On the hardness of learning ground state entanglement of geometrically local Hamiltonians

TL;DR

This paper proves that determining the entanglement structure of ground states in local Hamiltonians is computationally hard, with complexity linked to cryptographic problems, especially for geometrically local Hamiltonians in 1D and 2D.

Contribution

It establishes the cryptographic hardness of classifying ground state entanglement in local Hamiltonians, extending prior results to geometrically local cases and introducing novel pseudo-entanglement constructions.

Findings

Hardness is roughly factoring-hard in 1D.

Hardness is LWE-hard in 2D.

Ground state entanglement classification may be intractable.

Abstract

Characterizing the entanglement structure of ground states of local Hamiltonians is a fundamental problem in quantum information. In this work we study the computational complexity of this problem, given the Hamiltonian as input. Our main result is that to show it is cryptographically hard to determine if the ground state of a geometrically local, polynomially gapped Hamiltonian on qudits ($d=O(1)$) has near-area law vs near-volume law entanglement. This improves prior work of Bouland et al. (arXiv:2311.12017) showing this for non-geometrically local Hamiltonians. In particular we show this problem is roughly factoring-hard in 1D, and LWE-hard in 2D. Our proof works by constructing a novel form of public-key pseudo-entanglement which is highly space-efficient, and combining this with a modification of Gottesman and Irani's quantum Turing machine to Hamiltonian construction. Our work suggests that the problem of learning so-called "gapless" quantum phases of matter might be intractable.