  Random circuit sampling has become a popular means for demonstrating the
superiority of quantum computers over classical supercomputers. While quantum
chips are evolving rapidly, classical sampling algorithms are also getting
better and better. The major challenge is to generate bitstrings exhibiting an
XEB fidelity above that of the quantum chips. Here we present a classical
sampling algorithm for producing the probability distribution of any given
random quantum circuit, where the fidelity can be rigorously bounded.
Specifically, our algorithm performs rejection sampling after the introduced
very recently multi-tensor contraction algorithm. We show that the fidelity can
be controlled by partially contracting the dominant paths in the tensor network
and by adjusting the number of batches used in the rejection sampling. As a
demonstration, we classically produced 1 million samples with the fidelity
bounded by 0.2%, based on the 20-cycle circuit of the Sycamore 53-qubit quantum
chip. Though this task was initially estimated to take 10,000 years on the
Summit supercomputer, it took about 14.5 days using our algorithm on a
relatively small cluster with 32 GPUs (Tesla V100 16GB). Furthermore, we
estimate that for the Zuchongzhi 56-qubit 20-cycle circuit one can produce 1M
samples with fidelity 0.066% using the Selene supercomputer with 4480 GPUs
(Tesla A100 80GB) in about 4 days.
