  The ability to efficiently simulate random quantum circuits using a classical
computer is increasingly important for developing Noisy Intermediate-Scale
Quantum devices. Here we present a tensor network states based algorithm
specifically designed to compute amplitudes for random quantum circuits with
arbitrary geometry. Singular value decomposition based compression together
with a two-sided circuit evolution algorithm are used to further compress the
resulting tensor network. To further accelerate the simulation, we also propose
a heuristic algorithm to compute the optimal tensor contraction path. We
demonstrate that our algorithm is up to $2$ orders of magnitudes faster than
the Sch$\ddot{\text{o}}$dinger-Feynman algorithm for verifying random quantum
circuits on the $53$-qubit Sycamore processor, with circuit depths below $12$.
We also simulate larger random quantum circuits up to $104$ qubits, showing
that this algorithm is an ideal tool to verify relatively shallow quantum
circuits on near-term quantum computers.
