Commuting Local Hamiltonian Problem on 2D beyond qubits

• DOI: 10.48550/arxiv.2309.04910
• Publication Date: 2023-01-01
• Authors: Sandy Irani; Jiaqing Jiang
• Keywords: Hamiltonian (control theory); Constructive; Lemma (botany)

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