Abstract
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 + log(ε⁻¹)) up to error~ε, 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(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 error ε, with H the Hamiltonian and β the inverse temperature. For input states exhibiting certain coherence conditions -- e.g.,~ 0|^⊗ n e^-β(H + I) |+^⊗ n -- our method achieves exponential improvement in complexity (measured by circuit depth), O (2^-n/2 ε⁻¹ log β ), compared to the quantum singular value transformation-based approach, with complexity O (ε⁻¹ √β ). 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 .