Finite automata for caching in matrix product algorithms

TL;DR

This paper establishes a connection between matrix product states and complex weighted finite automata, providing a new perspective on efficient caching in matrix product algorithms and extending these techniques to higher dimensions.

Contribution

It introduces a diagrammatic approach linking matrix product factorizations with finite automata and generalizes these methods to multi-dimensional cases.

Findings

Automata can generate matrix product operators, enabling immediate factorizations.

The connection offers insights into efficient caching mechanisms.

Techniques are extended to multi-dimensional systems.

Abstract

A diagram is introduced for visualizing matrix product states which makes transparent a connection between matrix product factorizations of states and operators, and complex weighted finite state automata. It is then shown how one can proceed in the opposite direction: writing an automaton that ``generates'' an operator gives one an immediate matrix product factorization of it. Matrix product factorizations have the advantage of reducing the cost of computing expectation values by facilitating caching of intermediate calculations. Thus our connection to complex weighted finite state automata yields insight into what allows for efficient caching in matrix product algorithms. Finally, these techniques are generalized to the case of multiple dimensions.