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NLTS Hamiltonians from good quantum codes

Anurag Anshu, Nikolas P. Breuckmann, Chinmay Nirkhe

TL;DR

This work proves the No Low-Energy Trivial States (NLTS) conjecture by showing that constant-rate, linear-distance quantum LDPC codes yield NLTS Hamiltonians, meaning all low-energy states require nontrivial quantum circuit depth to prepare. The authors develop a clustering framework for approximate codewords and show that low-energy distributions in both X and Z bases are well-spread across disjoint clusters, which enforces nontrivial depth lower bounds. They further establish that good qLDPC codes satisfy the required clustering property by linking it to expansion properties of parity-check matrices, via balanced-product code constructions built from expander graphs and robust local codes. The results bridge quantum coding theory and circuit complexity, offering a constructive path toward NLTS and insights relevant to the quantum PCP program through explicit code families and expansion-based proofs.

Abstract

The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings [2014] posits that there exist families of Hamiltonians with all low energy states of non-trivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that the recently discovered families of constant-rate and linear-distance QLDPC codes correspond to NLTS local Hamiltonians.

NLTS Hamiltonians from good quantum codes

TL;DR

This work proves the No Low-Energy Trivial States (NLTS) conjecture by showing that constant-rate, linear-distance quantum LDPC codes yield NLTS Hamiltonians, meaning all low-energy states require nontrivial quantum circuit depth to prepare. The authors develop a clustering framework for approximate codewords and show that low-energy distributions in both X and Z bases are well-spread across disjoint clusters, which enforces nontrivial depth lower bounds. They further establish that good qLDPC codes satisfy the required clustering property by linking it to expansion properties of parity-check matrices, via balanced-product code constructions built from expander graphs and robust local codes. The results bridge quantum coding theory and circuit complexity, offering a constructive path toward NLTS and insights relevant to the quantum PCP program through explicit code families and expansion-based proofs.

Abstract

The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings [2014] posits that there exist families of Hamiltonians with all low energy states of non-trivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that the recently discovered families of constant-rate and linear-distance QLDPC codes correspond to NLTS local Hamiltonians.
Paper Structure (4 sections, 6 theorems, 27 equations)

This paper contains 4 sections, 6 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Proof of the NLTS theorem
  3. Proof that Property \ref{['property']} holds for families of good qLDPC codes
  4. Omitted Proofs

Key Result

Theorem 1

There exists a fixed constant $\epsilon > 0$ and an explicit family of $O(1)$-local frustration-free commuting Hamiltonians $\{\mathbf{H}^{(n)}\}_{n = 1}^\infty$ where $\mathbf{H}^{(n)} = \sum_{i = 1}^m h_i^{(n)}$ acts on $n$ particles and consists of $m = \Theta(n)$ local terms such that for any fa

Theorems & Definitions (10)

  • Theorem 1: No low-energy trivial states
  • Lemma 2
  • proof
  • Theorem 5: Formal statement of the NLTS theorem
  • Definition 6
  • Definition 7
  • Lemma 8
  • proof
  • Theorem 9
  • Theorem 10