Quantum Hamiltonian complexity and the detectability lemma

TL;DR

The paper introduces the Detectability Lemma (DL), a simple, combinatorial tool for analyzing local Hamiltonians that applies to frustration-free, gapped systems with ground energy $0$ and gap $\epsilon>0$. By constructing a local approximation to the ground-state projector through layered projections and causality cones, the authors derive a streamlined 1D area law and a concise, one-page proof of exponential decay of correlations, avoiding time-dependent Lieb-Robinson techniques. The DL is shown to be broadly applicable, offering a local alternative to LR arguments in several contexts and serving as a foundational tool for quantum Hamiltonian complexity, including potential routes toward quantum PCP and higher-dimensional area laws. The results underscore the value of combinatorial, locality-based methods for understanding entanglement and correlations in quantum many-body systems, and open avenues for connections to MAP (method of alternating projections) and measurement-based algorithms.

Abstract

Quantum Hamiltonian complexity studies computational complexity aspects of local Hamiltonians and ground states; these questions can be viewed as generalizations of classical computational complexity problems related to local constraint satisfaction (such as SAT), with the additional ingredient of multi-particle entanglement. This additional ingredient of course makes generalizations of celebrated theorems such as the PCP theorem from classical to the quantum domain highly non-trivial; it also raises entirely new questions such as bounds on entanglement and correlations in ground states, and in particular area laws. We propose a simple combinatorial tool that helps to handle such questions: it is a simplified, yet more general version of the detectability lemma introduced by us in the more restricted context on quantum gap amplification a year ago. Here, we argue that this lemma is applicable in much more general contexts. We use it to provide a simplified and more combinatorial proof of Hastings' 1D area law, together with a less than 1 page proof of the decay of correlations in gapped local Hamiltonian systems in any constant dimension. We explain how the detectability lemma can replace the Lieb-Robinson bound in various other contexts, and argue that it constitutes a basic tool for the study of local Hamiltonians and their ground states in relation to various questions in quantum Hamiltonian complexity.

Figures (2)

• Figure 1: Am illustration of a 1D system of two-local, nearest neighbors, interactions. The local terms ($H_1, H_2, \ldots$) can be arranged in two layers (even and odd), such that the terms in each layer do not overlap.
• Figure 2: An illustration of the expression $A^\ell B{ |{\Omega} \rangle }$ in a 1D system. $B$ is a local perturbation, applied to the ground state ${ |{\Omega} \rangle }$. The local terms underneath it correspond to the $P_i$ projections in $A^\ell$. The pink terms are the projections inside the causality cone of $B$. These terms are graph connected to $B$, and generally do not commute with it. The blue terms are outside the causality cone, and can therefore commute with $B$ and be absorbed by ${ |{\Omega} \rangle }$.

Theorems & Definitions (12)

• Lemma 1.1: Detectability Lemma (DL) in $1D$
• Lemma 3.1
• Lemma 3.2: The detectability lemma
• Theorem 4.1: Lieb-Robinson bound (LR bound) , adapted ref:Has10
• Theorem 5.1: Area Law for frustration free Hamiltonians in $1D$
• Lemma 5.2: Constant overlap with a product state
• Lemma 5.3: Constant overlap with a product state implies finite entropy
• Lemma 5.4
• Lemma 5.5: Existence of a distinguishing measurement
• Lemma 5.6: Distinguishing measurement implies large difference in entropies