Time-efficient implementation of quantum search with qudits

TL;DR

This paper introduces a simplified and more efficient method for implementing multi-valued Grover's quantum search using qudits, reducing complexity and interaction steps compared to previous approaches.

Contribution

It proposes a new scheme replacing the Hadamard gate with a simpler unitary matrix for qudit systems, enabling faster and more practical quantum search implementations.

Findings

The scheme reduces the number of control particles needed.

It allows implementation with a single interaction step in realistic systems.

The proposed matrix F simplifies the construction compared to the discrete Fourier transform.

Abstract

We propose a simpler and more efficient scheme for the implementation of the multi-valued Grover's quantum search. The multi-valued search generalizes the original Grover's search by replacing qubits with qudits --- quantum systems of $d$ discrete states. The qudit database is exponentially larger than the qubit database and thus it requires fewer particles to control. The Hadamard gate, which is the key elementary gate in the qubit implementation of Grover's search, is replaced by a $d$-dimensional (complex-valued) unitary matrix $F$, the only condition for which is to have a column of equal moduli elements irrespective of their phases; it can be realized through any physical interaction, which achieves an equal-weight superposition state. An example of such a transformation is the $d$-dimensional discrete Fourier transform, used in earlier proposals; however, its construction is much more costly than that of the far simpler matrix $F$. We present examples of how such a transform $F$ can be constructed in realistic qudit systems in a \emph{single} interaction step.