Variational Neural Quantum Statesfor the one-dimensional TransverseField Ising Model : Variationsbaserade neurala kvanttillstånd för den endi-mensionella Transverse Field Ising modellen
The Transverse Field Ising Model (TFIM) is an important quantum many-body model used to study the competition between spin-spin interactions and quantum fluctuations generated by a transverse field. Due to the exponential growth of the Hilbert space with increasing system size, numerical methods quickly become computationally infeasible motivating the use of approximate approaches. In this work Neural Quantum States were used as a variational method to approximate the ground-state properties of the one-dimensional TFIM. The NQS was represented using a feedforward Neural Network and optimized through Monte Carlo sampling together with stochastic reconfiguration. The implementation was done through the NetKet framework. The obtained results showed stable convergence of the variational energy and good agreement with exact diagonalization for the smaller system. The quantum phase transition was observed through the order parameter and spin-spin correlations. Finite-size effects were also observed when comparing the two system sizes. The smaller system more closely approaches the symmetric cat-like ground-state. In contrast, the larger system tends to remain near one symmetry-broken configuration. As a result, the smaller system showed stronger spin expectation fluctuations and a smoother behavior near the phase transition. The result demonstrates that the NQS approach is a good approximate method in studying the ground-state physics of the one-dimensional TFIM.
The Transverse Field Ising Model (TFIM) is an important quantum many-body model used to study the competition between spin-spin interactions and quantum fluctuations generated by a transverse field. Due to the exponential growth of the Hilbert space with increasing system size, numerical methods quickly become computationally infeasible motivating the use of approximate approaches. In this work Neural Quantum States were used as a variational method to approximate the ground-state properties of the one-dimensional TFIM. The NQS was represented using a feedforward Neural Network and optimized through Monte Carlo sampling together with stochastic reconfiguration. The implementation was done through the NetKet framework. The obtained results showed stable convergence of the variational energy and good agreement with exact diagonalization for the smaller system. The quantum phase transition was observed through the order parameter and spin-spin correlations. Finite-size effects were also observed when comparing the two system sizes. The smaller system more closely approaches the symmetric cat-like ground-state. In contrast, the larger system tends to remain near one symmetry-broken configuration. As a result, the smaller system showed stronger spin expectation fluctuations and a smoother behavior near the phase transition. The result demonstrates that the NQS approach is a good approximate method in studying the ground-state physics of the one-dimensional TFIM.
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.
Ising model has been successful in describing ferromagnetism and its phase transition to paramagnet. At the critical point, the free energy density function and its derivatives diverge. Their behaviour near the critical point are described by power-laws with associated critical exponents. In many cases, the critical exponents can be determined from analytic solutions via conventional renormalization groups methods or from Monte Carlo simulations. However, for quantum many-body systems, very few are tractable to analytical solutions. The quantum many-body wavefunction belongs to large dimensional Hilbert space that increases exponentially with system size. If the Hamiltonian is gapped and only local interaction is considered, then the wavefunction can be efficiently truncated. Tensor Network formalism provides a scheme to truncate the less important degrees of freedom via Singular Value Decomposition (SVD) of the density matrix. In this study, we investigated the thermodynamic properties and phase transition of one-dimensional transverse-field quantum Ising model (1D-tQIM) under the finite-size effect and random coupling strength. Starting with Matrix Product States (MPS) as a wavefunction ansatz, the Density Matrix Renormalization Group algorithm is applied to the MPS. The variational algorithm, which iteratively performs SVDs and truncation at each bond, approximates the ground state MPS wavefunction. All quantum observables are calculated from the contraction of the resultant ground state MPS. Although theoretically, divergence at critical points only happen in an infinite system, one can obtain the critical exponents through simulation of finite-size 1D-tQIM. Using the analytic solution as a benchmark, we compared the finite-size effects of the system using finite-size scaling analysis and MPS methods. The critical exponents of 1D-tQIM are independently calculated and compared with the analytical results. Thermodynamic quantities such as magnetization, susceptibility and correlation function are calculated for system sizes of 20, 40, 60, 80, 100 and 120 spins. We determined the respective critical exponents:
