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Fast mixing of weakly interacting fermionic systems at any temperature

Yu Tong, Yongtao Zhan

Abstract

We study the mixing time of a recently proposed efficiently implementable Lindbladian designed to prepare the Gibbs states in the setting of weakly interacting fermionic systems. We show that at any temperature, the Lindbladian spectral gap for even parity observables is lower bounded by a constant that is independent of the system size, when the interaction strength (e.g., the on-site interaction strength for the Fermi-Hubbard model) is below a constant threshold, which is also independent of the system size. This leads to a mixing time estimate that is at most linear in the system size, thus showing that the corresponding Gibbs states can be prepared efficiently on quantum computers.

Fast mixing of weakly interacting fermionic systems at any temperature

Abstract

We study the mixing time of a recently proposed efficiently implementable Lindbladian designed to prepare the Gibbs states in the setting of weakly interacting fermionic systems. We show that at any temperature, the Lindbladian spectral gap for even parity observables is lower bounded by a constant that is independent of the system size, when the interaction strength (e.g., the on-site interaction strength for the Fermi-Hubbard model) is below a constant threshold, which is also independent of the system size. This leads to a mixing time estimate that is at most linear in the system size, thus showing that the corresponding Gibbs states can be prepared efficiently on quantum computers.
Paper Structure (32 sections, 49 theorems, 275 equations)

This paper contains 32 sections, 49 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Main result.
  3. Proof ideas.
  4. Applications and implications.
  5. Notations.
  6. Organization.
  7. Problem setup
  8. The Hamiltonian
  9. The Lindbladian
  10. Background
  11. Third quantization
  12. Stability of free-fermion Hamiltonians
  13. The Lieb-Robinson bound
  14. Constructing the parent Hamiltonian
  15. Structure of the parent Hamiltonian
  16. ...and 17 more sections

Key Result

Theorem 1

Let $H=H_0+V$ be a Hamiltonian on a $D$-dimensional cubic lattice $\Lambda$ consisting of $n$ sites. $H_0$ is quadratic in Majorana operators and is $(1,r_0)$-geometrically local, and $V$ is parity-preserving and is $(U,r_0)$-geometrically local. Then for any inverse temperature $\beta>0$, there exi where $C,U_\beta>0$ are constants that depend only on $r_0,D,\beta$.

Theorems & Definitions (103)

  • Definition 1: $(\xi,r_0)$-geometrically local operators
  • Theorem 1: Main result
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Remark 1: $a$-fermions on the lattice $\Lambda$
  • Definition 2
  • Definition 3
  • Remark 2
  • Definition 4
  • ...and 93 more