Free Independence and Unitary Design from Random Matrix Product Unitaries

TL;DR

This paper demonstrates that random matrix product unitaries can approximate Haar randomness for certain observables, revealing insights into quantum chaos, operator independence, and the design of quantum circuits.

Contribution

It establishes that matrix product unitaries can serve as approximate unitary designs and exhibit freeness properties, connecting quantum chaos, operator independence, and design theory.

Findings

Reproduces Haar values of higher-order OTOCs with polynomial bond dimension.

Local observables thermalize in chaotic systems, requiring only polynomial resources.

Global observables approach Haar randomness with polynomial deviations, indicating approximate design properties.

Abstract

Unitary randomness underpins both fundamental tasks in quantum information and the modern theory of quantum chaos. On one side, a central concept is that of approximate unitary designs: circuits that look random according to small moments and for forward-in-time protocols. In a distinct setting, out-of-time-ordered correlators (OTOCs), intensely studied as a measure of information scrambling, have recently been shown to probe freeness between Heisenberg operators, the noncommutative generalization of statistical independence. Bridging these two concepts, we study the emergence of freeness in a random matrix product unitary ensemble. We prove that, with only polynomial bond dimension, these unitaries reproduce Haar values of higher-order OTOCs for local, finite-trace observables, while traceless observables instead require exponential resources. Indeed, local observables are precisely those predicted to thermalize in chaotic many-body systems according to the eigenstate thermalization hypothesis. Moreover, adding to previous literature, we show how random matrix product unitaries constitute approximate designs: we exactly compute the frame potential of the ensemble, showing convergence to the Haar value with polynomial deviations and so indicating that global observables are freely independent on-average. Our results highlight the need to refine previous notions of unitary design in the context of operator dynamics, guiding us towards protocols for quantum advantage and shedding light on the emergent complexity of chaotic many-body systems.