Solving cluster moment relaxation with hierarchical matrix

TL;DR

This paper introduces a hierarchical matrix approach to accelerate the solution of semidefinite relaxations for many-body problems, achieving near-linear time complexity and effective lower bounds on ground-state energy.

Contribution

It proposes a hierarchical matrix structure for SDP relaxations and an efficient update method, significantly speeding up computations for quantum many-body problems.

Findings

Achieves quadratic and near-linear per-iteration complexity.

Provides accurate lower bounds for quantum ground-state energy.

Demonstrates effectiveness on the quantum transverse field Ising model.

Abstract

Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a polynomial-time solvable semidefinite program (SDP) that provides a lower bound for the energy can be derived. In this paper, we propose accelerating the solution of such an SDP relaxation by imposing a hierarchical structure on the positive semidefinite (PSD) primal and dual variables. Furthermore, these matrices can be updated efficiently using the algebra of the compressed representations within an augmented Lagrangian method. We achieve quadratic and even near-linear time per-iteration complexity. Through experimentation on the quantum transverse field Ising model, we showcase the capability of our approach to provide a sufficiently accurate lower bound for the exact ground-state energy.