  In this work, we present an efficient rank-compression approach for the
classical simulation of Kraus decoherence channels in noisy quantum circuits.
The approximation is achieved through iterative compression of the density
matrix based on its leading eigenbasis during each simulation step without the
need to store, manipulate, or diagonalize the full matrix. We implement this
algorithm in an in-house simulator, and show that the low rank algorithm speeds
up simulations by more than two orders of magnitude over an existing
implementation of full rank simulator, and with negligible error in the target
noise and final observables. Finally, we demonstrate the utility of the low
rank method as applied to representative problems of interest by using the
algorithm to speed-up noisy simulations of Grover's search algorithm and
quantum chemistry solvers.
