Optimal stochastic modelling with unitary quantum dynamics

TL;DR

This paper introduces phase-enhanced quantum models for stochastic processes, demonstrating they can outperform classical models by requiring less past information and memory, revealing unique quantum advantages and complexity differences.

Contribution

The paper presents the most general class of unitary quantum models for causal simulation, showing advantages over previous methods in information efficiency and memory requirements.

Findings

Quantum models require less past information than classical models.

Quantum models can have smaller memory dimensions.

Quantum advantages depend on a trade-off between information storage and memory size.

Abstract

Identifying and extracting the past information relevant to the future behaviour of stochastic processes is a central task in the quantitative sciences. Quantum models offer a promising approach to this, allowing for accurate simulation of future trajectories whilst using less past information than any classical counterpart. Here we introduce a class of phase-enhanced quantum models, representing the most general means of causal simulation with a unitary quantum circuit. We show that the resulting constructions can display advantages over previous state-of-art methods - both in the amount of information they need to store about the past, and in the minimal memory dimension they require to store this information. Moreover, we find that these two features are generally competing factors in optimisation - leading to an ambiguity in what constitutes the optimal model - a phenomenon that does not manifest classically. Our results thus simultaneously offer new quantum advantages for stochastic simulation, and illustrate further qualitative differences in behaviour between classical and quantum notions of complexity.