Speeding up Lindblad dynamics via time-rescaling engineering

TL;DR

The paper presents a universal time-rescaling method to speed up Lindblad dynamics while preserving the trajectory in Hilbert space, using a rescaled Liouvillian tilde L(t) = L[ f(t) ] dot f(t) and a contraction parameter a > 1 that yields Delta t = Delta t_ref / a. An exact, Markovian fast process is obtained analytically without requiring knowledge of Liouvillian eigenvalues, and without introducing extra control fields beyond those of the reference protocol; the approach also admits a variant with time-independent environments. The method is validated on a driven two-level system in an amplitude-damping channel and on a dissipative two-site transverse-field Ising model, with control parameters scaled by dot f(t) to reproduce the same trajectory in a shorter time; a linear rescaling g(t) = a t further shows how to keep the environment time-independent while achieving faster dynamics by scaling the environment rates. Finally, the manuscript connects time-rescaling to quantum speed limits, showing that tilde t_QSL = t_QSL / a, which implies a whole family of brachistochrones across different dynamical constraints. This framework enhances quantum control and computation in noisy, many-body settings by providing analytic, locality-preserving fast protocols without requiring nonlocal interactions or environmental tailoring.

Abstract

We introduce a universal method for accelerating Lindblad dynamics that preserves the original trajectory through Hilbert space. The technique provides exact fast processes analytically, which are Markovian and do not require manipulation of the environment properties, by time-rescaling a reference dynamics. In particular, the engineered control protocols are based only on local interactions, and no additional control fields are required compared to the reference protocol. We demonstrate the scheme with two examples: a driven two-level system in an amplitude damping channel and the dissipative transverse field Ising model. We also show that, by starting with a reference process which is the fastest connecting two states under a certain constraint, the method provides other optimal processes satisfying modified constraints. Our approach can help advance techniques for quantum control and computation towards more complex noisy systems.

Figures (5)

• Figure 1: Time evolution of a reference ($a=1$) Lindblad dynamics and the corresponding TR dynamics for $a=2$ and $a=10$. In all cases, the trajectories from $\ket{\rho(0)}$ to $\ket{\rho(t_f)}$ in Hilbert space are the same. The time duration of the TR processes are $a$ times faster than the reference one.
• Figure 2: Spontaneous emission of a driven two-level system. Ground (blue) and excited (orange) state population dynamics of a driven two-level system under the action of the amplitude damping channel. The detuning $\delta$ and the Rabi frequency $\Omega$ of the reference dynamics (first column) are set as constants, with the decay rate $\gamma =1$. The results of the engineered TR dynamics are shown in the second and third columns for the $a=2$ and $a=10$ cases, respectively. The system starts out in the excited state in all cases.
• Figure 3: Dissipative transverse field Ising model. Time evolution of the populations of the states $\ket{00}$ (blue), $\ket{11}$ (black) and the sum of the populations of the states $\ket{01}$ and $\ket{10}$ (orange). For the reference dynamics (first column), the transverse magnetic-field strengths $h$ are set as constants, with the nearest-neighbor coupling strength $J=1.0$ and the decay rate $\gamma = 0.1$. The second and third columns show the results of the corresponding TR dynamics for $a=2$ and $a=10$, respectively. In all cases, we set $\ket{11}$ as the initial state.
• Figure 4: Driven two-level system subjected to the amplitude damping channel. Population dynamics of the ground (black) and excited (red) states are shown. The main plot displays the behavior of the accelerated dynamics, which has a time duration 15 times ($a = 15$) shorter than the reference process ($a = 1$), shown in the inset. For the reference case, we use the parameters $\delta = 0.5$, $\Omega = 7$, and $\gamma = 1.5$. In the accelerated process, the corresponding parameters are $\delta = 7.5$, $\Omega = 105$, and $\gamma = 22.5$.
• Figure 5: Dissipative transverse field Ising model. Time evolution of the populations of the states $\ket{00}$ (purple), $\ket{11}$ (black), and the sum of the populations of the states $\ket{01}$ and $\ket{10}$ (red). The main plot shows a fast process accelerated by a factor of 15 ($a = 15$) compared to the reference process ($a = 1$), which is shown in the inset. The parameters used in the reference process are $J = 3$, $h = 2$, and $\gamma = 0.2$. The corresponding parameters for the accelerated process are $J = 45$, $h = 30$, and $\gamma = 3$.