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

Hans Cuypers

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.

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

Abstract

Dynamical Lie algebras, i.e. Lie subalgebras of , 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 where is the size of the generating set.
Paper Structure (21 sections, 14 theorems, 73 equations, 3 figures)

This paper contains 21 sections, 14 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Pauli group and a quadratic $\mathbb{F}_2$-space
  3. Lie subalgebras and the geometry of elliptic lines
  4. Subspaces of the geometry of elliptic lines
  5. Lie subalgebras and subspaces
  6. Generators and frustration graphs
  7. An algorithm to determine the isomorphism type of a Pauli Lie algebra
  8. Relations to recent work
  9. Subalgebras generated by Pauli strings
  10. Minimal sets of generators
  11. Commutator graph
  12. Free-fermion Hamiltonians
  13. Cartan decomposition
  14. Discussion
  15. Quadratic spaces over the field with $2$ elements
  16. ...and 6 more sections

Key Result

Proposition 3.1

Figures (3)

  • Figure 1: A dual affine plane
  • Figure 2: The $32$ forbidden graphs.
  • Figure 3: A claw and its incidence graph with edges in gray.

Theorems & Definitions (24)

  • Example 2.1
  • Proposition 3.1
  • Theorem 4.1
  • Theorem 5.1
  • Remark 5.2
  • Proposition 5.3
  • Theorem 5.4
  • Example 5.5
  • Theorem 5.6: Classification of Lie subalgebras generated by Pauli strings
  • Theorem 6.1
  • ...and 14 more