  Simulating noisy quantum circuits is vital in designing and verifying quantum
algorithms in the current NISQ (Noisy Intermediate-Scale Quantum) era, where
quantum noise is unavoidable. However, it is much more inefficient than the
classical counterpart because of the quantum state explosion problem (the
dimension of state space is exponential in the number of qubits) and the
complex (non-unitary) representation of noises. Consequently, only noisy
circuits with up to about 50 qubits can be simulated approximately well. This
paper introduces a novel approximation algorithm for simulating noisy quantum
circuits when the noisy effectiveness is insignificant to improve the
scalability of the circuits that can be simulated. The algorithm is based on a
new tensor network diagram for the noisy simulation and uses the singular value
decomposition to approximate the tensors of quantum noises in the diagram. The
contraction of the tensor network diagram is implemented on Google's
TensorNetwork. The effectiveness and utility of the algorithm are demonstrated
by experimenting on a series of practical quantum circuits with realistic
superconducting noise models. As a result, our algorithm can approximately
simulate quantum circuits with up to 225 qubits and 20 noises (within about 1.8
hours). In particular, our method offers a speedup over the commonly-used
approximation (sampling) algorithm -- quantum trajectories method. Furthermore,
our approach can significantly reduce the number of samples in the quantum
trajectories method when the noise rate is small enough.
