  A recent quantum simulation of observables of the kicked Ising model on 127
qubits implemented circuits that exceed the capabilities of exact classical
simulation. We show that several approximate classical methods, based on sparse
Pauli dynamics and tensor network algorithms, can simulate these observables
orders of magnitude faster than the quantum experiment, and can also be
systematically converged beyond the experimental accuracy. Our most accurate
technique combines a mixed Schr\"{o}dinger and Heisenberg tensor network
representation with the Bethe free entropy relation of belief propagation to
compute expectation values with an effective wavefunction-operator sandwich
bond dimension >16,000,000, achieving an absolute accuracy, without
extrapolation, in the observables of <0.01, which is converged for many
practical purposes. We thereby identify inaccuracies in the experimental
extrapolations and suggest how future experiments can be implemented to
increase the classical hardness.
