  It is believed that random quantum circuits are difficult to simulate
classically. These have been used to demonstrate quantum supremacy: the
execution of a computational task on a quantum computer that is infeasible for
any classical computer. The task underlying the assertion of quantum supremacy
by Arute et al. (Nature, 574, 505--510 (2019)) was initially estimated to
require Summit, the world's most powerful supercomputer today, approximately
10,000 years. The same task was performed on the Sycamore quantum processor in
only 200 seconds.
  In this work, we present a tensor network-based classical simulation
algorithm. Using a Summit-comparable cluster, we estimate that our simulator
can perform this task in less than 20 days. On moderately-sized instances, we
reduce the runtime from years to minutes, running several times faster than
Sycamore itself. These estimates are based on explicit simulations of parallel
subtasks, and leave no room for hidden costs. The simulator's key ingredient is
identifying and optimizing the "stem" of the computation: a sequence of
pairwise tensor contractions that dominates the computational cost. This
orders-of-magnitude reduction in classical simulation time, together with
proposals for further significant improvements, indicates that achieving
quantum supremacy may require a period of continuing quantum hardware
developments without an unequivocal first demonstration.
