Demonstrating a quantum computational speed-up is a crucial milestone for near-term quantum technology. Recently, sampling protocols for quantum simulators have been proposed that have the potential to show such a quantum advantage, based on commonly made assumptions. The key challenge in the theoretical analysis of this scheme—as of other comparable schemes such as boson sampling—is to lessen the assumptions and close the theoretical loopholes, replacing them by rigorous arguments. In this work, we prove two open conjectures for a simple sampling protocol that is based on the continuous time evolution of a translation-invariant Ising Hamiltonian: anticoncentration of the generated probability distributions and average-case hardness of exactly evaluating those probabilities. The latter is proven building upon recently developed techniques for random circuit sampling. For the former, we exploit the insight that approximate 2-designs for the unitary group admit anticoncentration. We then develop new techniques to prove that the 2D time evolution of the protocol gives rise to approximate 2-designs. Our work provides the strongest theoretical evidence to date that Hamiltonian quantum simulators are classically intractable.
