Numerical simulation of quantum computing hardware and open quantum systems governed by the Lindblad equation is challenging due to the high dimensionality of the density matrix and the need to preserve fundamental physical properties. In our previous work, we developed an arbitrary-order, low-rank, completely positive and trace preserving (CPTP) method for the Lindblad equation with time-dependent Hamiltonians by nested Picard iteration (NPI). In this work, we develop Gregory NPI schemes, which are CPTP schemes constructed by Gregory-type quadrature on equispaced nodes. The methods, which are of order up to nine, substantially reduce the computational cost compared to our previously proposed NPI schemes with Gaussian quadrature rules, while retaining high-order accuracy and structure preservation. We analyze the stability of the resulting scheme for a physics-based test equation. Numerical experiments verify the convergence of the method and demonstrate the effectiveness of the low-rank approximation. We study the performance of a previously constructed CNOT gate for both closed and open quantum systems.
We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to
1019 degrees of freedom using only modest compute resources.
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.
Quantum control has been a cornerstone of quantum information science, driving major advances in quantum computing, quantum communication, and quantum sensing. Over the years, it has enabled the implementation of arbitrary completely positive and trace-preserving (CPTP) maps; an important next step is to extend control to Hermitian-preserving and trace-preserving (HPTP) maps, which underpin applications such as entanglement detection, quantum error mitigation, quantum simulation, and quantum machine learning. Here we present an efficient and fully constructive method for implementing arbitrary HPTP maps. Unlike existing methods that decompose an HPTP map into multiple CPTP maps or approximate it using bipartite Hamiltonians with large Hilbert spaces, our approach compiles a target HPTP map into a single executable CPTP map whose Kraus rank is guaranteed to be no larger than the intrinsic rank of the target HPTP map plus one, followed by simple classical post-processing. Numerical results for inverse noise channels used in quantum error mitigation, including bosonic photon loss, confirm substantial reductions in resources and highlight scalability in higher-dimensional settings. Together with our numerical benchmarks, these results validate the efficiency and versatility of the proposed framework, opening a route to broader quantum-information applications enabled by HPTP processing.
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.
The Lindblad master equation is a valuable tool in quantum mechanics, which describes the dynamics of open systems. In the scope of our research, it is combined with the one-dimensional Maxwell's equations to form the generalized Maxwell-Bloch equations. Since analytical solutions are not available in the general case, numerical methods have to be employed to solve the Lindblad equation. In this work, we focus on methods that are completely positive trace preserving (CPTP), i.e., that guarantee to preserve the properties of the density matrix. We review existing approaches and compare the most promising candidates in terms of computational performance.