In a belated revival of the Ghost Pontiff’s “Happy Paper Dance” ritual, I’d like to talk about the recent paper The k-local Pauli Commuting Hamiltonians Problem is in P by my student Jijiang (Johnny) Yan and his former advisor, Dave Bacon. The abstract is:
Given a Hamiltonian that is a sum of commuting few-body terms, the commuting Hamiltonian problem is to determine if there exists a quantum state that is the simultaneous eigenstate of all of these terms that minimizes each term individually. This problem is known to be in the complexity class quantum Merlin-Arthur, but is widely thought to not be complete for this class. Here we show that a limited form of this problem when the individual terms are all made up of tensor products of Pauli matrices is efficiently solvable on a classical computer and thus in the complexity class P. The problem can be thought of as the classical XOR-SAT problem over a symplectic vector space. This class of problems includes instance Hamiltonians whose ground states possess topological entanglement, thus showing that such entanglement is not always a barrier for the more general problem.
This result follows a long string of papers that discuss the complexity of finding the ground state energy of k-local Hamiltonians, usually modified by various adjectives like “commuting” or “frustration-free” or “Pauli” or “in d dimensions.” Typically, these problems are shown to be somewhere between NP and QMA, and the subtle differences between these relate to issues such as topological order and the quantum PCP conjecture. In fact, one specific inspiration for this paper was 1102.0770, which showed that 3-qubit (or even 3-qutrit) commuting Hamiltonians could not have topologically-ordered ground states, while 4-qubit commuting Hamiltonians include the toric code, and 2-qubit non-commuting Hamiltonians include things that look like the toric code.
This paper shows that, in the case of commuting Pauli Hamiltonians, the difference between 3-local and 4-local is not important from a complexity point of view; indeed, it is possible to efficiently find the ground state of even -local Hamiltonians.
At first this is shocking, but to see why it’s reasonable to expect this result, consider classical (commuting, Pauli) Hamiltonians. Determining whether these terms has a simultaneous ground state is equivalent to solving a linear system of equations over , which of course can be done in poly-time. This paper extends that to general Paulis, but the algorithm still involves solving linear systems of equations–this time over . It is one of my favorite examples of the power and simplicity of the Pauli matrices, tied perhaps with the elegant Wehner-Winter uncertainty relations for anti-commuting observables.