The principle of majorization: application to random quantum circuits

TL;DR

This paper investigates the principle of majorization in various classes of random quantum circuits, demonstrating its potential as an indicator of quantum complexity and distinguishing between different computational classes.

Contribution

The study applies the principle of majorization to diverse quantum circuits, revealing its effectiveness in differentiating circuit complexity and universality.

Findings

All circuit families satisfy decreasing majorization on average.

Fluctuations in Lorenz curves distinguish universal from non-universal circuits.

Majorization serves as an indicator of quantum dynamics complexity.

Abstract

We test the principle of majorization [J. I. Latorre and M. A. Martin-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 going to infinity) behaves like a random vector. However, clear differences appear in the fluctuations of the Lorenz curves associated to asymptotic states. The fluctuations of the Lorenz curves discriminate between universal and non-universal classes of random quantum circuits, and they also detect the complexity of some non-universal but not classically efficiently simulatable quantum random circuits. We conclude that majorization can be used as a indicator of complexity of quantum dynamics, as an alternative to, e.g., entanglement spectrum and out-of-time-order correlators (OTOCs).