Chaos and complexity by design

TL;DR

This paper establishes a quantitative connection between quantum chaos and pseudorandomness by linking out-of-time-order correlators, frame potential, and circuit complexity, revealing how chaos relates to randomness and complexity in quantum systems.

Contribution

It introduces a framework connecting chaos and pseudorandomness through generalized out-of-time-order correlators and frame potential, and derives bounds on quantum circuit complexity.

Findings

The norm squared of generalized out-of-time-order correlators is proportional to the frame potential.

These correlators determine the k-fold channel of unitary ensembles.

A lower bound on quantum circuit complexity is derived from the frame potential.

Abstract

We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the "frame potential," which is minimized by unitary $k$-designs and measures the $2$-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order $2k$-point correlators is proportional to the $k$th frame potential, providing a quantitative connection between chaos and pseudorandomness. Additionally, we prove that these $2k$-point correlators for Pauli operators completely determine the $k$-fold channel of an ensemble of unitary operators. Finally, we use a counting argument to obtain a lower bound on the quantum circuit complexity in terms of the frame potential. This provides a direct link between chaos, complexity, and randomness.