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arXiv:2101.03511 · Physical Review A

Invariant neural network ansatz for weakly symmetric open quantum lattices

Davide Nigro

Published Jun 3, 2021• 8 citations
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Abstract

We consider an open quantum system of $N$ interacting spin-1/2 particles on a $d$-dimensional lattice whose time evolution is (weakly) symmetric under the action of a finite group $G$ that is a subgroup of all the possible permutations of the lattice sites. We define a neural network representation for the system density operator that explicitly accounts for the system symmetries, and we show that its properties are preserved when using variational techniques such those recently used in the optimization of restricted-Boltzmann machines (RBMs). As a proof of principle, we demonstrate the validity of our approach by determining the steady state structure of the one-dimensional dissipative XYZ model in the presence of a uniform magnetic field and uniform dissipation. We first show a direct comparison between the results obtained by optimizing a RBM representation for the density operator and those obtained by means of the invariant neural network defined in the present paper, and then we show results concerning the convergence to the unique steady state configuration when using our approach only. Numerical results support the conjecture that the invariant Ansatz defined in this paper is more expressive that a standard RBM and can be optimized with lesser numerical efforts.

DOI
10.1103/physreva.103.062406

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We consider an open quantum system of NNN interacting spin-1/2 particles on a ddd-dimensional lattice whose time evolution is (weakly) symmetric under the action of a finite group GGG that is a subgroup of all the possible permutations of the lattice sites. We define a neural network representation for the system density operator that explicitly accounts for the system symmetries, and we show that its properties are preserved when using variational techniques such those recently used in the optimization of restricted-Boltzmann machines (RBMs). As a proof of principle, we demonstrate the validity of our approach by determining the steady state structure of the one-dimensional dissipative XYZ model in the presence of a uniform magnetic field and uniform dissipation. We first show a direct comparison between the results obtained by optimizing a RBM representation for the density operator and those obtained by means of the invariant neural network defined in the present paper, and then we show results concerning the convergence to the unique steady state configuration when using our approach only. Numerical results support the conjecture that the invariant Ansatz defined in this paper is more expressive that a standard RBM and can be optimized with lesser numerical efforts.
We investigate the properties of Lindblad equations on ddd-dimensional lattices supporting a unique steady-state configuration. We consider the case of a time evolution weakly symmetric under the action of a finite group GGG, which is also a symmetry group for the lattice structure. We show that in such case the steady state belongs to a relevant subspace, and provide an explicit algorithm for constructing an orthonormal basis of such set. As explicitly shown for a spin-1/2 system, the dimension of such subspace is extremely smaller than the dimension of the set of square operators. As a consequence, by projecting the dynamics within such set, the steady-state configuration can be determined with a considerably reduced amount of resources. We demonstrate the validity of our theoretical results by determining the exact structure of the steady-state configuration of the two-dimensional XYZXYZXYZ model in the presence of uniform dissipation, with and without magnetic fields, up to a number of sites equal to 12. Although in this work we consider explicitly only spin-1/2 systems, our approach can be exploited in the characterization of arbitrary spin systems and fermion and boson systems (with truncated Fock space), as well as many-particle systems with degrees of freedom having different statistical properties.
Abstract The variational wave functions based on neural networks have recently started to be recognized as a powerful ansatz to represent quantum many-body states accurately. In order to show the usefulness of the method among all available numerical methods, it is imperative to investigate the performance in challenging many-body problems for which the exact solutions are not available. Here, we construct a variational wave function with one of the simplest neural networks, the restricted Boltzmann machine (RBM), and apply it to a fundamental but unsolved quantum spin Hamiltonian, the two-dimensional J 1 – J 2 Heisenberg model on the square lattice. We supplement the RBM wave function with quantum-number projections, which restores the symmetry of the wave function and makes it possible to calculate excited states. Then, we perform a systematic investigation of the performance of the RBM. We show that, with the help of the symmetry, the RBM wave function achieves state-of-the-art accuracy both in ground-state and excited-state calculations. The study shows a practical guideline on how we achieve accuracy in a controlled manner.
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.
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.
The possibility to simulate the properties of many-body open quantum systems with a large number of degrees of freedom (d.o.f.) is the premise to the solution of several outstanding problems in quantum science and quantum information. The challenge posed by this task lies in the complexity of the density matrix increasing exponentially with the system size. Here, we develop a variational method to efficiently simulate the nonequilibrium steady state of Markovian open quantum systems based on variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the quantum master equation. We test the effectiveness of the method by modeling the two-dimensional dissipative XYZ spin model on a lattice.
We propose a neural-network variational quantum algorithm to simulate the time evolution of quantum many-body systems.Based on a modified restricted Boltzmann machine (RBM) wave function ansatz, the proposed algorithm can be efficiently implemented in near-term quantum computers with low measurement cost.Using a qubit recycling strategy, only one ancilla qubit is required to represent all the hidden spins in an RBM architecture.The variational algorithm is extended to open quantum systems by employing a stochastic Schrödinger equation approach.Numerical simulations of spin-lattice models demonstrate that our algorithm is capable of capturing the dynamics of closed and open quantum many-body systems with high accuracy without suffering from the vanishing gradient (or "barren plateau") issue for the considered system sizes.