Dynamical Lie algebras generated by Pauli strings and quadratic spaces over $\mathbb{F}_2$

TL;DR

This paper provides a unified mathematical framework for understanding Pauli Lie algebras and introduces an efficient algorithm to determine their structure based on a set of Pauli strings.

Contribution

It offers a comprehensive mathematical approach to recent results and presents a new algorithm for classifying dynamical Lie algebras generated by Pauli strings.

Findings

Unified mathematical framework for Pauli Lie algebras

Efficient algorithm with cubic time complexity

Algorithm determines isomorphism type of generated Lie algebra

Abstract

Dynamical Lie algebras, i.e. Lie subalgebras of $\mathfrak{su}(2^n)$, generated by Pauli strings have recently been studied intensively. They are also called Pauli Lie algebras or Hamiltonian Lie algebras. In this paper we provide a uniform mathematical approach to various recent results on Pauli Lie algebras. Moreover, we present an algorithm that on input of a set of Pauli strings determines the isomorphism type of the dynamical Lie algebra generated by these Pauli's in time $\mathcal{O}(\max(n,m)^3)$ where $m$ is the size of the generating set.