Regular language quantum states

TL;DR

This paper introduces regular language quantum states, a new family of many-body states built from regular languages, with a framework for their representation, recognition, and equivalence, connecting quantum physics and formal language theory.

Contribution

It develops a theoretical framework for regular language states, including their matrix product state representation, canonical form, and criteria for equivalence and shift-invariance.

Findings

Expressed regular language states as matrix product states.

Established criteria for recognizing and determining equivalence of these states.

Provided conditions for shift-invariance using tensor network theory.

Abstract

We introduce regular language states, a family of quantum many-body states. They are built from a special class of formal languages, called regular, which has been thoroughly studied in the field of computer science. They can be understood as the superposition of all the words in a regular language and encompass physically relevant states such as the GHZ-, W- or Dicke-states. By leveraging the theory of regular languages, we develop a theoretical framework to describe them. First, we express them in terms of matrix product states, providing efficient criteria to recognize them. We then develop a canonical form which allows us to formulate a fundamental theorem for the equivalence of regular language states, including under local unitary operations. We also exploit the theory of tensor networks to find an efficient criterion to determine when regular languages are shift-invariant.