Built on

E. B. Rozenbaum, L. A. Bunimovich and V. Galitski, “Early-Time Exponential Instabilities in Nonchaotic Quantum Systems,” Phys. Rev. Lett. 125
Original
1902
Earlier work this paper cites.
J. L. F. Barbón, E. Rabinovici, R. Shir and R. Sinha, “On The Evolution Of Operator Complexity Beyond Scrambling,” JHEP 10
Original
1907
Earlier work this paper cites.
A. Avdoshkin and A. Dymarsky, “Euclidean operator growth and quantum chaos,” Phys. Rev. Res. 2
Original
1911
Earlier work this paper cites.
T. Xu, T. Scaffidi and X. Cao, “Does scrambling equal chaos?,” Phys. Rev. Lett. 124
Original
1912
Earlier work this paper cites.
A. Dymarsky and A. Gorsky, “Quantum chaos as delocalization in Krylov space,” Phys. Rev. B 102
Original
1912
Earlier work this paper cites.
A. I. Larkin and Y. N. Ovchinnikov, “Quasiclassical method in the theory of superconductivity,” JETP 28, 6 (1969): 1200-1205
1969
Earlier work this paper cites.
Ya. G. Sinai, “Dynamical systems with elastic reflections,” Russ. Math. Surv. 25
1970
Earlier work this paper cites.
L. A. Bunimovich, “On ergodic properties of certain billiards,” Funct. Anal. Appl. 8
1974
Earlier work this paper cites.
J. Sanchez-Dehesa, “The spectrum of Jacobi matrices in terms of its associated weight function,” Journal of Computational and Applied Mathematics 4
1978
Earlier work this paper cites.
G. Benettin and J. -M. Strelcyn, “Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy,” Phys. Rev. A 17
1978
Earlier work this paper cites.
L. A. Bunimovich, “On the ergodic properties of nowhere dispersing billiards,” Commun. Math. Phys. 65
1979
Earlier work this paper cites.
S. W. McDonald and A. N. Kaufman, “Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories,” Phys. Rev. Lett. 42
1979
Earlier work this paper cites.
G. Casati, F. Valz-Gris, and I. Guarneri, “On the connection between quantization of nonintegrable systems and statistical theory of spectra,” Lett. Nuov. Cim. 28
1980
Earlier work this paper cites.
M. V. Berry, “Quantizing a classically ergodic system: Sinai’s billiard and the KKR method,” Annals Phys. 131
1981
Earlier work this paper cites.
S. G. Matinyan, G. K. Savvidy and N. G. Ter-Arutunian Savvidy, “CLASSICAL YANG-MILLS MECHANICS. NONLINEAR COLOR OSCILLATIONS,” Sov. Phys. JETP 53
1981
Earlier work this paper cites.
S. G. Matinyan, G. K. Savvidy and N. G. Ter-Arutunian Savvidy, “Stochasticity of Classical Yang-Mills Mechanics and Its Elimination by Higgs Mechanism. (In Russian),” JETP Lett. 34
1981
Earlier work this paper cites.
O. Bohigas, M. J. Giannoni and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52
1984
Earlier work this paper cites.

Similar

G. K. Savvidy, “Classical and Quantum Mechanics of Nonabelian Gauge Fields,” Nucl. Phys. B 246
1984
Cited alongside, same era.
S. Sachdev and J. Ye, “Gapless spin fluid ground state in a random, quantum Heisenberg magnet,” Phys. Rev. Lett. 70
1993
Cited alongside, same era.
T. S. Biro, S. G. Matinyan and B. Muller, “Chaos and gauge field theory,” World Sci. Lect. Notes Phys. 56
1994
Cited alongside, same era.
Beresford N. Parlett, “The Symmetric Eigenvalue Problem,” Society for Industrial and Applied Mathematics (1998)
1998
Cited alongside, same era.
T. Akutagawa, K. Hashimoto, T. Sasaki and R. Watanabe, “Out-of-time-order correlator in coupled harmonic oscillators,” JHEP 08
Original
2004
Cited alongside, same era.

Then

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, “Complexity, action, and black holes,” Phys. Rev. D 93
Original
2016
Later among the works it cites.
K. Hashimoto, K. Murata and R. Yoshii, “Out-of-time-order correlators in quantum mechanics,” JHEP 10
Original
2017
Later among the works it cites.
D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman, “A Universal Operator Growth Hypothesis,” Phys. Rev. X 9
Original
2019
Later among the works it cites.
A. Dymarsky and M. Smolkin, “Krylov complexity in conformal field theory,” Phys. Rev. D 104
Original
2021
Later among the works it cites.
K. Hashimoto, K. Murata, N. Tanahashi and R. Watanabe, “Bound on energy dependence of chaos,” Phys. Rev. D 106

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2023

Krylov complexity and chaos in quantum mechanics

Hashimoto, Koji, Murata, Keiju, Tanahashi, Norihiro et al.

