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arXiv:1803.08321 · Physical review. B./Physical review. B

Quenches near Ising quantum criticality as a challenge for artificial neural networks

Stefanie Czischek, Martin Gärttner, Thomas Gasenzer

Published Jul 31, 2018• 85 citations
arXivHTMLPDF

Abstract

The near-critical unitary dynamics of quantum Ising spin chains in transversal and longitudinal magnetic fields is studied using an artificial neural network representation of the wave function. A focus is set on strong spatial correlations which build up in the system following a quench into the vicinity of the quantum critical point. We compare correlations obtained by optimizing the parameters of the network states with analytical solutions in integrable cases and time-dependent density matrix renormalization group (tDMRG) simulations, as well as with predictions from a semiclassical discrete truncated Wigner analysis. While the semiclassical approach yields quantitatively correct results only at very short times and near zero transverse fields, the neural-network representation is applicable in a much wider regime. However, for quenches close to the quantum critical point the representation becomes inefficient. For nonintegrable models we show that in regimes where tDMRG is limited to short times due to extensive entanglement growth, also the neural-network parametrization converges only at short times.

DOI
10.1103/physrevb.98.024311

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The near-critical unitary dynamics of quantum Ising spin chains in transversal and longitudinal magnetic fields is studied using an artificial neural network representation of the wave function. A focus is set on strong spatial correlations which build up in the system following a quench into the vicinity of the quantum critical point. We compare correlations obtained by optimizing the parameters of the network states with analytical solutions in integrable cases and time-dependent density matrix renormalization group (tDMRG) simulations, as well as with predictions from a semiclassical discrete truncated Wigner analysis. While the semiclassical approach yields quantitatively correct results only at very short times and near zero transverse fields, the neural-network representation is applicable in a much wider regime. However, for quenches close to the quantum critical point the representation becomes inefficient. For nonintegrable models we show that in regimes where tDMRG is limited to short times due to extensive entanglement growth, also the neural-network parametrization converges only at short times.
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Exponential complexity of many-body wave functions limits accurate numerical simulations of real-time dynamics, especially beyond 1D, where rapid entanglement growth poses severe challenges. Neural Quantum States (NQS) have emerged as a powerful approach for real-time dynamics in 2D, but their scalability and accuracy in 3D have remained an open challenge. Here, we establish NQS as a scalable framework for 3D quantum dynamics by introducing a residual-based convolutional architecture tailored to cubic spin lattices. Focusing on the 3D transverse-field Ising model, we demonstrate that NQS reliably capture distinct quench regimes, including collapse-and-revival dynamics and, most challengingly, the dynamics following a sudden quench to the quantum critical point. We perform finite-rate quenches to the critical point on lattices containing up to 100010001000 qubits, an unprecedented system size for numerical simulations of real-time dynamics beyond 1D. This enables the first large-scale numerical demonstration of the 3D quantum Kibble--Zurek mechanism. The QKZM in 3D is particularly intriguing because it lies at the upper critical dimension of the Ising universality class, where the standard power laws are modified by logarithmic factors together with prominent sub-leading logarithmic corrections. By deriving these corrections from renormalization-group flow equations up to two-loop order, we obtain a robust data collapse across all simulated system sizes for the correlation function, the excess energy, and the quantum Fisher information, the latter revealing universal multipartite-entanglement dynamics. In all cases, we find compelling agreement with the expected scaling dimensions. Our findings establish NQS as a scalable and reliable tool for exploring nonequilibrium phenomena in 3D quantum matter and for providing numerical benchmarks for 3D quantum simulators.
The simulation of interacting many-body quantum systems is a central chal- lenge in condensed matter physics due to the exponential growth of the Hilbert space with system size. In this project, we implement a Neural Network Quantum State (NQS) ansatz based on a Complex Restricted Boltzmann Machine (RBM) to solve for the ground state of the 1D Transverse Field Ising Model (TFIM). Using Variational Monte Carlo (VMC) optimization via Stochastic Reconfiguration (SR), we successfully train the network to approximate the ground state wave function across the paramagnetic-ferromagnetic phase transition. We benchmark our results against Exact Diagonalization (ED) for small systems (N = 12), demonstrating high accuracy in ground state energy estimation. Furthermore, we investigate quantum correlations and order parameters, specifically the transverse magnetization mx and longitudinal spin-spin correlations Czz(r), highlighting the RBM’s ability to capture long-range order and critical behavior near the quantum critical point hx/J = 1.
Abstract Neural networks possess formidable representational power, rendering them invaluable in solving complex quantum many-body systems. While they excel at analyzing static solutions, nonequilibrium processes, including critical dynamics during a quantum phase transition, pose a greater challenge for neural networks. To address this, we utilize neural networks and machine learning algorithms to investigate time evolutions, universal statistics, and correlations of topological defects in a one-dimensional transverse-field quantum Ising model. Specifically, our analysis involves computing the energy of the system during a quantum phase transition following a linear quench of the transverse magnetic field strength. The excitation energies satisfy a power-law relation to the quench rate, indicating a proportional relationship between the excitation energy and the kink numbers. Moreover, we establish a universal power-law relationship between the first three cumulants of the kink numbers and the quench rate, indicating a binomial distribution of the kinks. Finally, the normalized kink-kink correlations are also investigated and it is found that the numerical values are consistent with the analytic formula.
Abstract The semi-classical discrete truncated Wigner approximation has recently been proposed as a simulation method for spin-1/2 systems. While it appears to provide a powerful approach which shows promising results in higher dimensions and for systems with long-range interactions, its performance is still not well understood in general. Here we perform a systematic benchmarking on the one-dimensional transverse-field Ising model and point to limitations of the approximation arising after sudden quenches into the quantum critical regime. Our procedure allows to identify the limitations of the semi-classical simulations and with that to determine the regimes and questions where quantum simulators can provide information which is inaccessible to semi-classics.
The quantum dynamics of many-qubit systems is an outstanding problem that has recently driven significant advances in both numerical methods and programmable quantum processing units. In this work, we employ a comprehensive toolbox of state-of-the-art numerical approaches to classically simulate the dynamics of the two-dimensional transverse field Ising model. Our methods include three different tensor network techniques -- matrix product states, tree-tensor networks, and two-dimensional tensor-networks under the belief propagation approximation -- as well as time-dependent variational Monte Carlo with Neural Quantum States. We focus on two paradigmatic dynamical protocols: (i) quantum annealing through a critical point and (ii) post-quench dynamics. Our extensive results show the quantitative predictions of various state-of-the-art numerical methods providing a benchmark for future numerical investigations and experimental studies with the aim to push the limitations on classical and QPUs. In particular, our work connects classical simulability to different regimes associated with quantum dynamics in Rydberg arrays - namely, quasi-adiabatic dynamics, the Kibble-Zurek mechanism, and quantum quenches.