Diffusion and Localization in Chaotic Billiards

Borgonovi, Fausto; Casati, Giulio; Li, Bao-Wen

doi:10.1103/physrevlett.77.4744

Cited by 115 publications

(136 citation statements)

References 11 publications

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“…The classical limit corresponds to k → ∞, T → 0, and K = kT = const. In this quantum model one can observe important physical phenomena like dynamical localization [8,9]. Indeed, due to quantum interference effects, the chaotic diffusion in momentum is suppressed, in a way similar to Anderson localization in disordered solids.…”

Section: Efficient Quantum Computing Of Complex Dynamicsmentioning

confidence: 99%

“…Indeed, due to quantum interference effects, the chaotic diffusion in momentum is suppressed, in a way similar to Anderson localization in disordered solids. Also in the vicinity of a broken KAM torus, cantori localization takes place, since a cantorus starts to act as a perfect barrier to quantum wave packet evolution, if the flux through it becomes less thanh [16,17,[8][9][10].…”

Section: Efficient Quantum Computing Of Complex Dynamicsmentioning

confidence: 99%

“…This model has rich and complex dynamics and finds various applications, e.g. for dynamical localization in billiards [8][9][10]. Our algorithm, based on the Quantum Fourier Transform (QFT) [11], simulates the dynamics of a system with N levels in O((log 2 N )…”

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confidence: 99%

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Efficient Quantum Computing of Complex Dynamics

et al. 2001

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We propose a quantum algorithm which uses the number of qubits in an optimal way and efficiently simulates a physical model with rich and complex dynamics described by the quantum sawtooth map. The numerical study of the effect of static imperfections in the quantum computer hardware shows that the main elements of the phase space structures are accurately reproduced up to a time scale which is polynomial in the number of qubits. The errors generated by these imperfections are more dangerous than the errors of random noise in gate operations.PACS numbers: 03.67. Lx, 05.45.Mt, 24.10.Cn When applied to computation, quantum mechanics opens completely new perspectives: a quantum computer, if constructed, could perform certain computations faster than classical computers, exploiting quantum mechanical features such as entanglement or superposition [1]. Shor discovered a quantum algorithm [2] which factorizes large integers exponentially faster than any known classical algorithm. It was also shown by Grover [3] that the search of an item in an unstructured database can be done with a square root speed up over any classical algorithm. However, at present only a few quantum algorithms are known which simulate physical systems with exponential efficiency. Such systems include certain many-body problems [4], spin lattices [5], and models of quantum chaos [6,7].In this Letter, we present a quantum algorithm which computes the time evolution of the quantum sawtooth map, exponentially faster than any classical computation. This model has rich and complex dynamics and finds various applications, e.g. for dynamical localization in billiards [8][9][10]. Our algorithm, based on the Quantum Fourier Transform (QFT) [11], simulates the dynamics of a system with N levels in O((log 2 N )2 ) operations per map iteration, while a classical computer, which performs Fast Fourier Transforms (FFT), requires O(N log 2 N ) operations. A further striking advantage of the algorithm is the optimum utilization of qubits: one needs only n q = log 2 N qubits (without any extra work space). We demonstrate that complex phase space structures can be simulated with less that 10 qubits, while about 40 qubits would allow one to make computations inaccessible to present-day supercomputers. This is particularly important, since experiments with few qubits are being performed at present [12,13], in particular the QFT was implemented on a three qubit nuclear magnetic resonance quantum computer [14]. For this reason the investigation of this interesting physical system will be accessible to first quantum computers, operating with few qubits and for which large-scale computations like integer factoring are not possible.The classical sawtooth map is given bywhere (n, θ) are conjugated action-angle variables (0 ≤ θ < 2π), and the bars denote the variables after one map iteration. Introducing the rescaled variable p = T n, one can see that the classical dynamics depends only on the single parameter K = kT , so that the motion is stable for −4 < K < 0 and com...

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Section: Efficient Quantum Computing Of Complex Dynamicsmentioning

confidence: 99%

Section: Efficient Quantum Computing Of Complex Dynamicsmentioning

confidence: 99%

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Efficient Quantum Computing of Complex Dynamics

et al. 2001

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“…For example, wave packets mimic to some extend the chaotic ray diffusion in phase space. However, destructive interference suppresses the chaotic diffusion on long time scales (Borgonovi et al, 1996;Casati et al, 1979;Fishmann et al, 1982;Frahm and Shepelyansky, 1997). This dynamical localization in phase space is closely related to real-space Anderson localization in disordered solids (Fishmann et al, 1982).…”

Section: Dynamical Localization and Scar Modesmentioning

confidence: 99%

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

2015

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“…[6] for pseudointegrable and in Refs. [7,[15][16][17] for chaotic billiards. Here we find that for our systems, there can be many energy windows, where the level statistics is comparatively close to Poisson statistics, and other energy intervals, where the behavior is close to Wigner statistics.…”

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Level statistics and eigenfunctions of pseudointegrable systems: Dependence on energy and genus number

Hlushchuk

Russ

2003

Phys. Rev. E

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We study the level statistics (second half moment I0 and rigidity ∆3) and the eigenfunctions of pseudointegrable systems with rough boundaries of different genus numbers g. We find that the levels form energy intervals with a characteristic behavior of the level statistics and the eigenfunctions in each interval. At low enough energies, the boundary roughness is not resolved and accordingly, the eigenfunctions are quite regular functions and the level statistics shows Poisson-like behavior. At higher energies, the level statistics of most systems moves from Poisson-like towards Wigner-like behavior with increasing g. Investigating the wavefunctions, we find many chaotic functions that can be described as a random superposition of regular wavefunctions. The amplitude distribution P (ψ) of these chaotic functions was found to be Gaussian with the typical value of the localization volume V loc ≈ 0.33. For systems with periodic boundaries we find several additional energy regimes, where I0 is relatively close to the Poisson-limit. In these regimes, the eigenfunctions are either regular or localized functions, where P (ψ) is close to the distribution of a sine or cosine function in the first case and strongly peaked in the second case. Also an interesting intermediate case between chaotic and localized eigenfunctions appears.

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Diffusion and Localization in Chaotic Billiards

Cited by 115 publications

References 11 publications

Efficient Quantum Computing of Complex Dynamics

Efficient Quantum Computing of Complex Dynamics

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

Level statistics and eigenfunctions of pseudointegrable systems: Dependence on energy and genus number

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