[Paper Review] Expressiveness and Learning of Hidden Quantum Markov Models

Sandesh Adhikary, Siddarth Srinivasan+2 more|arXiv (Cornell University)|Dec 2, 2019

Bayesian Modeling and Causal Inference37 references4 citations

TL;DR

This paper establishes that Hidden Quantum Markov Models (HQMMs) are a subclass of Observable Operator Models (OOMs) with guaranteed non-negative probabilities, and introduces a retraction-based optimization method on the Stiefel manifold that enables faster, more scalable learning of HQMMs. The approach outperforms prior methods in speed and accuracy, enabling training of larger HQMMs that were previously infeasible to learn.

ABSTRACT

Extending classical probabilistic reasoning using the quantum mechanical view of probability has been of recent interest, particularly in the development of hidden quantum Markov models (HQMMs) to model stochastic processes. However, there has been little progress in characterizing the expressiveness of such models and learning them from data. We tackle these problems by showing that HQMMs are a special subclass of the general class of observable operator models (OOMs) that do not suffer from the \emph{negative probability problem} by design. We also provide a feasible retraction-based learning algorithm for HQMMs using constrained gradient descent on the Stiefel manifold of model parameters. We demonstrate that this approach is faster and scales to larger models than previous learning algorithms.

Motivation & Objective

• To characterize the expressiveness of Hidden Quantum Markov Models (HQMMs) in relation to Observable Operator Models (OOMs) and NOOMs.
• To resolve the negative probability problem inherent in OOMs by showing HQMMs avoid it by design.
• To develop a practical, scalable learning algorithm for HQMMs that ensures feasibility of model parameters.
• To empirically validate the new learning method on synthetic and real-world datasets, demonstrating superior performance over prior approaches.
• To show that HQMMs can model arbitrary HMMs and outperform HMMs in classification tasks with comparable or fewer parameters.

Proposed method

• HQMMs are formulated using Kraus operators {K_i} satisfying the constraint ∑K_i†K_i = I, which ensures valid quantum states and avoids negative probabilities.
• The model parameters are represented as a matrix κ with κ†κ = I, lying on the Stiefel manifold, enabling constrained optimization.
• A retraction-based gradient descent algorithm is used to perform constrained optimization on the Stiefel manifold, ensuring feasibility of parameters during learning.
• The Liouville representation of HQMMs is derived, providing a unique and interpretable characterization of the model for direct comparison with OOMs.
• The method is implemented as COSM (Constrained Optimization on the Stiefel Manifold), which directly optimizes the likelihood on the manifold without iterative unitary corrections.
• The approach is compared to the prior GS (Gradient Search) method that used iterative unitary rotations, which is shown to be inefficient and prone to poor convergence.

Experimental results

Research questions

• RQ1 Can HQMMs be formally related to OOMs and NOOMs, and do they generalize these models?
• RQ2 Does the HQMM framework inherently avoid the negative probability problem of OOMs?
• RQ3 Can a scalable and efficient learning algorithm be developed for HQMMs that maintains parameter feasibility?
• RQ4 Do HQMMs outperform HMMs in modeling and classification tasks, especially on larger models?
• RQ5 Can HQMMs be trained on real-world datasets where previous methods failed due to computational infeasibility?

Key findings

• HQMMs are a proper subclass of OOMs and generalize NOOMs, with valid initial states characterized as the spectraplex of Hermitian positive semi-definite matrices with trace 1.
• The proposed COSM algorithm converges in under 300 seconds for large HQMMs, while the prior GS method failed to converge even after 2000 seconds.
• For a (6,6)-HMM, a (6,6,6)-HQMM trained with COSM successfully models the target HMM, confirming theoretical guarantees in practice.
• On the splice dataset, HQMMs trained with COSM outperform HMMs trained with EM, achieving higher classification accuracy with a larger number of parameters.
• The marginal gain in performance from increasing the Kraus rank w is largest when going from w=1 to w=2, but further gains are observed with higher n, indicating that both n and w must be tuned per dataset.
• The COSM method enables training of HQMMs with up to 100 hidden states, which was computationally infeasible with the prior GS method.