Learning Density Operator Latent Variable Models via Quantum Information Projection

Learning from incomplete information is a fundamental problem in statistics and machine learning. Often, the observed data depends on hidden variables that cannot be measured directly. For nearly five decades, the Expectation-Maximization (EM) algorithm has addressed this problem by alternating between estimating missing information and updating the model’s parameters. But can this principle be extended beyond classical probability?

Researchers at the Indian Institute of Science (IISc)— Adit Vishnu, Abhay Shastry (now at Chanakya University), Dhruva Kashyap (now at HP AI Lab), and Chiranjib Bhattacharyya—have developed Density Operator Expectation-Maximization (DO-EM), extending this fundamental learning principle to density operators, the mathematical objects used to represent quantum states.

An Operator ELBO

Classical EM relies on conditional probabilities and a mathematical construction called the Evidence Lower Bound (ELBO). Extending these ideas to density operators is challenging because operators need not commute, and classical conditioning has no straightforward quantum counterpart.

The researchers overcome this difficulty by deriving an operator analogue of ELBO using the data-processing inequality from quantum information theory. They formulate the expectation step as a quantum information projection: finding a joint density operator consistent with the observations while remaining closest to the current model. Combined with a parameter-update step, this yields the DO-EM algorithm.

Under suitable mathematical conditions, the information projection recovers the celebrated Petz recovery map, and the algorithm guarantees monotonic improvement in model likelihood.

Making Quantum BMs competitive

The researchers introduce classical–quantum latent variable models, combining classical observations with quantum hidden states. This allows quantum variants of Boltzmann machines to be trained on standard image datasets, scaling from the roughly dozen visible units studied in earlier work to models containing over 18,000 units.

Figure: Comparison of samples from a Deep Boltzmann Machine and a Quantum interleaved Deep Boltzmann Machine.

On MNIST, Fashion-MNIST, and CelebA, these models achieve 20–65% lower Fréchet Inception Distance (FID) than comparable classical models, indicating improved generative performance.

Beyond these empirical results, the work provides a framework for investigating whether a quantum structure offers greater representational power, more favourable learning dynamics, or both. By extending EM from probabilities to operators, it opens a new direction for learning with quantum models.

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