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 time t using significantly fewer than t queries 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 complexity O (t + (^-1) ) up to error~, 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 depth O ( (t + (^-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 ₁ | e^- (H + I) | ₂, up to additive error, with H the Hamiltonian and the inverse temperature. For input states exhibiting certain coherence conditions ---e. g. , ~ 0|^ n e^- (H + I) |+^ n---our method achieves exponential improvement in complexity (measured by circuit depth), O (2^-n/2 ^-1), compared to the quantum singular value transformation-based approach, with complexity O (^-1). We show how to apply this exponential improvement to applications such as the ground state overlap testing and amplitude estimation. For general | ₁ and | ₂, we also show how the level of improvement is changed with the coherence resource in | ₁ and | ₂.