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Low rank approximation for the numerical simulation of high dimensional Lindblad equations

Claude Le Bris, Pierre Rouchon

Published Jul 19, 2012• 0 citations
OpenAlex

Abstract

We suggest a general, simplified, computationally efficient approach to simulate high dimensional Lindblad equations. The approach is based on a low-rank approximation of the density. The practical efficiency is demonstrated on the example of oscillation revivals of atoms interacting resonantly with a slightly damped coherent quantized field.

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Low-rank numerical approximations for high-dimensional Lindblad equations

Claude Le Bris, Pierre Rouchon

A systematic numerical approach to approximate high-dimensional Lindblad equations is described. It is based on a deterministic rank $m$ approximation of the density operator, the rank $m$ being the only parameter to adjust. From a known initial density operator, this rank $m$ approximation gives at each time step an estimate of its largest $m$ eigenvalues with their associated eigenvectors. A numerical integration scheme is also proposed. Its numerical efficiency in the case of a rank $m=12$ approximation is demonstrated for oscillation revivals of 50 atoms interacting resonantly with a slightly damped coherent quantized field of 200 photons.

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We present a twofold contribution to the numerical simulation of Lindblad equations. First, an adaptive numerical approach to approximate Lindblad equations using low-rank dynamics is described: a deterministic low-rank approximation of the density operator is computed, and its rank is adjusted dynamically, using an on-the-fly estimator of the error committed when reducing the dimension. On the other hand, when the intrinsic dimension of the Lindblad equation is too high to allow for such a deterministic approximation, we combine classical ensemble averages of quantum Monte Carlo trajectories and a denoising technique. Specifically, a variance reduction method based on the consideration of a low-rank dynamics as a control variate is developed. Numerical tests for quantum collapse and revivals show the efficiency of each approach, along with the complementarity of the two approaches.

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We suggest a general, simplified, computationally efficient approach to simulate high dimensional Lindblad equations. The approach is based on a low-rank approximation of the density. The practical efficiency is demonstrated on the example of oscillation revivals of atoms interacting resonantly with a slightly damped coherent quantized field.
A systematic numerical approach to approximate high-dimensional Lindblad equations is described. It is based on a deterministic rank mmm approximation of the density operator, the rank mmm being the only parameter to adjust. From a known initial density operator, this rank mmm approximation gives at each time step an estimate of its largest mmm eigenvalues with their associated eigenvectors. A numerical integration scheme is also proposed. Its numerical efficiency in the case of a rank m=12m=12m=12 approximation is demonstrated for oscillation revivals of 50 atoms interacting resonantly with a slightly damped coherent quantized field of 200 photons.
We present a twofold contribution to the numerical simulation of Lindblad equations. First, an adaptive numerical approach to approximate Lindblad equations using low-rank dynamics is described: a deterministic low-rank approximation of the density operator is computed, and its rank is adjusted dynamically, using an on-the-fly estimator of the error committed when reducing the dimension. On the other hand, when the intrinsic dimension of the Lindblad equation is too high to allow for such a deterministic approximation, we combine classical ensemble averages of quantum Monte Carlo trajectories and a denoising technique. Specifically, a variance reduction method based on the consideration of a low-rank dynamics as a control variate is developed. Numerical tests for quantum collapse and revivals show the efficiency of each approach, along with the complementarity of the two approaches.
We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.
Abstract. The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems whose states are described by density matrices. These solution matrices are characterized by semipositiveness and trace preserving properties, which must be guaranteed in any physically meaningful numerical simulation. In this paper, novel full- and low-rank exponential Euler integrators are developed for approximating the Lindblad equation that preserve positivity and trace unconditionally. Theoretical results are presented that provide sharp error estimates for the two classes of exponential integration methods. Results of numerical experiments are discussed that illustrate the effectiveness of the proposed schemes, beyond present state-of-the-art capabilities.
The simulation of many-body open quantum systems is key to solving numerous outstanding problems in physics, chemistry, material science, and in the development of quantum technologies. Near-term quantum computers may bring considerable advantage for the efficient simulation of their static and dynamical properties, thanks to hybrid quantum-classical variational algorithms to approximate the dynamics of the density matrix describing the quantum state in terms of an ensemble average. Here, a variational quantum algorithm is developed to simulate the real-time evolution of the density matrix governed by the Lindblad master equation, under the assumption that the quantum state has a bounded entropy along the dynamics, entailing a low-rank representation of its density matrix. The algorithm encodes each pure state of the statistical mixture as a parametrized quantum circuit, and the associated probabilities as additional variational parameters stored classically, thereby requiring a significantly lower number of qubits than algorithms where the full density matrix is encoded in the quantum memory. Two variational ansatze are proposed, and their effectiveness is assessed in the simulation of the dynamics of a 2D dissipative transverse field Ising model. The results underscore the algorithm's efficiency in simulating the dynamics of open quantum systems in the low-rank regime with limited quantum resources on a near-term quantum device.