Arbitrary High Order Low-rank Completely Positive and Trace Preserving (CPTP) Schemes for Lindblad Equations with Time-dependent Hamiltonian

TL;DR

This work tackles numerical integration of open quantum dynamics governed by the Lindblad equation with time-dependent Hamiltonians, aiming to preserve complete positivity and trace at arbitrary high order. It introduces nested Picard iteration (NPI) to construct CPTP schemes in Kraus form, combining flow-operator approximations with Duhamel-based quadrature and low-rank truncation via truncated SVD. The approach yields a family of high-order, low-rank methods that are either explicit or implicit for the flow steps and remain efficient through low-rank representations, with stability properties established for single-qubit models. Numerical experiments on multi-qubit, qudit-cavity, and Jaynes–Cummings scenarios demonstrate high accuracy and computational efficiency, validating the method's potential for large-scale quantum simulations and optimal-control applications.

Abstract

In this paper, we develop a framework for designing arbitrary high order low-rank schemes for the Lindblad equation with time-dependent Hamiltonians. Our approach is based on nested Picard iterative integrators (NPI) and results in schemes in Kraus form that are completely positive and trace preserving (CPTP). The schemes are amenable to low rank formulations, making them suitable for problems where the matrix rank of the density matrix is small.

Figures (8)

• Figure 1: Stability region of the schemes corresponding to schemes with highest order flow operator taken as explicit schemes of order 1, 2, 3, 4 for the single-qubit test equation in Section \ref{['sec:teste1s']}: the interior domains bounded by the curves and the rightmost boundary at $\Delta t=0$.
• Figure 2: The graphs display the errors at the final time for methods (from top left to bottom right) of order one to four for example in Section \ref{['sec:test1']}. See the text for further explanation.
• Figure 3: Errors for different $\Delta t$ for the third and fourth-order implicit and explicit methods when used to compute the solution to the qubit-cavity problem.
• Figure 4: The rank as a function of time for the qubit-cavity problem when simulated using the fourth-order implicit (left) and explicit (right) method. The number of timesteps corresponds to the largest and smallest $\Delta t$ used in the experiments producing Figure \ref{['fig:QR1']}.
• Figure 5: Populations in the qubit (top, annotated by $A$) and cavity (bottom, annotated by $B$) as a function of time with coupling $\xi_{01} \neq0$ (left) and without coupling $\xi_{01}=0$ (right).