DO-EM: Density Operator Expectation Maximization

Adit Vishnu, Abhay Shastry, Dhruva Kashyap, Chiranjib Bhattacharyya

Keywords: Density Operators, Expectation-Maximization, Quantum Unsupervised Learning, Latent Variable Models

TL;DR: Training density operator latent variable models using DO-EM outperforms corresponding classical latent variable models.

Abstract: Density operators, quantum generalizations of probability distributions, are gaining prominence in machine learning due to their foundational role in quantum computing. Generative modeling based on density operator models (**DOMs**) is an emerging field, but existing training algorithms - such as those for the Quantum Boltzmann Machine - do not scale to real-world data, such as the MNIST dataset. The Expectation-Maximization algorithm has played a fundamental role in enabling scalable training of probabilistic latent variable models on real-world datasets. *In this paper, we develop an Expectation-Maximization framework to learn latent variable models defined through **DOMs** on classical hardware, with resources comparable to those used for probabilistic models, while scaling to real-world data.* However, designing such an algorithm is nontrivial due to the absence of a well-defined quantum analogue to conditional probability, which complicates the Expectation step. To overcome this, we reformulate the Expectation step as a quantum information projection (QIP) problem and show that the Petz Recovery Map provides a solution under sufficient conditions. Using this formulation, we introduce the Density Operator Expectation Maximization (DO-EM) algorithm - an iterative Minorant-Maximization procedure that optimizes a quantum evidence lower bound. We show that the **DO-EM** algorithm ensures non-decreasing log-likelihood across iterations for a broad class of models. Finally, we present Quantum Interleaved Deep Boltzmann Machines (**QiDBMs**), a **DOM** that can be trained with the same resources as a DBM. When trained with **DO-EM** under Contrastive Divergence, a **QiDBM** outperforms larger classical DBMs in image generation on the MNIST dataset, achieving a 40–60% reduction in the Fréchet Inception Distance.

Primary Area: Probabilistic methods (e.g., variational inference, causal inference, Gaussian processes)

Submission Number: 29239