Abstract The exact treatment of Markovian models of complex systems requires knowledge of probability distributions exponentially large in the number of components n. Mean-field approximations provide an effective reduction in complexity of the models, requiring only a number of phase space variables polynomial in system size. However, this comes at the cost of losing accuracy close to critical points in the systems dynamics and an inability to capture correlations in the system. In this work, we introduce a tunable approximation scheme for Markovian spreading models on networks based on matrix product states (MPSs). By controlling the bond dimensions of the MPS, we can investigate the effective dimensionality needed to accurately represent the exact 2 n dimensional steady-state distribution. We introduce the entanglement entropy as a measure of the compressibility of the system and find that it peaks just after the phase transition on the disordered side, in line with the intuition that more complex states are at the ‘edge of chaos’. We compare the accuracy of the MPS with exact methods on different types of small random networks and with Markov chain Monte Carlo methods for a simplified version of the railway network of the Netherlands with 55 nodes. The MPS provides a systematic way to tune the accuracy of the approximation by reducing the dimensionality of the systems state vector, leading to an improvement over second-order mean-field approximations for sufficiently large bond dimensions.
Abstract The exact treatment of Markovian models of complex systems requires knowledge of probability distributions exponentially large in the number of components n. Mean-field approximations provide an effective reduction in complexity of the models, requiring only a number of phase space variables polynomial in system size. However, this comes at the cost of losing accuracy close to critical points in the systems dynamics and an inability to capture correlations in the system. In this work, we introduce a tunable approximation scheme for Markovian spreading models on networks based on matrix product states (MPSs). By controlling the bond dimensions of the MPS, we can investigate the effective dimensionality needed to accurately represent the exact 2 n dimensional steady-state distribution. We introduce the entanglement entropy as a measure of the compressibility of the system and find that it peaks just after the phase transition on the disordered side, in line with the intuition that more complex states are at the ‘edge of chaos’. We compare the accuracy of the MPS with exact methods on different types of small random networks and with Markov chain Monte Carlo methods for a simplified version of the railway network of the Netherlands with 55 nodes. The MPS provides a systematic way to tune the accuracy of the approximation by reducing the dimensionality of the systems state vector, leading to an improvement over second-order mean-field approximations for sufficiently large bond dimensions.
The contact process is an emblematic model of a nonequilibrium system, containing a phase transition between inactive and active dynamical regimes. In the epidemiological context, the model is known as the susceptible-infected-susceptible model, and it is widely used to describe contagious spreading. In this work, we demonstrate how accurate and efficient representations of the full probability distribution over all configurations of the contact process on a one-dimensional chain can be obtained by means of matrix product states (MPSs). We modify and adapt MPS methods from many-body quantum systems to study the classical distributions of the driven contact process at late times. We give accurate and efficient results for the distribution of large gaps, and illustrate the advantage of our methods over Monte Carlo simulations. Furthermore, we study the large deviation statistics of the dynamical activity, defined as the total number of configuration changes along a trajectory, and investigate quantum-inspired entropic measures, based on the second Rényi entropy.
The contact process is an emblematic model of a non-equilibrium system, containing a phase transition between inactive and active dynamical regimes. In the epidemiological context, the model is known as the susceptible-infected-susceptible (SIS) model, and widely used to describe contagious spreading. In this work, we demonstrate how accurate and efficient representations of the full probability distribution over all configurations of the contact process on a one-dimensional chain can be obtained by means of Matrix Product States (MPS). We modify and adapt MPS methods from many-body quantum systems to study the classical distributions of the driven contact process at late times. We give accurate and efficient results for the distribution of large gaps, and illustrate the advantage of our methods over Monte Carlo simulations. Furthermore, we study the large deviation statistics of the dynamical activity, defined as the total number of configuration changes along a trajectory, and investigate quantum-inspired entropic measures, based on the second Rényi entropy.
Markov chain Monte Carlo (MCMC) is a powerful tool for sampling from complex probability distributions. Despite its versatility, MCMC often suffers from strong autocorrelation and the negative sign problem, leading to slowing down the convergence of statistical error. We propose a novel MCMC formulation based on tensor network representations to reduce the population variance and mitigate these issues systematically. By introducing stochastic projectors into the tensor network framework and employing Markov chain sampling, our method eliminates the systematic error associated with low-rank approximation in tensor contraction while maintaining the high accuracy of the tensor network method. We demonstrate the effectiveness of the proposed method on the two-dimensional Ising model, achieving an exponential reduction in statistical error with increasing bond dimension cutoff. Furthermore, we address the sign problem in systems with negative weights, showing significant improvements in average signs as bond dimension cutoff increases. The proposed framework provides a robust solution for accurate statistical estimation in complex systems, paving the way for broader applications in computational physics and beyond.
Abstract Tensor network methods as presented in our open source matrix product states software have opened up the possibility to study many-body quantum physics in one and quasi-one-dimensional systems in an easily accessible package similar to density functional theory codes but for strongly correlated dynamics. Here, we address methods which allow one to capture the full entanglement without truncation of the Hilbert space. Such methods are suitable for validation of and comparisons to tensor network algorithms, but especially useful in the case of new kinds of quantum states with high entanglement violating the truncation in tensor networks. Quantum cellular automata are one example for such a system, characterized by tunable complexity, entanglement, and a large spread over the Hilbert space. Beyond the evolution of pure states as a closed system, we adapt the techniques for open quantum systems simulated via the Lindblad master equation. We present three algorithms for solving closed-system many-body time evolution without truncation of the Hilbert space. Exact diagonalization methods have the advantage that they not only keep the full entanglement but also require no approximations to the propagator. Seeking the limits of a maximal number of qubits on a single core, we use Trotter decompositions or Krylov approximation to the exponential of the Hamiltonian. All three methods are also implemented for open systems represented via the Lindblad master equation built from local channels. We show their convergence parameters and focus on efficient schemes for their implementations including Abelian symmetries, e.g. symmetry used for number conservation in the Bose–Hubbard model or discrete symmetries in the quantum Ising model. We present the thermalization timescale in the long-range quantum Ising model as a key example of how exact diagonalization contributes to novel physics.
We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions.Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory.In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames.Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the Rényi entropies and negativities as well as their symmetry-resolved components.We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry.Although the cost is exponential in the subsystem size, it is sufficiently moderate so that-in contrast with other approaches-accurate results can be obtained on a personal computer for relatively large subsystem sizes.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.