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arXiv:1802.05931 · Physical Review A

Driven-dissipative quantum Monte Carlo method for open quantum systems

Alexandra Nagy, Vincenzo Savona

Published May 30, 2018• 42 citations
arXivHTMLPDF

Abstract

We develop a real-time full configuration-interaction quantum Monte Carlo approach to model driven-dissipative open quantum systems with Markovian system-bath coupling. The method enables stochastic sampling of the Liouville--von Neumann time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the nonequilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce the statistical error. Finally, we demonstrate the efficiency of our approach by applying it to the driven-dissipative two-dimensional $XYZ$ spin-$\frac{1}{2}$ model on a lattice.

DOI
10.1103/physreva.97.052129

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We develop a real-time full configuration-interaction quantum Monte Carlo approach to model driven-dissipative open quantum systems with Markovian system-bath coupling. The method enables stochastic sampling of the Liouville--von Neumann time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the nonequilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce the statistical error. Finally, we demonstrate the efficiency of our approach by applying it to the driven-dissipative two-dimensional XYZXYZXYZ spin-12\frac{1}{2}21​ model on a lattice.
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.
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.
Complex quantum phenomena can often be understood more easily by identifying a relevant system and an environment which interacts with the system. Numerically exact methods for these dissipative quantum systems are discussed with particular focus on their dynamics. Using the dissipative two‐state system and charge transport through a mesoscopic metallic island as illustrative examples, we present algorithms based on the importance sampling of Feynman path integrals as well as novel techniques relying on the numerical simulation of stochastic Liouville‐von Neumann equations.
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We address the real-time dynamics of lattice quantum spin models coupled to single or multiple Markovian dissipative reservoirs using the method of closed hierarchies of correlation functions. This approach allows us to solve a number of quantum spin models exactly in arbitrary dimensions, which is illustrated explicitly with two examples of driven-dissipative systems. We investigate their respective nonequilibrium steady states as well as the full real-time evolution on unprecedented system sizes. Characteristic time scales are derived analytically, which allows us to understand the nontrivial finite-size scaling of the dissipative gap. The corresponding scaling exponents are confirmed by solving numerically for the full real-time evolution of two-point correlation functions.
We develop a new implementation of the real-time full configuration interaction quantum Monte Carlo method, and apply it to the dynamics of the Heisenberg spin chain and the spectrum of the one-dimensional Fröhlich polaron. This is a general method for the numerical integration of the Schrödinger equation to evolve quantum systems in real time, introduced by Guther et al. as an extension of the imaginary-time version of the algorithm that finds ground-state properties. We introduce a new integrator for improved efficiency, and a scaling procedure for better control of simulations. We study the accuracy of the method, including the effect of discretising time and of the stochastic errors introduced by the Monte Carlo elements of the algorithm. Comparing the method to a hardware quantum simulator for the quench dynamics of a Heisenberg spin chain, we find that for this model the discretisation and stochastic errors grow quickly, limiting the effectiveness of the algorithm for finding accurate evolution over long times. We find that spectral information can be generated successfully using an energy-dependent damping, with either a tilted complex contour time evolution or a strong second-order damping. Specifically, we reproduce the low-energy spectrum of the Fr ̈ohlich polaron model and confirm the existence of a bound excited state.