Quantum annealing for combinatorial clustering

Abstract

Clustering is a powerful machine learning technique that groups "similar" data points based on their characteristics. Many clustering algorithms work by approximating the minimization of an objective function, namely the sum of within-thecluster distances between points. The straightforward approach involves examining all the possible assignments of points to each of the clusters. This approach guarantees the solution will be a global minimum, however the number of possible assignments scales quickly with the number of data points and becomes computationally intractable even for very small datasets. In order to circumvent this issue, cost function minima are found using popular local-search based heuristic approaches such as k-means and hierarchical clustering. Due to their greedy nature, such techniques do not guarantee that a global minimum will be found and can lead to sub-optimal clustering assignments. Other classes of global-search based techniques, such as simulated annealing, tabu search, and genetic algorithms may offer better quality results but can be too time consuming to implement. In this work, we describe how quantum annealing can be used to carry out clustering. We map the clustering objective to a quadratic binary optimization (QUBO) problem and discuss two clustering algorithms which are then implemented on commercially-available quantum annealing hardware, as well as on a purely classical solver "qbsolv." The first algorithm assigns N data points to K clusters, and the second one can be used to perform binary clustering in a hierarchical manner. We present our results in the form of benchmarks against well-known k-means clustering and discuss the advantages and disadvantages of the proposed techniques.

Ising-Based Louvain Method: Clustering Large Graphs with Specialized Hardware

“…Examples of these novel computing hardware include, adiabatic quantum computers, CMOS annealers, memristive circuits, and optical parametric oscillators, that are designed to solve optimization problems formulated as an Ising or QUBO mathematical model. Given that many well-known problems in graphs can easily be modeled in this form, there has been a growing interest in formulating and evaluating these problem and their subsequent QUBO models on the different specialized hardware platforms [4,15,21,32,34]. However, many real-world problems require significantly more variables than these devices can handle, thus hybrid methods are usually used.…”

A quantum annealing approach to ionic diffusion in solids

“…The quantum annealing technique 1 , 2 has been widely and successfully applied to challenging combinatorial optimizations 3 , including NP(Non-deterministic Polynomial time)-hard 4 and NP-complete problems 3 , 5 – 7 . Realistic problems such as the capacitated vehicle routing problem (CVRP), optimization of traffic quantity 8 – 11 , investment portfolio design 12 , scheduling problems 13 , and digital marketing 14 have recently been addressed by quantum annealing.…”

“…In the quantum annealing framework, an optimization problem is mapped into a quantum spin system described by the Ising Hamiltonian 1 , 2 . The problem is then solved by searching for optimal spin configurations minimizing the energy of the Hamiltonian.…”

Accuracy and minor embedding in subqubo decomposition with fully connected large problems: a case study about the number partitioning problem

“…Furthermore, with the release of an open-source suite spanning from the decomposing solver Qbsolv to the new Hybrid framework, D-Wave took a significant step forward towards gathering the attention from technology companies. As a matter of fact, with these technologies it is possible to close the gap between logical qubits representation encoded in the QUBO (Quadratic Unconstrained Binary Optimization) matrix and the physical embedding of the problem into the Chimera graph [13].…”