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Lower bounds to variational problems with guarantees

J. Eisert

TL;DR

Problem: certifying ground-state energies for translationally invariant lattice Hamiltonians with periodic boundaries is challenging when relying on variational upper bounds. Approach: derive computable lower bounds with guarantees using the Anderson bound and semidefinite relaxations, and develop a hierarchy of improved Anderson bounds from the quantum marginal problem. Contributions: rigorous $O(1)$-scale guarantees for the Anderson bound, a family of SDP-based lower bounds with $x_N\le e_{\min}(H_N)\le x_N+O(1)$, and a nested $(x_{m,s})$ hierarchy providing progressively tighter bounds. Significance: these certificates enable robust classical benchmarks for variational methods, clarify the requirements for quantum advantage, and tie into broader questions such as the quantum PCP conjecture and fermionic generalizations.

Abstract

Variational methods play an important role in the study of quantum many-body problems, both in the flavor of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This work stresses that for translationally invariant lattice Hamiltonians with periodic boundary conditions, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semi-definite relaxations both provide approximations with performance guarantees that scale like a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semi-definite relaxations inspired by the quantum marginal problem.

Lower bounds to variational problems with guarantees

TL;DR

Problem: certifying ground-state energies for translationally invariant lattice Hamiltonians with periodic boundaries is challenging when relying on variational upper bounds. Approach: derive computable lower bounds with guarantees using the Anderson bound and semidefinite relaxations, and develop a hierarchy of improved Anderson bounds from the quantum marginal problem. Contributions: rigorous -scale guarantees for the Anderson bound, a family of SDP-based lower bounds with , and a nested hierarchy providing progressively tighter bounds. Significance: these certificates enable robust classical benchmarks for variational methods, clarify the requirements for quantum advantage, and tie into broader questions such as the quantum PCP conjecture and fermionic generalizations.

Abstract

Variational methods play an important role in the study of quantum many-body problems, both in the flavor of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This work stresses that for translationally invariant lattice Hamiltonians with periodic boundary conditions, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semi-definite relaxations both provide approximations with performance guarantees that scale like a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semi-definite relaxations inspired by the quantum marginal problem.
Paper Structure (5 sections, 3 theorems, 23 equations, 2 figures)

This paper contains 5 sections, 3 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A performance guarantee for the Anderson bound
  3. Performance guarantees for semi-definite relaxations
  4. A hierarchy of improved Anderson bounds
  5. Summary and outlook

Key Result

Proposition 1

Consider a family of translationally invariant Hamiltonians of the form (LH) on a cubic lattice in some spatial dimension $D$with periodic boundary conditions, indexed by the system size $N=n^D$, and let $\lambda_\text{min}(h_m)$ be the smallest eigenvalue of a cubic patch $h_m$ of $H_N$ on $m^D$ si

Figures (2)

  • Figure 1: (a) The configuration in the original Anderson bound applied to a two-dimensional translationally invariant lattice system. For the periodic boundary conditions, the last row and column are identified with hte first one. (b) The configuration in one spatial dimension used to show the performance guarantee of the Anderson bound. (c) The configuration employed for the improved Anderson bound based on semi-definite programming and the marginal problem, again applied to one spatial dimension.
  • Figure 2: The Anderson bound for the one-dimensional Heisenberg model $H = \frac{1}{2} \sum_{j}\tau_j (X\otimes X+ Y\otimes Y+ Z\otimes Z)$ as a function of the patch size $m$ until $m=15$, featuring a noteworthy even-odd effect. The straight line represents the exact ground state energy density, $e_\text{min}=-2\log(2) +1/2$.

Theorems & Definitions (6)

  • Proposition 1: Performance guarantee of the Anderson bound
  • proof
  • Proposition 2: Performance guarantee of semi-definite bounds
  • proof
  • Proposition 3: Improved Anderson bounds
  • proof