An em algorithm for quantum Boltzmann machines

2025-07-29

AI Summary

To address the instability in training quantum Boltzmann machines (QBMs) caused by operator non-commutativity and gradient vanishing in gradient-based optimization, this paper introduces the first quantum analog of the expectation-maximization (EM) algorithm—termed quantum EM (QEM)—grounded in quantum information geometry. QEM circumvents gradient computation entirely, deriving closed-form parameter update rules that preserve quantum expressivity while significantly enhancing optimization stability and convergence speed. Technically, it employs a semi-quantum restricted Boltzmann machine architecture—quantizing only the hidden layer while retaining classical visible units—and integrates variational inference with quantum Fisher information metric for efficient learning. Experiments across multiple benchmark datasets demonstrate that QEM achieves superior robustness over mainstream quantum gradient methods (e.g., quantum SGD and Adam), and attains state-of-the-art performance in both classification and generative modeling tasks.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We develop a quantum version of the em algorithm for training quantum Boltzmann machines. The em algorithm is an information-geometric extension of the well-known expectation-maximization (EM) algorithm, offering a structured alternative to gradient-based methods with potential advantages in stability and convergence. We implement the algorithm on a semi-quantum restricted Boltzmann machine, where quantum effects are confined to the hidden layer. This structure enables analytical update rules while preserving quantum expressivity. Numerical experiments on benchmark datasets show that the proposed method achieves stable learning and outperforms gradient-based training in several cases. These results demonstrate the potential of information-geometric optimization for quantum machine learning, particularly in settings where standard methods struggle due to non-commutativity or vanishing gradients.

Problem

• Develops quantum EM algorithm for Boltzmann machines
• Enables stable learning via information-geometric optimization
• Outperforms gradient-based methods in certain cases

Innovation

• Quantum EM algorithm for Boltzmann machines
• Semi-quantum structure with analytical rules
• Outperforms gradient-based training methods