Simulating quantum chaos without chaos

TL;DR

This paper introduces 'pseudochaotic' quantum Hamiltonians that mimic chaotic ensembles computationally but lack traditional chaos signatures, challenging existing notions of quantum chaos and its complexity.

Contribution

It presents a new class of Hamiltonians that are computationally indistinguishable from chaotic models yet do not exhibit chaos signatures, and provides an efficient simulation algorithm for them.

Findings

Pseudochaotic Hamiltonians are indistinguishable from GUE ensemble computationally.

These Hamiltonians lack conventional chaos signatures like level repulsion and strong scrambling.

The work establishes limits on Hamiltonian learning and enhances understanding of quantum chaos complexity.

Abstract

Quantum chaos is a quantum many-body phenomenon that is associated with a number of intricate properties, such as level repulsion in energy spectra or distinct scalings of out-of-time ordered correlation functions. In this work, we introduce a novel class of "pseudochaotic" quantum Hamiltonians that fundamentally challenges the conventional understanding of quantum chaos and its relationship to computational complexity. Our ensemble is computationally indistinguishable from the Gaussian unitary ensemble (GUE) of strongly-interacting Hamiltonians, widely considered to be a quintessential model for quantum chaos. Surprisingly, despite this effective indistinguishability, our Hamiltonians lack all conventional signatures of chaos: it exhibits Poissonian level statistics, low operator complexity, and weak scrambling properties. This stark contrast between efficient computational indistinguishability and traditional chaos indicators calls into question fundamental assumptions about the nature of quantum chaos. We, furthermore, give an efficient quantum algorithm to simulate Hamiltonians from our ensemble, even though simulating Hamiltonians from the true GUE is known to require exponential time. Our work establishes fundamental limitations on Hamiltonian learning and testing protocols and derives stronger bounds on entanglement and magic state distillation. These results reveal a surprising separation between computational and information-theoretic perspectives on quantum chaos, opening new avenues for research at the intersection of quantum chaos, computational complexity, and quantum information. Above all, it challenges conventional notions of what it fundamentally means to actually observe complex quantum systems.