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Transforming the Lindblad Equation into a System of Linear Equations: Performance Optimization and Parallelization.

Iosif B. Meyerov, Козинов Евгений Александрович, Alexey Liniov, Valentin Volokitin, Igor Yusipov, Mikhail Ivanchenko, S. Denisov

Published Dec 3, 2019• 2 citations
OpenAlex

Abstract

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 $N^2 \times N^2$ matrices (instead of standard $N \times N$ 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 $O(N^5 log(N))$ for dense systems and $O(N^3 log(N))$ 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.

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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 NNN with the number of the system's parts, the computational complexity scales quadratically with NNN, since we have to deal now with super-operators represented by N2×N2N^2 \times N^2N2×N2 matrices (instead of standard N×NN \times NN×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))O(N^5 log(N))O(N5log(N)) for dense systems and O(N3log(N))O(N^3 log(N))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=2000N=2000N=2000 and dense systems of dimension N=200N=200N=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=200N=200N=200, which is impossible without the distribution of memory costs among cluster nodes.
With their constantly increasing peak performance and memory capacity, modern supercomputers offer new perspectives on numerical studies of open many-body quantum systems. These systems are often modeled by using Markovian quantum master equations describing the evolution of the system density operators. In this paper, we address master equations of the Lindblad form, which are a popular theoretical tools in quantum optics, cavity quantum electrodynamics, and optomechanics. By using the generalized Gell-Mann matrices as a basis, any Lindblad equation can be transformed into a system of ordinary differential equations with real coefficients. Recently, we presented an implementation of the transformation with the computational complexity, scaling as O(N5logN) for dense Lindbaldians and O(N3logN) for sparse ones. However, infeasible memory costs remains a serious obstacle on the way to large models. Here, we present a parallel cluster-based implementation of the algorithm and demonstrate that it allows us to integrate a sparse Lindbladian model of the dimension N=2000 and a dense random Lindbladian model of the dimension N=200 by using 25 nodes with 64 GB RAM per node.
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.
Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) ddtρt=Lρt\frac{\mathrm{d}}{\mathrm{d} t} ρ_t = \mathcal{L} ρ_tdtd​ρt​=Lρt​, where L\mathcal{L}L is a linear (super-)operator and ρtρ_tρt​ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state ρ∞ρ_\inftyρ∞​ (such that Lρ∞=0\mathcal{L} ρ_\infty = 0Lρ∞​=0), the first few excited states, and trajectories t↦ρtt\mapsto ρ_tt↦ρt​. In finite dimension, ρtρ_tρt​ is an n×nn\times nn×n matrix, L\mathcal{L}L thus typically costs n4n^4n4 to store explicitly as a dense matrix, and O(n6)O(n^6)O(n6) to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, L\mathcal{L}L usually costs only O(n3)O(n^3)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\mathcal{S}S, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map ΦΦΦ whose fixed point is directly related to ρ∞ρ_\inftyρ∞​, all the other eigenvalues having smaller magnitude. The map ΦΦΦ is thus well suited to iterative methods, and ρ∞ρ_\inftyρ∞​ can be found in a few Arnoldi iterations. Using the same inverse map S−1\mathcal{S}^{-1}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)O(n^3)O(n3) per iteration and offer state-of-the-art performance on CPU and GPU.
We consider the natural generalization of the Schrödinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an nnn-qubit system for time ttt within precision εεε. If the Lindbladian consists of poly(n)\mathrm{poly}(n)poly(n) operators that can each be expressed as a linear combination of poly(n)\mathrm{poly}(n)poly(n) tensor products of Pauli operators then the gate cost of our algorithm is O(t polylog(t/ε)poly(n))O(t\, \mathrm{polylog}(t/ε)\mathrm{poly}(n))O(tpolylog(t/ε)poly(n)). We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schrödinger evolution on a larger system and tracing out the ancillary system: the cost of such a \textit{reduction} incurs an efficiency overhead of O(t2/ε)O(t^2/ε)O(t2/ε) even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the "linear combinations of unitaries" construction that pertains to channels.
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.
Solving linear ordinary differential equations (ODEs) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed nonunitary dynamics into intrinsically unitary quantum circuits. In this Letter, we propose a new quantum algorithm for solving ODEs by harnessing open quantum systems. Specifically, we propose a novel technique called nondiagonal density matrix encoding, which leverages the inherent nonunitary dynamics of Lindbladians to encode general linear ODEs into the nondiagonal blocks of density matrices. This framework enables us to design quantum algorithms with both theoretical simplicity and high performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters under a plausible input model. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.