[Paper Review] Adiabatic Quantum Computing for Binary Clustering

Christian Bauckhage, Eduardo Barreto Brito+4 more|arXiv (Cornell University)|Jun 17, 2017

Quantum Computing Algorithms and Architecture11 references3 citations

TL;DR

This paper proposes an adiabatic quantum computing approach for binary clustering by reformulating the k-means objective into an Ising model suitable for D-Wave-type quantum processors. Using numerical simulations via QuTiP, it demonstrates that qubit systems can adiabatically evolve toward cluster solutions, showing feasibility of quantum adiabatic optimization for unsupervised learning.

ABSTRACT

Quantum computing for machine learning attracts increasing attention and recent technological developments suggest that especially adiabatic quantum computing may soon be of practical interest. In this paper, we therefore consider this paradigm and discuss how to adopt it to the problem of binary clustering. Numerical simulations demonstrate the feasibility of our approach and illustrate how systems of qubits adiabatically evolve towards a solution.

Motivation & Objective

• To bridge quantum computing and machine learning by adapting adiabatic quantum computation to unsupervised learning tasks.
• To address the challenge of mapping classical clustering objectives into Ising models compatible with current adiabatic quantum hardware.
• To demonstrate that binary clustering via k-means can be equivalently framed as maximizing between-cluster scatter, enabling Ising model formulation.
• To validate the quantum approach through numerical simulations of adiabatic evolution using QuTiP, providing insight into quantum state dynamics.

Proposed method

• Reformulate the k-means clustering objective for k=2 by maximizing between-cluster scatter $ S_B = 2n_1n_2\|\bm{\mu}_1 - \bm{\mu}_2\|^2 $, which is mathematically equivalent to minimizing within-cluster scatter.
• Express the maximization of $ S_B $ as a quadratic unconstrained binary optimization (QUBO) problem, which maps directly to the Ising Hamiltonian.
• Construct a qubit register where spin states represent cluster assignments (e.g., +1 for cluster 1, -1 for cluster 2), encoding data point assignments into the system's energy landscape.
• Define a time-dependent Hamiltonian that evolves adiabatically from a simple initial Hamiltonian to the final Ising Hamiltonian corresponding to the clustering problem.
• Simulate the adiabatic evolution using QuTiP, tracking the time evolution of quantum state amplitudes to observe convergence toward the ground state.
• Visualize the dynamics of basis state probabilities to analyze how the system evolves toward the optimal clustering configuration.

Experimental results

Research questions

• RQ1 Can the k-means clustering problem for k=2 be reformulated as an Ising model suitable for adiabatic quantum computation?
• RQ2 Does adiabatic evolution of a qubit register converge toward a quantum state that corresponds to a valid binary clustering solution?
• RQ3 How do the amplitudes of quantum basis states evolve during adiabatic evolution, and what insights do they provide into the clustering process?
• RQ4 What is the feasibility of using current adiabatic quantum hardware (e.g., D-Wave) to solve binary clustering problems via Ising model encoding?
• RQ5 Can the equivalence between minimizing within-cluster scatter and maximizing between-cluster scatter be leveraged to enable quantum optimization of clustering?

Key findings

• The problem of minimizing within-cluster scatter in k=2-means clustering is mathematically equivalent to maximizing between-cluster scatter $ S_B $, enabling a quantum-compatible reformulation.
• The between-cluster scatter $ S_B $ for k=2 simplifies to $ 2n_1n_2\|\bm{\mu}_1 - \bm{\mu}_2\|^2 $, which can be expressed as a quadratic form suitable for QUBO and Ising model encoding.
• Numerical simulations using QuTiP confirm that adiabatic evolution of the qubit system converges toward the ground state corresponding to the optimal clustering configuration.
• The simulation results show that the amplitudes of basis states evolve in a way that highlights the emergence of the correct cluster assignment, validating the quantum approach.
• The total scatter $ S_T $ is constant for a fixed dataset, and since $ S_T = S_W + \frac{1}{2n}S_B $, minimizing $ S_W $ is equivalent to maximizing $ S_B $, confirming the duality of objectives.
• The method enables the use of adiabatic quantum computing for unsupervised learning by transforming a classical optimization problem into a form native to current quantum annealing hardware.