction of qudit SWAP can be reduced. Such constructions would be useful in the case of high

Before introducing some known qudit gates, it is necessary to define important two-qubit gates, namely, CNOT and SWAP. One of the qubits of CNOT gate is known as control and the other one is called target. Whenever the control qubit is in

Two-qubit gates. (a) CNOT; (b) SWAP.

Now we introduce some qudit gates which are the different versions of qudit CNOT operation. Among the various versions,

Equivalence of the various qudit gates is identified and mentioned in this section. The equivalence of the operation can be checked by the action of the gates on the input states

It is known that the concatenation of three CNOT gates, can implement two-qubit SWAP operation. The CNOT swapping circuit is shown in Figure

Implementation of qubit SWAP using CNOT.

Before analyzing the qudit SWAP circuits, consider a special case. That is, we have SWAP circuits if one of the initial states is known. The circuit in Figure

Partial qudit SWAP circuit for one known input state. Note that a general dimension

Other possibilities of swapping the state

Partial qudit SWAP circuit for one known input state using

The equivalence of qudit gates identified in the last section can be exploited to simplify the existing circuits for the qudit SWAP gate.

The qudit SWAP gate circuit shown in Figure

Qudit SWAP circuit using

Simplified version of the circuit in Figure

Qudit SWAP circuit using

Simplified version of the circuit in Figure

The qudit SWAP can be implemented with a circuit consisting of three

Qudit SWAP circuit with

Simplified version of the circuit in Figure

Apart from simplifying the previously proposed circuits, the equivalence between the qudit gates can be employed to introduce more qudit SWAP circuits. The qudit SWAP can be constructed with three basic gates

Implementation of qudit SWAP using

Implementation of qudit SWAP using

Implementation of qudit SWAP using

Therefore the qudit

In this work, the existence of equivalence between various qudit gates is illustrated and these gates are useful in the construction of qudit SWAP circuit. The identification of equivalence of the qudit operations is useful to minimize the circuit complexity in the previously proposed qudit SWAP gate constructions. In particular, the number of single qudit gates is minimized in the qudit SWAP gate construction. These constructions are useful whenever the single qudit gates are difficult to implement in the experiment. Moreover, many new qudit SWAP circuits comprising of lesser number of gates are proposed. Notably, the proposed circuits have the property of symmetry thereby the inverted circuits also implement qudit SWAP operation. Hence, we argue that not only the three copies of controlled qudit gate (

The author declares that there is no conflict of interests regarding the publication of this paper.