Matrix Product States for Quantum Stochastic Modelling

TL;DR

This paper establishes a connection between stochastic process modeling and matrix product states (MPS) in quantum physics, enabling improved quantum predictive models and insights into quantum memory via entanglement.

Contribution

It introduces a novel link between stochastic processes and MPS, providing a systematic way to construct optimal quantum predictive models and analyze quantum memory.

Findings

Optimal predictive models correspond to MPS representations.

Quantum memory equals the entanglement across past-future bipartition.

MPS methods improve quantum stochastic modeling.

Abstract

In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process' future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.