  Near term quantum computers with a high quantity (around 50) and quality
(around 0.995 fidelity for two-qubit gates) of qubits will approximately sample
from certain probability distributions beyond the capabilities of known
classical algorithms on state-of-the-art computers, achieving the first
milestone of so-called quantum supremacy. This has stimulated recent progress
in classical algorithms to simulate quantum circuits. Classical simulations are
also necessary to approximate the fidelity of multiqubit quantum computers
using cross entropy benchmarking. Here we present numerical results of a
classical simulation algorithm to sample universal random circuits, on a single
workstation, with more qubits and depth than previously reported. For example,
circuits with $5 \times 9$ qubits of depth 37, $7 \times 8$ qubits of depth 27,
and $10 \times (\kappa > 10)$) qubits of depth 19 are all easy to sample. We
also show up to what depth the sampling, or estimation of observables, is
trivially parallelizable. The algorithm is related to the "Feynmann path"
method to simulate quantum circuits. For low-depth circuits, the algorithm
scales exponentially in the depth times the smaller lateral dimension, or the
treewidth, as explained in Boixo et. al., and therefore confirms the bounds in
that paper. In particular, circuits with $7 \times 7$ qubits and depth 40
remain currently out of reach. Follow up work on a supercomputer environment
will tighten this bound.
