Local Hamiltonians with Approximation-Robust Entanglement

Quantum entanglement is considered, by and large, to be a very delicate and non-robust phenomenon that is very hard to maintain in the presence of noise, or non-zero temperatures. In recent years however, and motivated, in part, by a quest for a quantum analog of the PCP theorem researches have tried to establish whether or not we can preserve quantum entanglement at "constant" temperatures that are independent of system size. This would imply that any quantum state with energy at most, say 0.05 of the total available energy of the Hamiltonian, would be highly-entangled. To this date, no such systems were found, and moreover, it became evident that even embedding local Hamiltonians on robust, albeit "non-physical" topologies, namely expanders, does not guarantee entanglement robustness. In this study, we indicate that such robustness may be possible after all: We construct an infinite family of O(1)-local Hamiltonians, corresponding to check terms of a quantum error-correcting code with the following property of inapproximability: any quantum state with energy at most 0.05 w.r.t. the total available energy cannot be even approximately simulated by classical circuits of bounded (sub-logarithmic) depth. In a sense, this implies that even providing a "witness" to the fact that the local Hamiltonian can be "almost" satisfied, already requires some measure of long-range entanglement. Our construction is but a first step in what, we believe, is a whole range of possible entanglement - robust local Hamiltonians. A natural next step, we believe, is to devise such local Hamiltonians that resist approximation in terms of bounded-depth quantum circuits (e.g. NLTS), and even find such robust forms of entanglement that are useful for some computation.