Exponential Lindbladian fast forwarding and exponential amplification of certain Gibbs state properties

We investigate Lindbladian fast-forwarding and its applications to estimating Gibbs state properties. Fast-forwarding refers to the ability to simulate a system of timetusing significantly fewer thantqueries or circuit depth. While various Hamiltonian systems are known to circumvent the no fast-forwarding theorem, analogous results for dissipative dynamics, governed by Lindbladians, remain largely unexplored. We first present a quantum algorithm for simulating purely dissipative Lindbladians with unitary jump operators, achieving additive query complexityO(t+log⁡(ε-1))up to errorε, improving previous algorithms. When the jump operators have certain structures (i.e. block-diagonal Paulis), the algorithm can be modified to achieve exponential fast-forwarding, attaining circuit depthO(log⁡(t+log⁡(ε-1))), while preserving query complexity via parallel access. Using these fast-forwarding techniques, we develop a quantum algorithm for estimating Gibbs state properties of the form⟨ψ1|e-β(H+I)|ψ2⟩, up to additive errorϵ, withHthe Hamiltonian andβthe inverse temperature. For input states exhibiting certain coherence conditions-e.g.⟨0|⊗ne-β(H+I)|+⟩⊗n-our method achieves exponential improvement in complexity (measured by circuit depth),O(2-n/2ϵ-1log⁡β),compared to the quantum singular value transformation-based approach, with complexityO~(ϵ-1β). We show how to apply this exponential improvement to applications such as the ground state overlap testing and amplitude estimation. For general|ψ1⟩and|ψ2⟩, we also show how the level of improvement is changed with the coherence resource in|ψ1⟩and|ψ2⟩.