Scalable semidefinite programming approach to variational embedding for quantum many-body problems

TL;DR

This paper introduces a scalable semidefinite programming method for variational embedding in quantum many-body problems, improving computational efficiency and leveraging system symmetries.

Contribution

It develops an efficient, parallelizable algorithm for quantum embedding SDPs that exploits translation invariance to reduce complexity.

Findings

Algorithm achieves better scalability for large systems.

Parallel local updates improve computational efficiency.

Exploiting translation invariance reduces problem complexity.

Abstract

In quantum embedding theories, a quantum many-body system is divided into localized clusters of sites which are treated with an accurate `high-level' theory and glued together self-consistently by a less accurate `low-level' theory at the global scale. The recently introduced variational embedding approach for quantum many-body problems combines the insights of semidefinite relaxation and quantum embedding theory to provide a lower bound on the ground-state energy that improves as the cluster size is increased. The variational embedding method is formulated as a semidefinite program (SDP), which can suffer from poor computational scaling when treated with black-box solvers. We exploit the interpretation of this SDP as an embedding method to develop an algorithm which alternates parallelizable local updates of the high-level quantities with updates that enforce the low-level global constraints. Moreover, we show how translation invariance in lattice systems can be exploited to reduce the complexity of projecting a key matrix to the positive semidefinite cone.