The quantum phase transitions provide a paradigm for studying collective quantum phenomena that are a result of competing non-commuting interactions. This paper will study the ground state properties and quantum critical dynamics of the one-dimensional transverse field Ising model through a combined perspective that includes exact diagonalisation, variational quantum eigensolver (VQE) simulations, and simulations on realistic physical quantum devices. We focus on a lattice of four spins, where we calculate the ground-state energies, magnetic order parameters and correlation functions at uniformly applied conditions, which is repeated by all systems. Precise diagonalisation provides both a benchmark, which is symmetry-conserving, and a depth-two, physics inspired variational approximation, which provides simulations accessible to hardware. The circuits that have been optimised identically are then placed on the IQM Garnet quantum processor, using a resource-efficient batched protocol. We find that the ground-state energies of shallow variational circuits are reliably captured by the circuit over the entire parameter space; the magnetic arrangement parameters and observables sensitive to correlation signal significantly more noise. The error analysis of quantitative analysis reveals a strong broadening of critical crossover on hardware, which is consistent with the noise attenuation of long-range correlations. These findings highlight the current capabilities as well as the fundamental limitations of noisy intermediate-scale quantum systems in modelling quantum critical phenomena as a benchmark to future enhancements in obtaining quantum hardware and quantum algorithms development.
Recent research has demonstrated the usefulness of neural networks as variational ansatz functions for quantum many-body states. However, high-dimensional sampling spaces and transient autocorrelations confront these approaches with a challenging computational bottleneck. Compared to conventional neural networks, physical-model devices offer a fast, efficient and inherently parallel substrate capable of related forms of Markov chain Monte Carlo sampling. Here, we demonstrate the ability of a neuromorphic chip to represent the ground states of quantum spin models by variational energy minimization. We develop a training algorithm and apply it to the transverse field Ising model, showing good performance at moderate system sizes (N≤10). A systematic hyperparameter study shows that scalability to larger system sizes mainly depends on sample quality, which is limited by temporal parameter variations on the analog neuromorphic chip. Our work thus provides an important step towards harnessing the capabilities of neuromorphic hardware for tackling the curse of dimensionality in quantum many-body problems.
Abstract Along the way initiated by Carleo and Troyer [G. Carleo and M. Troyer, Science 355 (2017) 602], we construct the neural-network quantum state of transverse-field Ising model (TFIM) by an unsupervised machine learning method. Such a wave function is a map from the spin-configuration space to the complex number field determined by an array of network parameters. To get the ground state of the system, values of the network parameters are calculated by a Stochastic Reconfiguration (SR) method. We provide for this SR method an understanding from action principle and information geometry aspects. With this quantum state, we calculate key observables of the system, the energy, correlation function, correlation length, magnetic moment, and susceptibility. As innovations, we provide a high efficiency method and use it to calculate entanglement entropy (EE) of the system and get results consistent with previous work very well .
Neural Quantum States (NQS) are powerful tools used to represent complex quantum many-body states in an increasingly wide range of applications. However, despite their popularity, at present only a rudimentary understanding of their limitations exists. In this work, we investigate the dependence of NQS on the choice of the computational basis, focusing on restricted Boltzmann machines. Considering a family of rotated Hamiltonians corresponding to the paradigmatic transverse-field Ising model, we discuss the properties of ground states responsible for the dependence of NQS performance, namely the presence of ground state degeneracies as well as the uniformity of amplitudes and phases, carefully examining their interplay. We identify that the basis-dependence of the performance is linked to the convergence properties of a cluster or cumulant expansion of multi-spin operators -- providing a framework to directly connect physical, basis-dependent properties, to performance itself. Our results provide insights that may be used to gauge the applicability of NQS to new problems and to identify the optimal basis for numerical computations.