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arXiv:1910.09241 · Physical Review A

Complexity of the steady state of weakly symmetric open quantum lattices

Davide Nigro

Published Feb 19, 2020• 8 citations
arXivHTMLPDF

Abstract

We investigate the properties of Lindblad equations on $d$-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 $G$, 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 $XYZ$ 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.

DOI
10.1103/physreva.101.022109

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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.
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.
This work is concerned with determination of the steady-state structure of time-independent Lindblad master equations, especially those possessing more than one steady state. The approach here is to treat Lindblad systems as generalizations of unitary quantum mechanics, extending the intuition of symmetries and conserved quantities to the dissipative case. We combine and apply various results to obtain an exhaustive characterization of the infinite-time behavior of Lindblad evolution, including both the structure of the infinite-time density matrix and its dependence on initial conditions. The effect of the environment in the infinite-time limit can therefore be tracked exactly for arbitrary state initialization and without knowledge of dynamics at intermediate time. As a consequence, sufficient criteria for determining the steady state of a Lindblad master equation are obtained. These criteria are knowledge of the initial state, a basis for the steady-state subspace, and all conserved quantities. We give examples of two-qubit dissipation and single-mode ddd-photon absorption where all quantities are determined analytically. Applications of these techniques to quantum information, computation, and feedback control are discussed.
Many-body open quantum systems are exposed to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. As the exact treatment of such models is generally unfeasible, it is favourable to formulate an approximate description by means of the dynamics of the reduced density operator of the system. When studying models with a weak coupling to a memoryless environment, the existence of such solution is granted and the reduced dynamics is governed by the Lindblad quantum master equation. In recent years, open quantum systems have evolved into a major field of studies. Focus of these studies are the characterization of emergent phenomena and dissipative phase transitions, as well as the ongoing debate about whether quantum computing schemes are still hard to simulate classically --- and thus achieve quantum supremacy --- when in presence of some degree of noise-induced decoherence. However, the challenge posed by this task lies in the complexity of the density matrix that increases exponentially with the system size, and the quest for efficient numerical methods is a research field that is still in its infancy. Here, we first develop a real-time full configuration interaction quantum Monte Carlo technique that stems from a class of methods generally known as projector Monte Carlo. The approach enables the stochastic sampling of the Lindblad time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the non-equilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce statistical error. We demonstrate the efficiency of the approach by applying it to the dissipative two-dimensional XYZ spin-1/2 model on a lattice. As the importance sampling of projector approaches is often combined with variational results, we then introduce a novel method that is based on the 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 Lindblad quantum master equation. We test the effectiveness of the method by modeling the steady state of the dissipative two-dimensional XYZ spin-1/2 model through a dissipative phase transition, and also the real-time dynamics of the dissipative Ising model. In addition, we discuss the application of the developed methods and the open questions of the field.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
We provide a rigorous construction of Markovian master equations for a wide class of quantum systems that encompass quadratic models of finite size, linearly coupled to an environment modeled by a set of independent thermal baths. Our theory can be applied for both fermionic and bosonic models in any number of physical dimensions and does not require any particular spatial symmetry of the global system. We show that, for nondegenerate systems under a full secular approximation, the effective Lindblad operators are the normal modes of the system, with coupling constants that explicitly depend on the transformation matrices that diagonalize the Hamiltonian. Both the dynamics and the steady-state (guaranteed to be unique) properties can be obtained with a polynomial amount of resources in the system size. We also address the particle and energy current flowing through the system in a minimal two-bath scheme and find that they hold the structure of Landauer's formula, being thermodynamically consistent.
We consider finite-dimensional many-body quantum systems described by time-independent Hamiltonians and Markovian master equations, and present a systematic method for constructing smaller-dimensional, reduced models that e x a c t l y reproduce the time evolution of a set of initial conditions or observables of interest. Our approach exploits Krylov operator spaces and their extension to operator algebras, and may be used to obtain reduced linear models of minimal dimension, well-suited for simulation on classical computers, or reduced quantum models that preserve the structural constraints of physically admissible quantum dynamics, as required for simulation on quantum computers. Notably, we prove that the reduced quantum-dynamical generator is still in Lindblad form. By introducing a new type of observable-dependent symmetries , we show that our method provides a non-trivial generalization of techniques that leverage symmetries, unlocking new reduction opportunities. We quantitatively benchmark our method on paradigmatic open many-body systems of relevance to condensed-matter and quantum-information physics. In particular, we demonstrate how our reduced models can quantitatively describe decoherence dynamics in central-spin systems coupled to structured environments, magnetization transport in boundary-driven dissipative spin chains, and unwanted error dynamics on information encoded in a noiseless quantum code.