Principle of majorization: Application to random quantum circuits
Raúl O. Vallejos, Fernando de Melo, Gabriel G. Carlo · Physical Review A · 2021
We test the principle of majorization J. I. Latorre and M. A. Mart\'{\i}n-Delgado, Phys. Rev. A 66, 022305 (2002). in random circuits. Three classes of circuits were considered: (i) universal, (ii) classically simulatable, and (iii) neither universal nor classically simulatable. The studied families are: {cnot, H, T}, {cnot, H, not}, {cnot, H, S} (Clifford), matchgates, and IQP (instantaneous quantum polynomial-time). We verified that all the families of circuits satisfy on average the principle of decreasing majorization. In most cases the asymptotic state (number of gates $\ensuremath{\rightarrow}\ensuremath{\infty}$) behaves like a random vector. However, clear differences appear in the fluctuations of the Lorenz curves associated with asymptotic states. The fluctuations of the Lorenz curves discriminate between universal and nonuniversal classes of random quantum circuits, and they also detect the complexity of some nonuniversal but not classically efficiently simulatable quantum random circuits. We conclude that majorization can be used as an indicator of complexity of quantum dynamics, as an alternative to, e.g., entanglement spectrum and out-of-time-order correlators.