Understand

Recently, Krylov complexity was proposed as a measure of complexity and chaoticity of quantum systems.

We consider the stadium billiard as a typical example of the quantum mechanical system obtained by quantizing a classically chaotic system, and numerically evaluate Krylov complexity for operators and states.
Despite no exponential growth of the Krylov complexity, we find a clear correlation between variances of Lanczos coefficients and classical Lyapunov exponents, and also a correlation with the statistical distribution of adjacent spacings of the quantum energy levels.
This shows that the variances of Lanczos coefficients can be a measure of quantum chaos.

Built on

E. B. Rozenbaum, L. A. Bunimovich and V. Galitski, “Early-Time Exponential Instabilities in Nonchaotic Quantum Systems,” Phys. Rev. Lett. 125
Original
1902
Earlier work this paper cites.
J. L. F. Barbón, E. Rabinovici, R. Shir and R. Sinha, “On The Evolution Of Operator Complexity Beyond Scrambling,” JHEP 10
Original
1907
Earlier work this paper cites.
A. Avdoshkin and A. Dymarsky, “Euclidean operator growth and quantum chaos,” Phys. Rev. Res. 2
Original
1911
Earlier work this paper cites.
T. Xu, T. Scaffidi and X. Cao, “Does scrambling equal chaos?,” Phys. Rev. Lett. 124
Original
1912
Earlier work this paper cites.
A. Dymarsky and A. Gorsky, “Quantum chaos as delocalization in Krylov space,” Phys. Rev. B 102
Original
1912
Earlier work this paper cites.
A. I. Larkin and Y. N. Ovchinnikov, “Quasiclassical method in the theory of superconductivity,” JETP 28, 6 (1969): 1200-1205
1969
Earlier work this paper cites.
Ya. G. Sinai, “Dynamical systems with elastic reflections,” Russ. Math. Surv. 25
1970
Earlier work this paper cites.
L. A. Bunimovich, “On ergodic properties of certain billiards,” Funct. Anal. Appl. 8
1974
Earlier work this paper cites.
J. Sanchez-Dehesa, “The spectrum of Jacobi matrices in terms of its associated weight function,” Journal of Computational and Applied Mathematics 4
1978
Earlier work this paper cites.
G. Benettin and J. -M. Strelcyn, “Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy,” Phys. Rev. A 17
1978
Earlier work this paper cites.
L. A. Bunimovich, “On the ergodic properties of nowhere dispersing billiards,” Commun. Math. Phys. 65
1979
Earlier work this paper cites.
S. W. McDonald and A. N. Kaufman, “Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories,” Phys. Rev. Lett. 42
1979
Earlier work this paper cites.
G. Casati, F. Valz-Gris, and I. Guarneri, “On the connection between quantization of nonintegrable systems and statistical theory of spectra,” Lett. Nuov. Cim. 28
1980
Earlier work this paper cites.
M. V. Berry, “Quantizing a classically ergodic system: Sinai’s billiard and the KKR method,” Annals Phys. 131
1981
Earlier work this paper cites.
S. G. Matinyan, G. K. Savvidy and N. G. Ter-Arutunian Savvidy, “CLASSICAL YANG-MILLS MECHANICS. NONLINEAR COLOR OSCILLATIONS,” Sov. Phys. JETP 53
1981
Earlier work this paper cites.
S. G. Matinyan, G. K. Savvidy and N. G. Ter-Arutunian Savvidy, “Stochasticity of Classical Yang-Mills Mechanics and Its Elimination by Higgs Mechanism. (In Russian),” JETP Lett. 34
1981
Earlier work this paper cites.
O. Bohigas, M. J. Giannoni and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52
1984
Earlier work this paper cites.

Similar

G. K. Savvidy, “Classical and Quantum Mechanics of Nonabelian Gauge Fields,” Nucl. Phys. B 246
1984
Cited alongside, same era.
S. Sachdev and J. Ye, “Gapless spin fluid ground state in a random, quantum Heisenberg magnet,” Phys. Rev. Lett. 70
1993
Cited alongside, same era.
T. S. Biro, S. G. Matinyan and B. Muller, “Chaos and gauge field theory,” World Sci. Lect. Notes Phys. 56
1994
Cited alongside, same era.
Beresford N. Parlett, “The Symmetric Eigenvalue Problem,” Society for Industrial and Applied Mathematics (1998)
1998
Cited alongside, same era.
T. Akutagawa, K. Hashimoto, T. Sasaki and R. Watanabe, “Out-of-time-order correlator in coupled harmonic oscillators,” JHEP 08
Original
2004
Cited alongside, same era.