Measuring the Lyapunov exponent using quantum mechanics

Cucchietti, Fernando M.; Lewenkopf, Caio; Mucciolo, Eduardo R.; Pastawski, Horacio M.; Vallejos, Raúl O.

doi:10.1103/physreve.65.046209

Cited by 99 publications

(112 citation statements)

References 33 publications

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“…Extensive investigations have been performed in recent years to understand the decaying behaviors of the LE [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. In chaotic systems, roughly speaking, the LE has a Gaussian decay [23] below a perturbative border and has an exponential decay M L (t) ∝ exp(−Γt) above the border.…”

Section: Lochmidt Echo and Fidelity For The Dicke Model At Qptmentioning

confidence: 99%

“…The dramatic change of the wave function at a QPT implies a fast decrease of the fidelity when approaching the critical point. Another metric quantity useful for characterizing the occurrence of a QPT is the quantum Loschmidt echo (LE) [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], which provides a measure to the stability of the quantum motion under * Electronic address: wgwang@ustc.edu.cn two slightly different Hamiltonians. This quantity exhibits a dramatic change in its decaying behavior in the neighborhood of critical points.…”

Section: Introductionmentioning

confidence: 99%

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Complexity and instability of quantum motion near a quantum phase transition

et al. 2014

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We show that the number of harmonics of the Wigner function, recently proposed as a measure of quantum complexity, can also be used to characterize quantum phase transitions. The non-analytic behavior of this quantity in the neighborhood of a quantum phase transition is illustrated by means of the Dicke model and is compared to two well-known measures of the (in)stability of quantum motion: the quantum Loschmidt echo and fidelity.

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Section: Lochmidt Echo and Fidelity For The Dicke Model At Qptmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Complexity and instability of quantum motion near a quantum phase transition

et al. 2014

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“…Moreover, a relationship between this quantity and the Lyapunov exponent that characterizes the classical chaos has been established analytically by using the semiclassical theory by Jalabert and Pastawski [3], which has been confirmed numerically in several models [4][5][6][7][8][9]. Therefore, this quantity has attracted a great attention from the community of quantum chaos [10][11][12][13], It has been shown that several regimes exist.…”

Section: Introductionmentioning

confidence: 99%

Crossover of quantum Loschmidt echo from golden-rule decay to perturbation-independent decay

Wang

2002

Phys. Rev. E

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We study the crossover of the quantum Loschmidt echo (or fidelity) from the golden rule regime to the perturbation-independent exponential decay regime by using the kicked top model. It is shown that the deviation of the perturbation-independent decay of the averaged fidelity from the Lyapunov decay results from quantum fluctuations in individual fidelity, which are caused by the coherence in the initial coherent states. With an averaging procedure suppressing the quantum fluctuations effectively, the perturbation-independent decay is found to be close to the Lyapunov decay. We also show that the Fourier transform of the fidelity is determined directly by the initial state and the eigenstates of the Floquet operators of the two classically chaotic systems concerned. The absolute value part and the phase part of the Fourier transform of the fidelity are found to be divided into several correlated parts, which is a manifestation of the coherence of the initial coherent state. In the whole crossover region, some important properties of the fidelity, such as the exponent of its exponential decay and the short initial time within which the fidelity almost does not change, are found to be closely related to properties of the central part of its Fourier transform. PACS number(s): 05.45. Mt, 05.45.Pq, 42.50.Md, 76.60.Lz

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“…This profile plays a fundamental role also in the analysis of system stability for few-body systems subject to a sudden perturbation [10], such as the onset of an external field, and has been studied extensively in the context of quantum chaos and dynamical localization [11,12]. Quite generally the LDOS is related to the survival probability of the unperturbed eigenstate [4,10,13], and there has been considerable recent effort to understand the conditions under which the LDOS width determines the rate of fidelity decay under imperfect motion-reversal ("Loschmidt echo") [13,14,15,16].A number of theoretical methods have been devised to characterize the LDOS for complex systems. These methods include banded random matrix models [3,17,18,19], models of a single-level with constant couplings to a "picketfence" spectrum [4,20], and perturbative techniques with partial summations over diagrams to infinite order [21].…”

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confidence: 99%

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confidence: 99%

Estimation of the local density of states on a quantum computer

et al. 2004

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We report an efficient quantum algorithm for estimating the local density of states (LDOS) on a quantum computer. The LDOS describes the redistribution of energy levels of a quantum system under the influence of a perturbation. Sometimes known as the "strength function" from nuclear spectroscopy experiments, the shape of the LDOS is directly related to the survivial probability of unperturbed eigenstates, and has recently been related to the fidelity decay (or "Loschmidt echo") under imperfect motion-reversal. For quantum systems that can be simulated efficiently on a quantum computer, the LDOS estimation algorithm enables an exponential speed-up over direct classical computation.PACS numbers: 05.45.Mt, 03.67.Lx A major motivation for the physical realization of quantum information processing is the idea, intimated by Feynman, that the dynamics of a wide class of complex quantum systems may be simulated efficiently by these techniques [1]. For a quantum system with Hilbert space size N , an efficient simulation is one that requires only Polylog(N ) gates. This situation should be contrasted with direct simulation on a classical processor, which requires resources growing at least as N 2 . However, complete measurement of the final state on a quantum processor requires O(N 2 ) repetitions of the quantum simulation. Similarly, estimation of the eigenvalue spectrum of a quantum system admitting a Polylog(N ) circuit decomposition requires a phase-estimation circuit that grows as O(N ) [2]. As a result there still remains the important problem of devising methods for the efficient readout of those characteristic properties that are of practical interest in the study of complex quantum systems. In this Letter we introduce an efficient quantum algorithm for estimating, to 1/Polylog(N ) accuracy, the local density of states (LDOS), a quantity of central interest in the description of both many-body and complex fewbody systems. We also determine the class of physical problems for which the LDOS estimation algorithm provides an exponential speed-up over known classical algorithms given this finite accuracy.The LDOS describes the profile of an eigenstate of an unperturbed quantum system over the eigenbasis of perturbed version of the same quantum system. In the context of manybody systems the LDOS was introduced to describe the effect of strong two-particle interactions on the single particle (or single hole) eigenstates [3,4,5,6,7]. More recently, the LDOS has been studied to characterize the effect of imperfections (due to residual interactions between the qubits) in the operation of quantum computers [8,9]. This profile plays a fundamental role also in the analysis of system stability for few-body systems subject to a sudden perturbation [10], such as the onset of an external field, and has been studied extensively in the context of quantum chaos and dynamical localization [11,12]. Quite generally the LDOS is related to the survival probability of the unperturbed eigenstate [4,10,13], and there has been considerable re...

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Measuring the Lyapunov exponent using quantum mechanics

Cited by 99 publications

References 33 publications

Complexity and instability of quantum motion near a quantum phase transition

Complexity and instability of quantum motion near a quantum phase transition

Crossover of quantum Loschmidt echo from golden-rule decay to perturbation-independent decay

Estimation of the local density of states on a quantum computer

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