Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first show that any variational approximation to the dynamics of a quantum system based on the Dirac action principle leads to a classical Hamiltonian dynamics for the variational parameters. Since this Hamiltonian is generically nonlinear and nonintegrable, the dynamics thus generated can be chaotic, in distinction to the exact quantum evolution. We then restrict attention to a system of two biquadratically coupled quantum oscillators and study two variational schemes, the leading order large N (four canonical variables) and Hartree (six canonical variables) approximations. The chaos seen in the approximate dynamics is an artifact of the approximations: this is demonstrated by the fact that its onset occurs on the same characteristic time scale as the breakdown of the approximations when compared to numerical solutions of the time-dependent Schrodinger equation.
We explore quantum chaos diagnostics of variational circuit states at random parameters and study their correlation with the circuit expressibility and the optimization of control parameters. By measuring the operator spreading coefficient and the eigenvalue spectrum of the modular Hamiltonian of the reduced density matrix, we identify the universal structure of random matrix models in high-depth circuit states. We construct different layer unitaries corresponding to the GOE and GUE distributions and quantify their VQA performance. Our study also highlights a potential tension between the OTOC…
The exponential growth or decay with time of the out-of-time-order commutator (OTOC) is one widely used diagnostic of many-body chaos in spatially-extended systems. In studies of many-body classical chaos, it has been noted that one can define a velocity-dependent Lyapunov exponent, $λ({\bf v})$, which is the growth or decay rate along "rays" at that velocity. We examine the behavior of $λ({\bf v})$ for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by $λ({\bf \hat n}v_B({\bf \hat n}))=0$, with a generally directi…
Dynamical localization in chaotic systems: spectral statistics and localization measure in the kicked rotator as a paradigm for time-dependent and time-independent systems
It is proposed to improve the quality of a variational description of a closed quantum system by adding ficticious dissipation that reduces the entanglement. The proposal is implemented for a small Bose-Hubbard chain, which shows chaotic behavior and associated fast growth of quantum fluctuations. For appropriately chosen dissipation, good agreement with the truncated Wigner approximation (which is very accurate for the chosen system parameters) is found.
Russian Quantum Center（俄罗斯量子中心）；University of New South Wales（新南威尔士大学）；Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（美国国家标准与技术研究院/马里兰大学量子信息与计算机科学联合中心）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；QuTech, Delft Technical University（代尔夫特理工大学量子技术研究所）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；National University of Science and Technology MISIS（国立科技大学米西斯）
Understanding properties of quantum matter is an outstanding challenge in science. In this paper, we demonstrate how machine-learning methods can be successfully applied for the classification of various regimes in single-particle and many-body systems. We realize neural network algorithms that perform a classification between regular and chaotic behavior in quantum billiard models with remarkably high accuracy. We use the variational autoencoder for autosupervised classification of regular/chaotic wave functions, as well as demonstrating that variational autoencoders could be used as a tool f…
We present a mean-photon-number dependent variational method, which works well in whole coupling regime if the photon energy is dominant over the spin-flipping, to evaluate the properties of the Rabi model for both the ground state and the excited states. For the ground state, it is shown that the previous approximate methods, the generalized rotating-wave approximation (only working well in the strong coupling limit) and the generalized variational method (only working well in the weak coupling limit), can be recovered in the corresponding coupling limits. The key point of our method is to ta…
Relation between the eigenfrequencies of Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian in a time-dependent variational approach
A recent proposal by Hallam et al. suggested using the chaotic properties of the semiclassical equations of motion, obtained by the time dependent variational principle (TDVP), as a characterization of quantum chaos. In this paper, we calculate the Lyapunov spectrum of the semiclassical theory approximating the quantum dynamics of a strongly interacting Rydberg atom array, which lead to periodic motion. In addition, we calculate the effect of quantum fluctuations around this approximation, and obtain the escape rate from the periodic orbit. We compare this rate to the rate extracted from the e…
We study various aspects of the mass deformed SYK model which can escape the interiors of pure boundary state black holes. SYK boundary states are given by a simple local boundary condition on the Majorana fermions and then evolved in Euclidean time in the SYK Hamiltonian. We study the ground state of this mass deformed SYK model in detail. We also use SYK boundary states as a variational approximation to the ground state of the mass deformed SYK model. We compare variational approximation with the exact ground state results and they showed a good agreement. We also study the time evolution of…
The time-dependent variational principle is used to optimize the linear and nonlinear parameters of Gaussian basis functions to solve the time-dependent Schrodinger equation in 1 and 3 dimensions for a one-body soft Coulomb potential in a laser field. The accuracy is tested comparing the solution to finite difference grid calculations using several examples. The approach is not limited to one particle systems and the example presented for two electrons demonstrates the potential to tackle larger systems using correlated basis functions.
