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A recent quantum simulation of observables of the kicked Ising model on 127qubits [Nature 618, 500 (2023)] implemented circuits that exceed thecapabilities of exact classical simulation. We show that several approximateclassical methods, based on sparse Pauli dynamics and tensor networkalgorithms, can simulate these observables orders of magnitude faster than thequantum experiment, and can also be systematically converged beyond theexperimental accuracy. Our most accurate technique combines a mixedSchr\"odinger and Heisenberg tensor network representation with the Bethe freeentropy relation of belief propagation to compute expectation values with aneffective 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 identifyinaccuracies in the experimental extrapolations and suggest how futureexperiments can be implemented to increase the classical hardness.

Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance