Commuting Local Hamiltonian Problem on 2D beyond qubits

We study the complexity of local Hamiltonians in which the terms pairwise commute. Commuting local Hamiltonians (CLHs) provide a way to study the role of non-commutativity in the complexity of quantum systems and touch on many fundamental aspects of quantum computing and many-body systems, such as the quantum PCP conjecture and the area law. Despite intense research activity since Bravyi and Vyalyi introduced the CLH problem two decades ago [BV03], its complexity remains largely unresolved; it is only known to lie in NP for a few special cases. Much of the recent research has focused on the physically motivated 2D case, where particles are located on vertices of a 2D grid and each term acts non-trivially only on the particles on a single square (or plaquette) in the lattice. In particular, Schuch [Sch11] showed that the CLH problem on 2D with qubits is in NP. Aharonov, Kenneth and Vigdorovich~[AKV18] then gave a constructive version of this result, showing an explicit algorithm to construct a ground state. Resolving the complexity of the 2D CLH problem with higher dimensional particles has been elusive. We prove two results for the CLH problem in 2D: (1) We give a non-constructive proof that the CLH problem in 2D with qutrits is in NP. As far as we know, this is the first result for the commuting local Hamiltonian problem on 2D beyond qubits. Our key lemma works for general qudits and might give new insights for tackling the general case. (2) We consider the factorized case, also studied in [BV03], where each term is a tensor product of single-particle Hermitian operators. We show that a factorized CLH in 2D, even on particles of arbitrary finite dimension, is equivalent to a direct sum of qubit stabilizer Hamiltonians. This implies that the factorized 2D CLH problem is in NP. This class of CLHs contains the Toric code as an example.