Hamiltonian Complexity and QMA-Completeness

One of the cornerstone results in computational complexity theory is the Cook-Levin Theorem [1]: the theorem that establishes that Boolean satisfiability problems are NP-complete. A Boolean satisfiability problem (SAT) asks the following question: given a set of clauses built from literals (variables that take values “TRUE” and “FALSE”), Boolean operators and parentheses, does there exist an assignment of values for the literals such that all clauses evaluate to TRUE? The Cook-Levin theorem demonstrates, in some sense, that satisfiability problems are the stereotypical set of NP problems. Specifically, SAT is a problem in NP and there is at most a polynomial overhead in reduction of other languages in NP to a SAT problem. It is also worth pursuing what restrictions can be made on the clauses such that this property of NP-completeness remains; for example, if each clause contains at most k literals (“k-SAT”). It was shown by Karp [2] that 3-SAT is NP-complete, but 2-SAT was shown by Krom [3] to be in P. This sharp jump in problem complexity between 2-SAT and 3-SAT was generalised in a pleasing result known as Schaefer’s dichotomy theorem [4], which demonstrates this step-change in complexity when the constraints imposed are drawn from a particular set. This narrative plays out in a surprisingly similar way when the complexity classes under consideration are related to the power of quantum computers rather than classical ones. The quantum equivalent of NP is known as QMA (an initialism of “Quantum Merlin-Arthur”, a name inspired by the “call and response” type protocol that problems in this class are solved by). In this class, a sender with unbounded computational resources (Merlin) gives a poly-sized proof of an affirmative answer to a decision problem to a receiver (Arthur). Arthur’s verification protocol must then lie in BQP; that is, he must be able to verify or reject Merlin’s proof with a quantum computer with polynomial resources in polynomial time. One may then ask whether there exists any problem that is QMAcomplete, and whether these are natural quantum equivalents of problems that are known to be NP-complete. Interestingly, in the same sense that a computational problem in NP can be encoded as a satisfiability problem for a set of constraints, a computational problem in QMA can be encoded as the problem of finding a ground-state of some local Hamiltonian given a set of constraints. This problem, k-LocalHamiltonian, is the quantum analogue of the Boolean satisfiability problems introduced above (in particular, the analogue of k-MaxCSP, the problem of identifying the maximum number of contraints satisified when clauses have at most k literals in a general constraint satisfaction problem). The k-LocalHamiltonian problem was initially introduced by Kitaev [5]. In direct analogy with k-MaxCSP, which is NP-complete for k ≥ 2, it can be shown that k-LocalHamiltonian is QMA-complete for k ≥ 2 [6]. In addition to the constraint of locality, it may be of interest to consider constraints on the Hamiltonian based on the physical quantum system it describes. This is of great importance for adiabatic quantum computers, as implementation of any QMA-complete Hamiltonian is sufficient for universal adiabatic quantum computing [7]. In [7], it was demonstrated that the Ising model can be rendered QMA-complete with appropriate additional couplings, and in [8] it was shown that the Heisenberg and Hubbard models are QMA-complete in the case of physically local couplings. In a manner analogous to Schaefer’s dichotomy theorem, these constraints have been recently generalised in [9] to give a dichotomy in computational complexity for physically realisable local Hamiltonians. The structure of this essay is to expand on this narrative; first, by giving a concise definition of both QMA and the k-LocalHamiltonian problem and then by considering Schaefer’s dichotomy theorem and its quantum equivalent. Finally, the relevance of this work to adiabatic quantum computing will be outlined.