  A recent quantum simulation of observables of the kicked Ising model on 127
qubits [Nature 618, 500 (2023)] 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\"odinger 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.

Quantum Physics