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arXiv:quant-ph/0701044 · Physical Review A

Fractal fidelity as a signature of quantum chaos

Franco Pellegrini, Simone Montangero

Published Nov 29, 2007• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.76.052327

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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.
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Quantum chaos is the study of quantum systems whose classical description is chaotic. How does chaos manifest itself in the quantum world? In this spirit, we study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled-tops, where chaos and entanglement arise from the same physical mechanism. The long-time entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section - it is nearly uniform in the chaotic sea, and reproduces the detailed structure of the regular islands. The uniform value in the chaotic sea is explained by the random state conjecture. As classically chaotic dynamics take localized distributions in phase space to random distributions, quantized versions take localized coherent states to pseudo-random states in Hilbert space. Such random states are highly entangled, with an average value near that of the maximally entangled state. Continuing on our journey to find the footprints of chaos in the quantum world, we explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. We find an increase in the rate of information gain and hence higher fidelities in the process when the Floquet maps employed increase in chaoticity. We make predictions for the information gain using random matrix theory in the fully chaotic regime and show a remarkable agreement between the two. The last part of this thesis is devoted to the study of the nature of quantum correlations themselves. We present an operational interpretation of quantum discord based on the quantum state merging protocol. Quantum discord is the markup in the cost of quantum communication in the process of quantum state merging, if one discards relevant prior information.
We numerically analyze quantum survival probability fluctuations in an open, classically chaotic system. In a quasiclassical regime and in the presence of classical mixed phase space, such fluctuations are believed to exhibit a fractal pattern, on the grounds of semiclassical arguments. In contrast, we work in a classical regime of complete chaoticity and in a deep quantum regime of strong localization. We provide evidence that fluctuations are still fractal, due to the slow, purely quantum algebraic decay in time produced by dynamical localization. Such findings considerably enlarge the scope of the existing theory.
More than four decades of research on chaos in isolated quantum systems have led to the identification of universal signatures -- such as level repulsion and eigenstate thermalization -- that serve as cornerstones in our understanding of complex quantum dynamics. The emerging field of dissipative quantum chaos explores how these properties manifest in open quantum systems, where interactions with the environment play an essential role. We report the first experimental detection of dissipative quantum chaos and integrability by measuring the complex spacing ratios (CSRs) of open many-body quantum systems implemented on a high-fidelity superconducting quantum processor. Employing gradient-based tomography, we retrieve a ``donut-shaped'' CSR distribution for chaotic dissipative circuits, a hallmark of level repulsion in open quantum systems. For an integrable circuit, spectral correlations vanish, evidenced by a sharp peak at the origin in the CSR distribution. As we increase the depth of the integrable dissipative circuit, the CSR distribution undergoes an integrability-to-chaos crossover, demonstrating that intrinsic noise in the quantum processor is a dissipative chaotic process. Our results reveal the universal spectral features of dissipative many-body systems and establish present-day quantum computation platforms, which are predominantly used to run unitary simulations, as testbeds to explore dissipative many-body phenomena.