Quantum NP - A Survey
Dorit Aharonov, Tomer Naveh
TL;DR
The paper surveys Kitaev’s foundational result that quantum analogues of NP, encapsulated by the class QMA, can be characterized via the Local Hamiltonian problem, a quantum generalization of SAT. It presents a detailed circuit-to-Hamiltonian construction showing that a quantum verifier’s computation can be encoded into a local Hamiltonian whose ground-state energy reflects acceptance, and proves that 5-local Hamiltonians are QMA-hard, with a complete framework built around history states and clock registers. The discussion highlights amplification, complexity relations to classical classes, and a suite of open questions including the possibility of lower locality sufficiency and a quantum analogue of the PCP theorem. Overall, the work establishes a deep connection between quantum computation, spectral properties of local Hamiltonians, and quantum complexity theory, shaping subsequent research in quantum verification and Hamiltonian complexity.
Abstract
We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3-SAT, namely local Hamiltonians, is QMA complete. The result builds upon the classical Cook-Levin proof of the NP completeness of SAT, but differs from it in several fundamental ways, which we highlight. This result raises a rich array of open problems related to quantum complexity, algorithms and entanglement, which we state at the end of this survey. This survey is the extension of lecture notes taken by Naveh for Aharonov's quantum computation course, held in Tel Aviv University, 2001.