Ancilla-Free Fast-Forwarding Lindbladian Simulation Algorithms by Hamiltonian Twirling
Minbo Gao, Zhengfeng Ji, Chenghua Liu
TL;DR
The paper addresses the challenge of efficiently simulating Markovian open quantum systems by revealing a structural link between Lindbladian dynamics and Hamiltonian twirling channels. It shows that for infinitely divisible distributions, the twirling channel $\Phi_{H,\mathcal{D}}$ equals $e^{\mathcal{L}}$ with a Lindbladian generator commuting with $H$, and in the Gaussian case provides an exact Lindbladian representation as a Gaussian average of unitaries. This yields an ancilla-free fast-forwarding algorithm achieving a diamond-norm error $\varepsilon$ in time $O(\sqrt{t\log(1/\varepsilon)})$, with extensions to multiple Hermitian jumps and non-Gaussian (compound Poisson) twirls. The work further connects Gaussian twirling to continuous-variable quantum phase estimation and discusses the Lévy-Khintchine framework to classify when such twirling realizations exist, offering a flexible pathway for tunable dephasing and non-Gaussian smoothing in dissipative quantum simulations.
Abstract
Simulation of open quantum systems is an area of active research in quantum algorithms. In this work, we revisit the connection between Markovian open-system dynamics and averages of Hamiltonian real-time evolutions, which we refer to as Hamiltonian twirling channels. Focusing on the class of Lindbladians with a single Hermitian jump operator $H$ recently studied in Shang et al. (arXiv:2510.06759), we show that the time-$t$ evolution map can be expressed exactly a Gaussian twirl over the unitary orbit ${\{\mathrm{e}^{-\mathrm{i} Hs}\}}_{s\in\mathbb{R}}$. This structural insight allows us to design a fast-forwarding algorithm for Lindbladian simulation that achieves diamond-norm error $\varepsilon$ with time complexity $O\big(\sqrt{t\log(1/\varepsilon)}\big)$ -- matching the performance of Shang et al. while requiring no auxiliary registers or controlled operations. The resulting ancilla-free and control-free algorithm is therefore more amenable to near-term experimental implementation. By purifying the Gaussian twirl procedure and performing a conjugate measurement, we derive a continuous-variable quantum phase estimation algorithm. In addition, by applying the Lévy-Khintchine representation theorem, we clarify when and how a dissipative dynamics can be realized using Hamiltonian twirling channels. Guided by the general theory, we explore Hamiltonian twirling with compound Poisson distributions and their potential algorithmic implications.