We develop a variational approach to simulating the dynamics of open quantum many-body systems using deep autoregressive neural networks. The parameters of a compressed representation of a mixed quantum state are adapted dynamically according to the Lindblad master equation by employing a time-dependent variational principle. We illustrate our approach by solving the dissipative quantum Heisenberg model in one and two dimensions for up to 40 spins and by applying it to the simulation of confinement dynamics in the presence of dissipation.
In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schrödinger equation. The method is based on: i) the definition of a linearly independent working space, and ii) a variational construction of the propagator over finite time-steps. The second point allows the method to properly account for changes in the dimensionality of the working space along the time evolution. In particular, the time evolution is represented by a semi-unitary transformation. Tests are done on a quartic double-well potential with Gaussian…
We propose an improved scheme to do the time dependent variational principle (TDVP) in finite matrix product states (MPS) for two-dimensional systems or one-dimensional systems with long range interactions. We present a method to represent the time-evolving state in a MPS with its basis enriched by state-averaging with global Krylov vectors. We show that the projection error is significantly reduced so that precise time evolution can still be obtained even if a larger time step is used. Combined with the one-site TDVP, our approach provides a way to dynamically increase the bond dimension whil…
We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. We show that the spread of the wave function in the Hilbert space is determined by the survival probability which is known to have pre-exponential, exponential, and long-term power-law limiting behaviors. This result relies on complexity of the wave functions and thus is generic to many systems. In the chaotic limit modeled by the Gaussian Orthogonal Ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the c…
The eigenstate thermalisation hypothesis resolves the paradox of emergent thermal or classical behaviour in a closed quantum system by focussing upon local observations. This permits the remainder of the system to act as a bath, thermalisation arising due to a process of de-phasing that gradually reveals the thermal nature of local observables measured in an eigenstate. This is very different from thermalisation in closed classical systems, which is driven by dynamical chaos. We show how quantum thermalisation in closed systems can be recast in a way that is directly related to classical therm…
Ab-initio variational wave functions for the time-dependent many-electron Schrödinger equation
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Center for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院量子科学与工程中心）
The projection of time-dependent variational principle (TDVP) for matrix product states enables us to perform long-time simulations of one-dimensional quantum systems with the conservation of the total energy and the norm of wave functions. We compare long-time dynamics after a quantum quench simulated by TDVP with those by the exact diagonalization method in order to evaluate the performance of TDVP. We show that in a nonintegrable model the projection of TDVP clearly improves the long-time behaviors of global observables included in the Hamiltonian, such as the kinetic and interaction energi…
Dirac-Frenkel variational method with Davydov D2 trial wavefunction is extended by introducing a thermalization algorithm and applied to simulate dynamics of a general open quantum system. The algorithm allows to control temperature variations of a harmonic finite size bath, when in contact with the quantum system. Thermalization of the bath vibrational modes is realised via stochastic scatterings, implemented as a discrete-time Bernoulli process with Poisson statistics. It controls bath temperature by steering vibrational modes' evolution towards their canonical thermal equilibrium. Numerical…
