Hamiltonian complexity

TL;DR

This paper surveys the intersection of Hamiltonian physics and computational complexity, introducing the simulation problem for families of quantum Hamiltonians and observables. It highlights key hardness results, including QMA-completeness for ground-state energies and NP-hardness in classical analogs, as well as regimes where efficient simulation is possible via Lieb-Robinson bounds and low-dimensional variational classes like MPS/DMRG. The quantum PCP conjecture is presented as a central guiding problem, with partial progress through gap amplification techniques. The discussion also covers area laws, entanglement structure, and the outlook for extending these ideas to continuous degrees of freedom.

Abstract

In recent years we've seen the birth of a new field known as hamiltonian complexity lying at the crossroads between computer science and theoretical physics. Hamiltonian complexity is directly concerned with the question: how hard is it to simulate a physical system? Here I review the foundational results, guiding problems, and future directions of this emergent field.