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.

翻译：刻画局域哈密顿量基态的纠缠结构是量子信息中的一个基本问题。本文研究在给定哈密顿量的情况下，该问题的计算复杂性。我们的主要结果表明，对于量子比特（$d=O(1)$）上的几何局域、多项式能隙哈密顿量，判断其基态具有近面积律还是近体积律纠缠在密码学上是困难的。这改进了 Bouland 等人（arXiv:2311.12017）先前针对非几何局域哈密顿量的结果。具体而言，我们证明该问题在一维情况下大致是因数分解困难的，在二维情况下是 LWE 困难的。我们的证明通过构造一种新颖的、具有高度空间效率的公钥伪纠缠形式，并将其与 Gottesman 和 Irani 的量子图灵机到哈密顿量构造方法的改进相结合来实现。我们的工作表明，学习所谓“无能隙”量子物态相的问题可能是难以处理的。