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.
Abstract In this work, we present an efficient rank-compression approach for the classical simulation of Kraus decoherence channels in noisy quantum circuits. The approximation is achieved through iterative compression of the density matrix based on its leading eigenbasis during each simulation step without the need to store, manipulate, or diagonalize the full matrix. We implement this algorithm using an in-house simulator and show that the low-rank algorithm speeds up simulations by more than two orders of magnitude over existing implementations of full-rank simulators, and with negligible error in the noise effect and final observables. Finally, we demonstrate the utility of the low-rank method as applied to representative problems of interest by using the algorithm to speed up noisy simulations of Grover’s search algorithm and quantum chemistry solvers.
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.
We develop LREI, short for Low-Rank Eigenmode Integration, a memory- and time-efficient numerical scheme for solving quantum Landau–Lifshitz (q-LL) and quantum Lan- dau–Lifshitz–Gilbert (q-LLG) equations, which govern spin dynamics in open quantum systems. Although the system size grows exponentially with the number of spins, our approach benefits from the low-rank structure of the density matrix and the sparsity of system Hamiltonians to avoid costly full matrix computations. By representing density matrices in terms of their low-rank factors and using Krylov subspace techniques for partial eigendecompositions, we reduce the per-step complexity of Runge–Kutta and Adams–Bashforth schemes from O(N 3) to O(r2N ), where N = 2n is the Hilbert space dimension for n spins and r ? N is the effective rank of the density matrix. Likewise, the memory footprint is reduced from O(N 2) to O(rN ), since no full N × N matrices are ever formed. Among several technical improvements we applied, one key idea was to handle the computation of the action of the invariant subspace of the density matrix associated with its zero eigenvalues. This was accomplished by applying Householder reflectors constructed for the dominant eigenspace, thereby enabling the entire solution process to proceed without ever forming any large matrices. As an example, we can now evolve a time step of a twenty-spin system—corresponding to a density matrix size exceeding one million—in just a few seconds on a standard laptop. In addition, both classes of Runge-Kutta and Adams-Bashforth techniques are reformulated to preserve the physical properties of the density matrix throughout the time evolution. The new low-rank algorithm makes it possible to simulate much larger spin systems that were previously computationally infeasible. This, in turn, provides a powerful tool for comparing the q-LL and q-LLG dynamics, assessing the validity of each model, and exploring how quantum features such as correlations and entanglement evolve across different regimes of system size and damping.
We introduce a model-independent method for the efficient simulation of low-entropy systems, whose dynamics can be accurately described with a limited number of states. Our method leverages the time-dependent variational principle to efficiently integrate the Lindblad master equation, dynamically identifying and modifying the low-rank basis over which we decompose the system's evolution. By dynamically adapting the dimension of this basis, and thus the rank of the density matrix, our method maintains optimal representation of the system state, offering a substantial computational advantage over existing adaptive low-rank schemes in terms of both computational time and memory requirements. We demonstrate the efficacy of our method through extensive benchmarks on a variety of model systems, with a particular emphasis on multiqubit bosonic codes, a promising candidate for fault-tolerant quantum hardware. Our results highlight the method's versatility and efficiency, making it applicable to a wide range of systems characterized by arbitrary degrees of entanglement and moderate entropy throughout their dynamics. We provide an implementation of the method as a Julia package, making it readily available to use. Published by the American Physical Society 2024
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.