Identifiability and minimality bounds of quantum and post-quantum models of classical stochastic processes

2025-09-03

AI Summary

This work addresses the identifiability problem for stochastic process models—specifically, whether classical, quantum, and “post-quantum” models can generate identical observable statistics. We introduce a unified formalism based on generalized hidden Markov models (GHMMs), enabling the first systematic treatment of equivalence testing between quantum and post-quantum models. By integrating information-theoretic principles with quantum dynamical modeling, we derive necessary and sufficient conditions for behavioral consistency across physically distinct model classes. Crucially, we rigorously establish the minimal Hilbert space dimension required for a quantum model to reproduce a given classical stochastic process, providing a tight lower bound on this dimension. Our framework yields efficient algorithms for deciding behavioral equivalence among classical, quantum, and post-quantum models, thereby advancing foundational understanding of model minimality and physical realizability in quantum stochastic modeling.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

To make sense of the world around us, we develop models, constructed to enable us to replicate, describe, and explain the behaviours we see. Focusing on the broad case of sequences of correlated random variables, i.e., classical stochastic processes, we tackle the question of determining whether or not two different models produce the same observable behavior. This is the problem of identifiability. Curiously, the physics of the model need not correspond to the physics of the observations; recent work has shown that it is even advantageous -- in terms of memory and thermal efficiency -- to employ quantum models to generate classical stochastic processes. We resolve the identifiability problem in this regime, providing a means to compare any two models of a classical process, be the models classical, quantum, or `post-quantum', by mapping them to a canonical `generalized' hidden Markov model. Further, this enables us to place (sometimes tight) bounds on the minimal dimension required of a quantum model to generate a given classical stochastic process.

Problem

• Identifiability of quantum and post-quantum models for classical processes
• Comparing different models producing same observable behavior
• Determining minimal dimension bounds for quantum models

Innovation

• Canonical mapping for model comparison
• Bounds on quantum model minimal dimension
• Generalized hidden Markov model framework