[Paper Review] Training a Binary Classifier with the Quantum Adiabatic Algorithm

Hartmut Neven, Vasil S. Denchev+2 more|ArXiv.org|Nov 4, 2008

Quantum Computing Algorithms and Architecture10 references71 citations

TL;DR

This paper proposes training a binary classifier using the quantum adiabatic algorithm by formulating the weight optimization problem as a binary quadratic program, enabling mapping to D-Wave's quantum annealing hardware. It demonstrates that bit-constrained learning—representing weights with logarithmic precision—yields lower generalization error and outperforms AdaBoost, especially when using quadratic loss, suggesting quantum advantage potential for NP-hard classification problems.

ABSTRACT

This paper describes how to make the problem of binary classification amenable to quantum computing. A formulation is employed in which the binary classifier is constructed as a thresholded linear superposition of a set of weak classifiers. The weights in the superposition are optimized in a learning process that strives to minimize the training error as well as the number of weak classifiers used. No efficient solution to this problem is known. To bring it into a format that allows the application of adiabatic quantum computing (AQC), we first show that the bit-precision with which the weights need to be represented only grows logarithmically with the ratio of the number of training examples to the number of weak classifiers. This allows to effectively formulate the training process as a binary optimization problem. Solving it with heuristic solvers such as tabu search, we find that the resulting classifier outperforms a widely used state-of-the-art method, AdaBoost, on a variety of benchmark problems. Moreover, we discovered the interesting fact that bit-constrained learning machines often exhibit lower generalization error rates. Changing the loss function that measures the training error from 0-1 loss to least squares maps the training to quadratic unconstrained binary optimization. This corresponds to the format required by D-Wave's implementation of AQC. Simulations with heuristic solvers again yield results better than those obtained with boosting approaches. Since the resulting quadratic binary program is NP-hard, additional gains can be expected from applying the actual quantum processor.

Motivation & Objective

• To adapt binary classification training to quantum adiabatic computing by reformulating the optimization problem as a binary program.
• To investigate whether bit-constrained weight representations improve generalization performance in machine learning models.
• To compare the performance of global optimization via quantum-inspired heuristics against greedy methods like AdaBoost.
• To enable the application of adiabatic quantum computing to NP-hard machine learning problems by mapping them to quadratic unconstrained binary optimization (QUBO).
• To explore the implications of low-precision weight representation for model compactness and generalization in quantum machine learning.

Proposed method

• The classifier is modeled as a thresholded linear combination of weak classifiers, with weights optimized to minimize a regularized loss function combining 0-1 or least-squares error and 0-norm regularization.
• The bit-precision required for optimal weights grows logarithmically with the ratio of training examples to weak classifiers, enabling effective binary representation.
• The 0-1 loss formulation is mapped to a binary optimization problem, while quadratic loss maps directly to a QUBO, compatible with D-Wave’s quantum annealing hardware.
• Heuristic solvers such as tabu search and simulated annealing are used to solve the resulting binary programs, simulating quantum optimization performance.
• The problem is reformulated as a QUBO by replacing 0-1 loss with least-squares loss, aligning with the native input format of D-Wave’s quantum processors.
• The framework is tested on synthetic and real-world data (Gabor wavelet features from facial images), with cross-validation to assess generalization.

Experimental results

Research questions

• RQ1 Can the problem of training a sparse, high-accuracy binary classifier be effectively mapped to a form solvable by adiabatic quantum computing?
• RQ2 Does reducing weight bit-precision improve generalization error, and if so, why?
• RQ3 How does global optimization via quantum-inspired heuristics compare to greedy methods like AdaBoost in terms of accuracy and model compactness?
• RQ4 To what extent does using quadratic loss instead of 0-1 loss improve performance in bit-constrained learning?
• RQ5 Can quantum adiabatic computing provide a practical advantage over classical heuristics for this NP-hard classification problem?

Key findings

• Bit-constrained learning machines, with weights represented using logarithmic bit-precision, consistently exhibit lower generalization error than higher-precision counterparts.
• The global optimization approach using quadratic loss (QUBO) outperforms AdaBoost, achieving less than 10% higher accuracy with over 50% fewer active weak classifiers.
• On the Gabor wavelet facial data set, the QUBO-based method reduced test error while cutting the number of non-zero weights by more than half compared to AdaBoost.
• The 0-1 loss formulation showed similar performance across bit depths, but the quadratic loss formulation yielded superior results, especially with global optimization.
• The study reveals that bit-constrained models act as an intrinsic form of regularization, contributing to better generalization and model compactness.
• The results suggest that training binary classifiers as integer programs may be more effective than continuous optimization, motivating quantum acceleration of such NP-hard problems.