Ancilla-Free Fast-Forwarding Lindbladian Simulation Algorithms by Hamiltonian Twirling

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.