Robust Quantum Circuit for Clique Problem with Intermediate Qudits

TL;DR

This paper presents an improved quantum circuit for solving the $k$-clique and maximum clique problems using higher-dimensional qudits, reducing cost and depth compared to existing qubit-based circuits.

Contribution

It introduces a novel quantum circuit implementation utilizing intermediate qudits, specifically ququarts, to enhance efficiency for clique problems.

Findings

Reduced circuit cost and depth using ququarts

First application of intermediate qudits for clique problems

Demonstrated advantages over traditional qubit-based circuits

Abstract

Clique problem has a wide range of applications due to its pattern matching ability. There are various formulation of clique problem like $k$-clique problem, maximum clique problem, etc. The $k$-Clique problem, determines whether an arbitrary network has a clique or not whereas maximum clique problem finds the largest clique in a graph. It is already exhibited in the literature that the $k$-clique or maximum clique problem (NP-problem) can be solved in an asymptotically faster manner by using quantum algorithms as compared to the conventional computing. Quantum computing with higher dimensions is gaining popularity due to its large storage capacity and computation power. In this article, we have shown an improved quantum circuit implementation for the $k$-clique problem and maximum clique problem (MCP) with the help of higher-dimensional intermediate temporary qudits for the first time to the best of our knowledge. The cost of state-of-the-art quantum circuit for $k$-clique problem is colossal due to a huge number of $n$-qubit Toffoli gates. We have exhibited an improved cost and depth over the circuit by applying a generalized $n$-qubit Toffoli gate decomposition with intermediate ququarts (4-dimensional qudits).