We consider the problem of estimating the maximal energy of quantum p-local
spin glass random Hamiltonians, the quantum analogues of widely studied
classical spin glass models. Denoting by Eβ(p) the (appropriately
normalized) maximal energy in the limit of a large number of qubits n, we
show that Eβ(p) approaches 2log6β as p increases. This value is
interpreted as the maximal energy of a much simpler so-called Random Energy
Model, widely studied in the setting of classical spin glasses.
Our most notable and (arguably) surprising result proves the existence of
near-maximal energy states which are product states, and thus not entangled.
Specifically, we prove that with high probability as nββ, for any
E<Eβ(p) there exists a product state with energy β₯E at sufficiently
large constant p. Even more surprisingly, this remains true even when
restricting to tensor products of Pauli eigenstates. Our approximations go
beyond what is known from monogamy-of-entanglement style arguments -- the best
of which, in this normalization, achieve approximation error growing with n.
Our results not only challenge prevailing beliefs in physics that extremely
low-temperature states of random local Hamiltonians should exhibit
non-negligible entanglement, but they also imply that classical algorithms can
be just as effective as quantum algorithms in optimizing Hamiltonians with
large locality -- though performing such optimization is still likely a hard
problem.
Our results are robust with respect to the choice of the randomness
(disorder) and apply to the case of sparse random Hamiltonian using Lindeberg's
interpolation method. The proof of the main result is obtained by estimating
the expected trace of the associated partition function, and then matching its
asymptotics with the extremal energy of product states using the second moment
method.Comment: Added a disclaimer about error in current draf