Neural network quantum states such as ansatz wave functions have shown a great deal of promise for finding the ground state of spin models. Recently, work has focused on extending this idea to mixed states for simulating the dynamics of open systems. Most approaches so far have used a purification ansatz where a copy of the system Hilbert space is added, which when traced out gives the correct density matrix. Here we instead present an extension of the restricted Boltzmann machine which directly represents the density matrix in Liouville space. This allows the compact representation of states which appear in mean-field theory. We benchmark our approach on two different versions of the dissipative transverse-field Ising model, which show our ansatz is able to compete with other state-of-the-art approaches. Published by the American Physical Society 2024
Neural network quantum states such as ansatz wave functions have shown a great deal of promise for finding the ground state of spin models. Recently, work has focused on extending this idea to mixed states for simulating the dynamics of open systems. Most approaches so far have used a purification ansatz where a copy of the system Hilbert space is added, which when traced out gives the correct density matrix. Here we instead present an extension of the restricted Boltzmann machine which directly represents the density matrix in Liouville space. This allows the compact representation of states which appear in mean-field theory. We benchmark our approach on two different versions of the dissipative transverse-field Ising model, which show our ansatz is able to compete with other state-of-the-art approaches. Published by the American Physical Society 2024
Abstract In this work we apply deep neural networks to find the non-equilibrium steady state solution to correlated open quantum many-body systems. Motivated by the ongoing search to find more powerful representations of (mixed) quantum states, we design a simple prototypical convolutional neural network and show that parametrizing the density matrix directly with more powerful models can yield better variational ansatz functions and improve upon results reached by neural density operator based on the restricted Boltzmann machine. Hereby we give up the explicit restriction to positive semi-definite density matrices. However, this is fulfilled again to good approximation by optimizing the parameters. The great advantage of this approach is that it opens up the possibility of exploring more complex network architectures that can be tailored to specific physical properties. We show how translation invariance can be enforced effortlessly and reach better results with fewer parameters. We present results for the dissipative one-dimensional transverse-field Ising model and a two-dimensional dissipative Heisenberg model compared to exact values.
The utilization of artificial neural networks to represent quantum many-body wave functions has garnered significant attention, with enormous recent progress for both ground states and nonequilibrium dynamics. However, quantifying state complexity within this neural quantum states framework remains elusive. In this study, we address this key open question from a complementary point of view: Which states are simple to represent with neural quantum states? Concretely, we show on a general level that ground states of mean-field theories with permutation symmetry require only a limited number of independent neural network parameters. We analytically establish that, in the thermodynamic limit, convergence to the ground state of the fully connected transverse-field Ising model (TFIM), the mean-field Ising model, can be achieved with just one single parameter. Expanding our analysis, we explore the behavior of the one-parameter ansatz under breaking of the permutation symmetry. For that purpose, we consider the TFIM with tunable long-range interactions, characterized by an interaction exponent α . We show analytically that the one-parameter ansatz for the neural quantum state still accurately captures the ground state for a whole range of values for 0≤α≤1 , implying a mean-field description of the model in this regime. Published by the American Physical Society 2024
One of the main challenges of quantum many-body physics is that the dimensionality of the Hilbert space grows exponentially with the system size, which makes it extremely difficult to solve the Schrodinger equations of the system. But typically, many physical systems have a simplified internal structure which makes the parameters needed to characterize their ground states exponentially smaller. This makes many numerical methods possible in capture the physics of the system. Among these modern numerical techniques, neural networks, which show great power in approximating functions and extracting features of the big data, are now attracting many interests. Neural network representation of quantum many-body states shows great potential in solving some traditionally difficult quantum problems involving large number of freedoms. In this work, we briefly review the progress of using the artificial neural network to build quantum many-body ansatz states. We take Boltzmann machine representation as a prototypical example to illustrate various aspects of neural network representation of quantum many-body states. We first briefly review the classical neural networks, then we illustrate how to use neural networks to represent quantum states and density operators. Some physical properties of the neural network states, including entanglement features, representational power, and the relation with tensor network states, are discussed. For applications, we briefly review the progress of many-body calculating based on neural network states, neural network state approach to tomography, and also the classical simulation of quantum computing based on Boltzmann machine states. At the end of the work, some outlooks and open problems are given.
Machine learning is actively being explored for its potential to design, validate, and even hybridize with near-term quantum devices. A central question is whether neural networks can provide a tractable representation of a given quantum state of interest. When true, stochastic neural networks can be employed for many unsupervised tasks, including generative modeling and state tomography. However, to be applicable for real experiments, such methods must be able to encode quantum mixed states. Here, we parametrize a density matrix based on a restricted Boltzmann machine that is capable of purifying a mixed state through auxiliary degrees of freedom embedded in the latent space of its hidden units. We implement the algorithm numerically and use it to perform tomography on some typical states of entangled photons, achieving fidelities competitive with standard techniques.
Artificial neural netwokrs have proven to be a powerful tool with applications in various fields: from data processing and optical character recognition to traffic control or prediction of dynamic systems evolution. Recently, they have demonstrated to be a promising approach to solving quantum mechanical spin systems where neural network is used as an Ansatz representing the wave function. In the case of Heisenberg model, an architecture called restricted Boltzmann machine was successful, thus in this work, we are going to focus on its generalization, J1-J2 model in one as well as two dimensions, in order to find out whether restricted Boltzmann machines can be efficiently used also for frustrated spin systems.
We present a general variational approach to determine the steady state of open quantum lattice systems via a neural-network approach. The steady-state density matrix of the lattice system is constructed via a purified neural-network Ansatz in an extended Hilbert space with ancillary degrees of freedom. The variational minimization of cost functions associated to the master equation can be performed using a Markov chain Monte Carlo sampling. As a first application and proof of principle, we apply the method to the dissipative quantum transverse Ising model.