Well-conditioned iterative methods for large open quantum systems

Solving the Lindblad Master Equation (ME) for large open quantum systems is computationally demanding because the Liouvillian superoperator scales as O(n^4) to store and O(n^6) to invert exactly. While iterative methods are appealing, they often fail due to the poor conditioning of the Lindbladian. The authors propose a new approach based on the decomposition of the Lindbladian into a no-jump generator (S) and a jump generator (K). By leveraging the fact that S can be inverted efficiently as a continuous Lyapunov equation, they develop a preconditioner and an auxiliary CPTP map that make iterative methods like Arnoldi and GMRES highly efficient.

The authors demonstrate that the steady state of the Lindblad equation can be found as the fixed point of a CPTP map, which is well-suited for Arnoldi iterations. Furthermore, they introduce a right-preconditioner based on the no-jump resolvent, which ensures that the preconditioned operator is a strict contraction for any positive shift. This allows for the efficient computation of the low-lying spectrum via shift-invert Arnoldi and enables an implicit time integrator that is competitive for stiff systems. These methods scale as O(n^3) per iteration and show significant speedups on both CPU and GPU compared to standard dense solvers or existing sparse direct methods.

This work provides a scalable framework for characterizing large open quantum systems, such as those found in quantum error correction (e.g., cat qubits) or bosonic systems, where standard exact diagonalization is prohibitive. By avoiding the explicit construction of the n^2 x n^2 Liouvillian, the authors enable the simulation of systems with Hilbert space dimensions in the thousands, offering a robust alternative to existing sparse solvers and time-evolution methods.