Diagonal, Switching, and Locally Optimal Unraveling for efficient quantum trajectories in metastable open quantum systems

Abstract

Open quantum systems can display metastability, with multiple long-lived states coexisting. Examples include the bistability of bright and dim manifolds in nonlinear driven resonators, or the long-lived coherent states of opposite phases in parametrically-driven systems, the building block of cat-state-based quantum computers. Metastability often emerges at system sizes where density-matrix evolution is impractical. Quantum trajectories mitigate this issue by evolving wavefunctions rather than density matrices. Metastable states then emerge as attractors towards which the system rapidly relaxes until a rare switching event brings it to another attractor. Simulating them remains costly, as one must resolve fast intra-attractor dynamics over timescales long enough to gather switching statistics. To tackle this problem, we introduce the Diagonal, Switching, and Locally Optimal Unraveling (DiSLOU) method, which combines three optimization layers: (i) Switching among locally optimal unravelings that minimize the number of quantum jumps in each metastable region. (ii) Diagonal propagation: the no-jump evolution is accelerated by diagonalizing the non-Hermitian effective Hamiltonian once per attractor. (iii) Local reduced eigenspaces: the no-jump state can be faithfully propagated in a truncated subspace of only slowly decaying eigenmodes. We benchmark DiSLOU on the driven-dissipative Kerr resonator, finding no measurable loss of accuracy and the largest speedup in the bistable regime, where standard trajectories are most expensive. We then apply the method to the more challenging problem of accurately simulating rare bit-flip events in the cat manifold emerging in a two-mode dissipatively stabilized parametric oscillator, obtaining a performance improvement of over one order of magnitude. An implementation of DiSLOU in Julia is available as an open-source package.

Quantum trajectories reduce the memory cost of Lindblad evolution, but metastable switching remains expensive: simulations must process fast dynamics and many jumps while waiting for rare transitions. This paper introduces DiSLOU, a trajectory solver targeting this separation of timescales. The method combines three layers. First, it switches between a finite set of shifted-jump unravelings chosen to approximately cancel the mean jump amplitudes near each metastable attractor. Second, it diagonalizes each gauge’s effective non-Hermitian Hamiltonian once, allowing event-driven no-jump propagation without ODE stepping. Third, it retains slowly decaying eigenmodes locally, checking projection residuals after jumps and reverting to the full space when necessary. The first two layers preserve the Lindblad dynamics exactly; the third is approximate. Benchmarks compare ensemble dynamics with master-equation evolution or standard trajectories for a Kerr resonator and two-mode dissipative cat systems, including coupling to a two-level system, with Hilbert-space dimensions up to $N=2720$. Agreement is reported within sampling uncertainty. Long Kerr trajectories show roughly two orders of magnitude acceleration, and detuned two-mode examples show two to three orders in median propagation time. For a resonant cat, estimating a rare quadrature-decay rate at 20% relative precision takes over tenfold less propagation time despite requiring more trajectories. Most timing benchmarks exclude spectral preparation. A Julia implementation is provided. Limitations include cubic-cost dense preparation, ill-conditioned eigenvectors, and reduced benefits for short evolutions or broadly exploring dynamics. Projection residuals do not bound accumulated trajectory error; tolerance convergence must be checked.