Quantum Hamiltonian Complexity

TL;DR

This survey introduces Quantum Hamiltonian Complexity (QHC), linking local quantum constraints to computational problems and situating ground-state computation as a central complexity task. It synthesizes quantum information tools (density matrices, tensor networks, area laws) with computer-science results (QMA-completeness of LH, Quantum SAT, and structure lemmas for commuting Hamiltonians), emphasizing 1D area laws and the role of tensor-network representations (MPS, MERA) in both physics and algorithms. Key results highlighted include Kitaev’s LH framework, Bravyi’s 2-SAT in P, and 1D area laws with AGSP-based proofs, alongside a discussion of Hubbard-type models and quantum simulations. The paper also surveys recent developments, challenges (e.g., quantum PCP), and the rich interplay between physics-inspired techniques and computational complexity in higher dimensions. Overall, it provides a bridge between physical intuition and algorithmic hardness in local quantum constraint systems, with practical implications for understanding entanglement, simulation, and the limits of classical descriptions of quantum systems.

Abstract

Constraint satisfaction problems are a central pillar of modern computational complexity theory. This survey provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity, which includes the study of quantum constraint satisfaction problems. Over the past decade and a half, this field has witnessed fundamental breakthroughs, ranging from the establishment of a "Quantum Cook-Levin Theorem" to deep insights into the structure of 1D low-temperature quantum systems via so-called area laws. Our aim here is to provide a computer science-oriented introduction to the subject in order to help bridge the language barrier between computer scientists and physicists in the field. As such, we include the following in this survey: (1) The motivations and history of the field, (2) a glossary of condensed matter physics terms explained in computer-science friendly language, (3) overviews of central ideas from condensed matter physics, such as indistinguishable particles, mean field theory, tensor networks, and area laws, and (4) brief expositions of selected computer science-based results in the area. For example, as part of the latter, we provide a novel information theoretic presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.

Figures (7)

• Figure 1: (a) A single tensor $M(i_1,i_2,i_3)$. (b) Two tensors $M(i_1,i_2,i_3)$ and $N(j_1,j_2,j_3)$ contracted on the edge $(M,N)$.
• Figure 2: Five tensor networks, studied in Question \ref{['q:tens1']}.
• Figure 3: Demonstrating the linear map view of tensor networks.
• Figure 4: A MERA network on $8$ sites. The circle vertices represent "disentangling" unitaries. The square vertices represent isometries. (a) The tensor network view. (b) The quantum circuit view. (c) The causal cone of the site labeled $s$.
• Figure 5: A quantum lattice system partitioned into two parts $L$ and $\overline L$. If the represented quantum state obeys an area law, the amount of entanglement between $L$ and $\overline L$ is bounded above by a quantity that is proportional to the surface area $|\partial L|$, rather than the volume $|L|$.

Theorems & Definitions (17)

• Definition 2.1: QMA
• Theorem 7.1
• Lemma 7.2: Kitaev KSV02, Geometric Lemma, Lemma 14.4
• Theorem 7.3
• Lemma 7.4: Structure Lemma BV05
• Lemma 7.5
• Lemma 7.6
• Corollary 7.7
• Lemma 7.8: KLV00BV05
• Definition 7.1: Quantum $2$-SAT ($2$-QSAT) B06