Entropy Constraints for Ground Energy Optimization

Abstract

We study the use of von Neumann entropy constraints for obtaining lower bounds on the ground energy of quantum many-body systems. Known methods for obtaining certificates on the ground energy typically use consistency of local observables and are expressed as semidefinite programming relaxations. The local marginals defined by such a relaxation do not necessarily satisfy entropy inequalities that follow from the existence of a global state. Here, we propose to add such entropy constraints that lead to tighter convex relaxations for the ground energy problem. We give analytical and numerical results illustrating the advantages of such entropy constraints. We also show limitations of the entropy constraints we construct: they are implied by doubling the number of sites in the relaxation and as a result they can at best lead to a quadratic improvement in terms of the matrix sizes of the variables. We explain the relation to a method for approximating the free energy known as the Markov Entropy Decomposition method.

References in corpus (20)

• An Area Law for One Dimensional Quantum Systems
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Matrix product states represent ground states faithfully
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• Quantum Hamiltonian Complexity
• Convergent relaxations of polynomial optimization problems with non-commuting variables
• Semidefinite approximations of the matrix logarithm
• Variational Benchmarks for Quantum Many-Body Problems
• Markov entropy decomposition: a variational dual for quantum belief propagation
• A new inequality for the von Neumann entropy
• Quantum Approximate Markov Chains are Thermal
• Solving condensed-matter ground-state problems by semidefinite relaxations
• Lower Bounds for Ground States of Condensed Matter Systems
• Product-state Approximations to Quantum Ground States
• On an Extension Problem for Density Matrices
• An Improved Approximation Algorithm for Quantum Max-Cut
• Beyond product state approximations for a quantum analogue of Max Cut
• Lower bounds to variational problems with guarantees
• Entropic bounds for the quantum marginal problem
• Optimizing quantum circuit parameters via SDP

Cited by in corpus (5)

• Certified algorithms for equilibrium states of local quantum Hamiltonians
• QSlack: A slack-variable approach for variational quantum semi-definite programming
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• On the set of reduced states of translation invariant, infinite quantum systems
• Bootstrap Method in Theoretical Physics