  Towards the efficient simulation of near-term quantum devices using tensor
network states, we introduce an improved real-space parallelizable
matrix-product state (MPS) compression method. This method enables efficient
compression of all virtual bonds in constant time, irrespective of the system
size, with controlled accuracy, while it maintains the stability of the
wavefunction norm without necessitating sequential renormalization procedures.
In addition, we introduce a parallel regauging technique to partially restore
the deviated canonical form, thereby improving the accuracy of the simulation
in subsequent steps. We further apply this method to simulate unitary quantum
dynamics and introduce a parallel time-evolving block-decimation (pTEBD)
algorithm. We employ the pTEBD algorithm for extensive simulations of typical
one- and two-dimensional quantum circuits, involving over 1000 qubits. The
obtained numerical results unequivocally demonstrate that the pTEBD algorithm
achieves the same level of simulation precision as the current state-of-the-art
MPS algorithm but in polynomially shorter time, exhibiting nearly perfect weak
scaling performance on a modern supercomputer.
