We analyze the fidelity of a quantum simulation and we show that it displays fractal fluctuations iff 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.
In this work a classical linear harmonic oscillator, evolving during a small time interval (so that simple non-linear, second order Taylor approximation of the dynamics is satisfied) and restarting (by a mechanism) in a strictly chosen series of the time moments, is considered. It is shown that given linear oscillator behaves dynamically analogously to discrete logistic equation, which, as an especial case, includes chaotic (fractal) behaviour too. (All this refers too on the analogous quantum systems, e.g. ammonia molecule with simple vibration dynamics of the atoms many times perturbed by me…
Magic state distillation protocols have a complicated non-linear nature. Analysis of protocols is therefore usually restricted to one-parameter families of states, which aids tractability. We show that if we lift this one-parameter restriction and embrace the complexity, distillation exhibits fractal properties. By studying these fractals we demonstrate that some protocols are more effective when not restricted. Low fidelity states that are usually worthless for distillation are now usable, and fewer iterations of the protocols are needed to reach high fidelity.
We show that the recently introduced operator fidelity metric provides a natural tool to investigate the cross-over to quantum chaotic behaviour. This metric is an information-theoretic measure of the global stability of a unitary evolution against perturbations. We use random matrix theory arguments to conjecture that the operator fidelity metric can be used as an "order parameter" to discriminates phases with regular behaviour from quantum chaotic ones. A numerical study of the onset of chaotic in the Dicke model is given in order to support the conjecture
In [PRE \textbf{90}, 022920 (2014), arXiv:1404.6738] a study of the ground state fidelity of the Dicke model as a function of the coupling parameter is presented. Abrupt jumps of the fidelity in the superradiant phase are observed and are assumed to be related to the transition to chaos. We show that this conclusion results from a misinterpretation of the numerics. In fact, if the parity symmetry is taken into account, the unexpected jumps disappear.
Simultaneous measurement of several noncommuting observables is modeled by using semigroups of completely positive maps on an algebra with a non-trivial center. The resulting piecewise-deterministic dynamics leads to chaos and to nonlinear iterated function systems (quantum fractals) on complex projective spaces.
The operator fidelity is a measure of the information-theoretic distinguishability between perturbed and unperturbed evolutions. The response of this measure to the perturbation may be formulated in terms of the operator fidelity susceptibility (OFS), a quantity which has been used to investigate the parameter spaces of quantum systems in order to discriminate their regular and chaotic regimes. In this work we numerically study the OFS for a pair of non-linearly coupled two-dimensional harmonic oscillators, a model which is equivalent to that of a hydrogen atom in a uniform external magnetic f…
We explore the effect of a system's symmetries on fidelity decay behavior. Chaos-like exponential fidelity decay behavior occurs in non-chaotic systems when the system possesses symmetries and the applied perturbation is not tied to a classical parameter. Similar systems without symmetries exhibit faster-than-exponential decay under the same type of perturbation. This counter-intuitive result, that extra symmetries cause the system to behave in a chaotic fashion, may have important ramifications for quantum error correction.
We study the coupled translational, electronic, and field dynamics of the combined system "a two-level atom + a single-mode quantized field + a standing-wave ideal cavity". We derive Hamilton -- Schrödinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initia…
Matrix logistic map: fractal spectral distributions and transfer of chaos
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences（波兰科学院理论与应用信息研究所）；Institute of Physics, Jagiellonian University（雅盖隆大学物理研究所）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）
Mean fidelity amplitude and parametric energy--energy correlations are calculated exactly for a regular system, which is subject to a chaotic random perturbation. It turns out that in this particular case under the average both quantities are identical. The result is compared with the susceptibility of chaotic systems against random perturbations. Regular systems are more susceptible against random perturbations than chaotic ones.
We study the quantum critical behaviour of the Dicke Hamiltonian, with finite number of atoms and explore the signature of quantum chaos using measures like the ground state fidelity and the Loschmidt echo. We show that both these quantities clearly point to the chaotic nature of the system in the super-radiant phase.
We deal with a system comprising a nonlinear (Kerr-like) oscillator excited by a series of ultra-short external pulses. We introduce the fidelity-based entropic parameter that can be used as an indicator of quantum chaos. Moreover, we propose to use the fidelity-like parameter comprising the information about the mean number of photons in the system. We shall concentrate on the long-time behaviour of the parameters discussed, showing that for deep chaos cases the quantum fidelities behave chaotically in the classical sense despite their strictly quantum character.
Departing from classical concepts of ergodic theory, formulated in terms of probability densities, measures describing the chaotic behavior and the loss of information in quantum open systems are proposed. As application we discuss the chaotic outcomes of continuous measurement processes in the EEQT framework. Simultaneous measurement of four noncommuting spin components is shown to lead to a chaotic jump on quantum spin sphere and to generate specific fractal images - nonlinear ifs (iterated function system). The model is purely theoretical at this stage, and experimental confirmation of the …
Perfect state transfer is possible in modulated spin chains, imperfections however are likely to corrupt the state transfer. We study the robustness of this quantum communication protocol in the presence of disorder both in the exchange couplings between the spins and in the local magnetic field. The degradation of the fidelity can be suitably expressed, as a function of the level of imperfection and the length of the chain, in a scaling form. In addition the time signal of fidelity becomes fractal. We further characterize the state transfer by analyzing the spectral properties of the Hamilton…
Recent work has connected the type of fidelity decay in perturbed quantum models to the presence of chaos in the associated classical models. We demonstrate that a system's rate of fidelity decay under repeated perturbations may be measured efficiently on a quantum information processor, and analyze the conditions under which this indicator is a reliable probe of quantum chaos and related statistical properties of the unperturbed system. The type and rate of the decay are not dependent on the eigenvalue statistics of the unperturbed system, but depend on the system's eigenvector statistics in …
We show that the mechanism of quantum freeze of fidelity decay for perturbations with zero time-average, recently discovered for a specific case of integrable dynamics [New J. Phys. 5 (2003) 109], can be generalized to arbitrary quantum dynamics. We work out explicitly the case of chaotic classical counterpart, for which we find semi-classical expressions for the value and the range of the plateau of fidelity. After the plateau ends, we find explicit expressions for the asymptotic decay, which can be exponential or Gaussian depending on the ratio of the Heisenberg time to the decay time. Arbit…
We show that the three-junction SQUID device designed for the Josephson flux qubit can be used to study the dynamics of quantum chaos when operated at high energies. We determine the parameter region where the system is classically chaotic. We calculate numerically the fidelity or Loschmidt echo (LE) in the quantum dynamics under perturbations in the magnetic field and in the critical currents, and study different regimes of the LE. We discuss how the LE could be observed experimentally considering both the preparation of the initial state and the measurement procedure.
Integrability and Chaos via fractal analysis of Spectral Form Factors: Gaussian approximations and exact results
Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；University of Southern California（南加州大学）；Istituto Nazionale di Fisica Nucleare (INFN)（意大利国家核物理研究所）；Scuola Superiore Meridionale（南方高等学院）
Weyl's law approximates the number of states in a quantum system by partitioning the energetically accessible phase-space volume into Planck cells. Here we show that typical resonances in generic open quantum systems follow a modified, fractal Weyl law, even though their classical dynamics is not globally chaotic but also contains domains of regular motion. Besides the obvious ramifications for quantum decay, this delivers detailed insight into quantum-to-classical correspondence, a phenomenon which is poorly understood for generic quantum-dynamical systems.
