Advantage of utilizing nonlocal magic resource in Haar-random circuits

Abstract: In design and simulation of quantum circuits with multiple units, the computational ability is greatly limited by quickly increasing entanglement, and the ordinary sampling methods normally exhibit low efficiency. Herein, we uncover an intrinsic scaling law of the nonlocal magic resource and the bond dimension of matrix product states in Haar-random quantum circuits, that is, the nonlocal magic resource is converged on a bond dimension in logarithmic scale with the system size. It means, in the practical simulations of quantum circuits, merely small bond dimension suffices to bear with the dynamics of stabilizer R\'{e}nyi entropy with rank $n=1,2$. On the other hand, the entanglement converges on a bond dimension exponentially scaled in the system size. This remarkable difference reveals that, while the simulation of entanglement on a classical computer is limited, the utilization of nonlocal magic resource as a characterization could make the simulation power much stronger. Furthermore, the intrinsic scaling enables an information separation between the nonlocal magic resource and the extra entanglement, achieving an indication that it is inappropriate to regard the entanglement as the driving force of the growth and spreading of nonlocal magic resource.