A proposal for a conditional definition of the Swap Gate

TL;DR

This paper proposes a new conditional definition for the SWAP gate in quantum computing, based on the concept of conditional quantum dynamics, extending the paradigm used for other gates like CNOT.

Contribution

It introduces a novel conditional 'If-Then' definition for the SWAP gate, expanding the framework of conditional quantum operations.

Findings

Defines a conditional SWAP gate based on qubit states

Classifies SWAP gate into two classes based on qubit conditions

Provides a theoretical foundation for conditional quantum gate design

Abstract

In order to realize quantum logical operations, Quantum Computation (QC) requires that its basic tools and concepts obey the laws of physics. One of the fundamental concepts in QC is the conditional quantum dynamics \cite{barenco}, some times called controlled-unitary operation \cite{kumar}, which is established by the conditional "If-Then" sentence. The best know example is the c-not gate, which operates on the computational basis as follows: \textit{If the control qubit is set to $% |1>$, then apply the single qubit quantum NOT gate on the target. Otherwise, if the control qubit is set to $|0>$, then the target qubit is unchanged}. This gate represents the paradigm for the conditional quantum dynamics, where the flipping of the target qubit is conditioned to the state of the control qubit. Other gates have been defined in a similar way of conditional evolution; for instance, the control phase gates \cite{paulo1}. However, to the best of our knowledge, such conditional quantum dynamics has not been yet used to define the SWAP gate. Here we propose a possible conditional definition, in the form of If-Then sentence, to construct a SWAP gate in the case of two qubits. This definition suggests a classification in two classes, which depends upon the number of qubits that have to undergo a conditional quantum dynamics.