Interpreting Quantum Learning Models via Stochastic Processes

Quantum machine learning (QML) models, which rely on coherent quantum evolution and measurement, typically lack a classical probabilistic interpretation. Unlike classical Markovian processes, quantum dynamics generally violate the Chapman-Kolmogorov divisibility condition, meaning the probability of an outcome cannot be decomposed into a sum over intermediate configurations. This paper investigates whether quantum processes can be mapped onto stochastic dynamics over a configuration space and identifies the trade-offs inherent in such representations.

The authors develop a framework to represent quantum learning models as stochastic processes by mapping quantum states to probability distributions over a fixed set of operators (POVMs). They analyze two primary representations:

1. Informationally Complete (IC) POVMs: These representations ensure the dynamics are Markovian (first-order) but force the transition kernels to be quasi-stochastic, meaning they can contain negative values.
2. Projective Representations: By restricting the configuration space to orthogonal projectors, the transition kernels remain positive (stochastic), but the dynamics generally become non-Markovian, requiring memory of past configurations to satisfy the Chapman-Kolmogorov condition.

The study establishes a fundamental trade-off between positivity and divisibility in quantum dynamics. If one demands a Markovian representation, one must accept negative probabilities (quasi-stochasticity). If one demands a positive stochastic representation, one must accept non-Markovianity (dependence on past states). The authors further apply this to Projective Simulation—a model of agency and learning—to show how quantum deliberation can be approximated by finite-order stochastic kernels, effectively recovering classical machine learning behavior in specific regimes.