Fat fractals in quantum chaos

M Uleysky, L Kon’kov, S Prants

This hypothesis explores the application of the McGinty Equation to fractal quantum chaos and dynamical systems, proposing that the chaotic behavior of quantum systems exhibits fractal properties. The primary objective is to understand how fractal geometry influences the dynamics of quantum systems, including quantum chaos, phase space structures, and the evolution of quantum states, providing new insights into the interplay between quantum mechanics and chaos theory.

Urban Duh, Marko Žnidarič

We describe a chaos scenario valid for chaotic many-body quantum systems, the simulation of which is one of the prime candidates for demonstrating quantum advantage, and clarify its importance for classical truncation-based simulations. By studying the spectral properties of the truncated propagator, we, for the first time, identify fractal structures responsible for many-body quantum chaos, analogous to the fractality in classical chaos emerging from the time evolution of initially smooth long-wavelength densities into ever finer short-wavelength densities. Namely, we show that the leading slowest-decaying Ruelle-Pollicott eigenoperators have a non-trivial fractal dimension quantifying their non-locality, visible also in the divergence of their condition numbers. Furthermore, we find that unitarity imposes an asymptotic equality between the temporal decay rate of local correlations and this many-body fractal dimension. With this insight, the scenario for many-body quantum chaos becomes clear: over time, local operators evolve toward increasingly non-local ones with a quantifiable fractal structure, thereby naturally leading to effective non-unitary relaxation on the subspace of local operators -- a kind of many-body Kolmogorov cascade in the space of operators -- which enables accurate truncation-based simulations. Our predictions are demonstrated numerically in various one- and two-dimensional quantum circuits, with and without conservation laws, as well as by exact results in dual-unitary circuits.

Franco Pellegrini, Simone Montangero

We analyze the fidelity of a quantum simulation and we show that it displays fractal fluctuations if and only if the simulated dynamics is chaotic. This analysis allows us to investigate a given simulated dynamics without any prior knowledge. In the case of integrable dynamics, the appearance of fidelity fractal fluctuations is a signal of a highly corrupted simulation. We conjecture that fidelity fractal fluctuations are a signature of the appearance of quantum chaos. Our analysis can be realized already by a few-qubit quantum processor.

Anonymous