Local Hamiltonians Whose Ground States are Hard to Approximate

TL;DR

This paper constructs 16-local Hamiltonians with ground states where even a small fraction of qubits must be highly entangled, providing evidence for the robustness of quantum entanglement and supporting key conjectures in quantum complexity.

Contribution

It introduces a family of local Hamiltonians with ground states that are globally entangled despite local marginals, advancing understanding of entanglement robustness and conjectures like qLDPC and NLTS.

Findings

Ground states require high entanglement even in small qubit subsets

Provides evidence supporting the qLDPC and NLTS conjectures

Introduces a new lower bound on vertex expansion of low-depth quantum circuits

Abstract

Ground states of local Hamiltonians can be generally highly entangled: any quantum circuit that generates them (even approximately) must be sufficiently deep to allow coupling (entanglement) between any pair of qubits. Until now this property was not known to be "robust" - the marginals of such states to a subset of the qubits containing all but a small constant fraction of them may be only locally entangled, and hence approximable by shallow quantum circuits. In this work we construct a family of 16-local Hamiltonians for which any 1-10^{-9} fraction of qubits of any ground state must be highly entangled. This provides evidence that quantum entanglement is not very fragile, and perhaps our intuition about its instability is an artifact of considering local Hamiltonians which are not only local but spatially local. Formally, it provides positive evidence for two wide-open conjectures in condensed-matter physics and quantum complexity theory which are the qLDPC conjecture, positing the existence of "good" quantum LDPC codes, and the NLTS conjecture due to Freedman and Hastings positing the existence of local Hamiltonians in which any low-energy state is highly-entangled. Our Hamiltonian is based on applying the hypergraph product by Tillich and Zemor to a classical locally testable code. A key tool in our proof is a new lower bound on the vertex expansion of the output of low-depth quantum circuits, which may be of independent interest.