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

Adam Bouland, Chenyi Zhang, Zixin Zhou

TL;DR

This work shows it is cryptographically hard to determine if the ground state of a geometrically local, polynomially gapped Hamiltonian on qudits has near-area law vs near-volume law entanglement, and suggests that the problem of learning so-called"gapless"quantum phases of matter might 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.

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

TL;DR

This work shows it is cryptographically hard to determine if the ground state of a geometrically local, polynomially gapped Hamiltonian on qudits has near-area law vs near-volume law entanglement, and suggests that the problem of learning so-called"gapless"quantum phases of matter might 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 () 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.

Paper Structure

This paper contains 53 sections, 32 theorems, 119 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Proof sketch
  3. Discussion and open problems
  4. Outline of the paper
  5. Preliminaries
  6. Notation
  7. Independent hash functions
  8. Quantum Turing machine
  9. Entanglement measures and properties
  10. Continuity properties
  11. Entanglement entropy for phase states
  12. Cutwise orthogonality
  13. Local Hamiltonians
  14. Construction of lossy functions
  15. DCRA based lossy function
  16. ...and 38 more sections

Key Result

Theorem 1.1

It is hard for a classical computer to determine if a 1D local Hamiltonian on qudits has a ground state with 1D near area law vs. near volume law entanglement.

Figures (10)

  • Figure 1: The tree structure of $h(x)$
  • Figure 2: This example demonstrates how the final dilution step works: The space complexity is $15$, and the input track has $5$ non-blank cells (aka the input length is $5$). The dilution phase inserts $2$ blank cells between non-blank ones, making them evenly distributed.
  • Figure 3: The initial configuration of the $(k+2)$-track TM
  • Figure 4: The configuration of track 0 and track 1 on a chain with $n=3$ in the first five iterations of the head. The order is from top to bottom and then from left to right.
  • Figure 5: The configuration of track 0 to track 2 on a chain with $n=3$ in the fifth iteration of the first clock arrow. The order is from top to bottom and then from left to right.
  • ...and 5 more figures

Theorems & Definitions (68)

  • Theorem 1.1: Informal
  • Theorem 1.2: Informal
  • Definition 1: $r$-wise independent function family
  • Lemma 2.1
  • Definition 2: von Neumann entropy
  • Definition 3
  • Definition 4: Binary entropy function
  • Lemma 2.2: Continuity of the conditional von Neumann entropy winter2016
  • Definition 5: $T$-matrix associated with phase states, Definition 2.9 of bouland2024public
  • Lemma 2.3: Lemma 2.10 of bouland2024public
  • ...and 58 more