On stability of k-local quantum phases of matter

The current theoretical framework for topological phases of matter is based on the thermodynamic limit of a system with geometrically local interactions. A natural question is to what extent the notion of a phase of matter remains well-defined if we relax the constraint of geometric locality, and replace it with a weaker graph-theoretic notion of k-locality. As a step towards answering this question, we analyze the stability of the energy gap to perturbations for Hamiltonians corresponding to general quantum low-density parity-check codes, extending work of Bravyi and Hastings [Commun. Math. Phys. 307, 609 (2011)]. A corollary of our main result is that if there exist constants ε1,ε2>0 such that the size Γ(r) of balls of radius r on the interaction graph satisfy Γ(r)=O(exp(r1−ε1)) and the local ground states of balls of radius r≤ρ∗=O(log(n)1+ε2) are locally indistinguishable, then the energy gap of the associated Hamiltonian is stable against local perturbations. This gives an almost exponential improvement over the D-dimensional Euclidean case, which requires Γ(r)=O(rD) and ρ∗=O(nα) for some α>0. The approach we follow falls just short of proving stability of finite-rate qLDPC codes, which have ε1=0; we discuss some strategies to extend the result to these cases. We discuss implications for the third law of thermodynamics, as k-local Hamiltonians can have extensive zero-temperature entropy.