Quantum-Assisted Graph Clustering and Quadratic Unconstrained D-ary Optimisation

TL;DR

This paper explores quantum-assisted algorithms for graph clustering, focusing on Max-Cut and Max-D-Cut problems, proposing new Hamiltonians, and extending to qudit-based quantum devices with numerical evaluations and circuit constructions.

Contribution

It introduces a quantum Ising model framework for Max-Cut and Max-D-Cut, develops Hamiltonians for these problems, and designs qudit circuits for quantum approximate optimization.

Findings

Numerical evaluations demonstrate quantum algorithms' potential for graph clustering.

Extension of quantum Max-Cut to Max-3-Cut with Hamiltonian formulation.

Preliminary insights into implementing these algorithms on qudit-based quantum hardware.

Abstract

Of late, we are witnessing spectacular developments in Quantum Information Processing with the availability of Noisy Intermediate-Scale Quantum devices of different architectures and various software development kits to work on quantum algorithms. Different problems, which are hard to solve by classical computation, but can be sped up (significantly in some cases) are also being populated. Leveraging these aspects, this paper examines unsupervised graph clustering by quantum algorithms or, more precisely, quantum-assisted algorithms. By carefully examining the two cluster Max-Cut problem within the framework of quantum Ising model, an extension has been worked out for max 3-cut with the identification of an appropriate Hamiltonian. Representative results, after carrying out extensive numerical evaluations, have been provided including a suggestion for possible futuristic implementation with qutrit devices. Further, extrapolation to more than 3 classes, which can be handled by qudits, of both annealer and gate-circuit varieties, has also been touched upon with some preliminary observations; quantum-assisted solving of Quadratic Unconstrained D-ary Optimisation is arrived at within this context. As an additional novelty, a qudit circuit to solve max-d cut through Quantum Approximate Optimization algorithm is systematically constructed.