Rapidly growing performance and memory capacity of modern supercomputers open new perspectives for numerical studies of open quantum systems. These systems are in contact with their environments and their modeling is typically based on Markovian kinetic equations, describing the evolution of the system density operators. Additional to the exponential growth of the system dimension
N with the number of the system's parts, the computational complexity scales quadratically with
N, since we have to deal now with super-operators represented by
N2×N2 matrices (instead of standard
N×N matrices of operators). In this paper we consider the so-called Lindblad equation, a popular tool to model dynamics of open systems in quantum optics and superconducting quantum physics. Using the generalized Gell-Mann matrices as a basis, we transform the original Lindblad equation into a system of ordinary differential equations (ODEs) with real coefficients. Earlier, we proposed an implementation of this idea with the complexity of computations scaling as
O(N5log(N)) for dense systems and
O(N3log(N)) for sparse systems. However, infeasible memory costs remained an obstacle on the way to large models. Here we present a new parallel algorithm based on cluster manipulations with large amount of data needed to transform the original Lindblad equation into an ODE system. We demonstrate that the algorithm allows us to integrate sparse systems of dimension
N=2000 and dense systems of dimension
N=200 by using up to 25 nodes with 64 GB RAM per node. We also managed to perform large-scale supercomputer sampling to study the spectral properties of an ensemble of random Lindbladians for
N=200, which is impossible without the distribution of memory costs among cluster nodes.
Dynamics of an open N-state quantum system is often modeled with a Markovian master equation describing the evolution of the system density operator. By using generators of SU(N) group as a basis, the density operator can be transformed into a real-valued "coherence-vector." A generator of the dissipative evolution, so-called "Lindbladian," can be expanded over the same basis and recast in the form of a real matrix. Together, these expansions result is a nonhomogeneous system of N^{2}-1 real-valued linear ordinary differential equations. Now one can, e.g., implement standard high-performance algorithms to integrate the system of equations forward in time while being sure in exact preservation of the trace (norm) and Hermiticity of the density operator. However, when performed in a straightforward way, the expansion turns to be an operation of the time complexity O(N^{10}). The complexity can be reduced when the number of dissipative operators is independent of N, which is often the case for physically meaningful models. Here we present an algorithm to transform quantum master equation into a system of real-valued differential equations and propagate it forward in time. By using a specific scalable model, we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with N=10^{3} states on a single node of a computer cluster.
The time evolution of Markovian open quantum systems is governed by Lindblad master equations, whose solution can be formally written as the Lindbladian exponential acting on the initial density matrix. By expanding this Lindbladian exponential into the Taylor series, we propose a generic method for integrating Lindblad master equations. In this method, the series is truncated, retaining a finite number of terms, and the iterative actions of Lindbladian on the density matrix follow the corresponding master equation. Our method offers significant improvements in numerical efficiencies both in memory cost and computation time, especially for systems with many degrees of freedom. Moreover, our proposed method can be integrated seamlessly with tensor networks. Two illustrative examples, a two-level system exhibiting damped Rabi oscillations and a driven dissipative Heisenberg chain, are used to demonstrate the validity of our method. The superiority of our method is benchmarked with detailed performance tests.
Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME)
dtdρt=Lρt, where
L is a linear (super-)operator and
ρt is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state
ρ∞ (such that
Lρ∞=0), the first few excited states, and trajectories
t↦ρt. In finite dimension,
ρt is an
n×n matrix,
L thus typically costs
n4 to store explicitly as a dense matrix, and
O(n6) to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However,
L usually costs only
O(n3) to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution
S, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map
Φ whose fixed point is directly related to
ρ∞, all the other eigenvalues having smaller magnitude. The map
Φ is thus well suited to iterative methods, and
ρ∞ can be found in a few Arnoldi iterations. Using the same inverse map
S−1 as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like
O(n3) per iteration and offer state-of-the-art performance on CPU and GPU.
In this paper, we investigate the problem of simulating open system dynamics governed by the well-known Lindblad master equation. In our prequel paper [Patel and Wilde, Open Sys. & Info. Dyn., 30:2350010 (2023)], we introduced an input model in which Lindblad operators are encoded into pure quantum states, called program states, and we also introduced a method, called wave matrix Lindbladization, for simulating Lindbladian evolution by means of interacting the system of interest with these program states. Therein, we focused on a simple case in which the Lindbladian consists of only one Lindblad operator and a Hamiltonian. Here, we extend the method to simulating general Lindbladians and other cases in which a Lindblad operator is expressed as a linear combination or a polynomial of the operators encoded into the program states. We propose quantum algorithms for all these cases and also investigate their sample complexity, i.e., the number of program states needed to simulate a given Lindbladian evolution approximately. Finally, we demonstrate that our quantum algorithms provide an efficient route for simulating Lindbladian evolution relative to full tomography of encoded operators, by proving that the sample complexity for tomography is dependent on the dimension of the system, whereas the sample complexity of wave matrix Lindbladization is dimension independent.
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.