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

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 $\mathcal{O}\left(t+\log ({\epsilon }^{-1})\right)$ up to error~$\epsilon$, 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 $\mathcal{O}\left(\log \left(t+\log ({\epsilon }^{-1})\right)\right)$, 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 $⟨\ddot{I}{ˆ}_1|e^{-{\hat{I}}^2(H+I)}|\ddot{I}{ˆ}_2⟩$, up to additive error $\hat{I}\mu$, with $H$ the Hamiltonian and ${\hat{I}}^2$ the inverse temperature. For input states exhibiting certain coherence conditions -- e.g.,~$⟨0{|}^{\otimes n}{e}^{-{\hat{I}}^2(H+I)}|+{⟩}^{\otimes n}$ -- our method achieves exponential improvement in complexity (measured by circuit depth), $\mathcal{O}(2^{-n/2}\hat{I}{\mu }^{-1}\log {\hat{I}}^2),$ compared to the quantum singular value transformation-based approach, with complexity $\tilde{\mathcal{O}}(\hat{I}{\mu }^{-1}{\sqrt{\hat{I}}}^2)$. We show how to apply this exponential improvement to applications such as the ground state overlap testing and amplitude estimation. For general $|\ddot{I}{ˆ}_1⟩$ and $|\ddot{I}{ˆ}_2⟩$, we also show how the level of improvement is changed with the coherence resource in $|\ddot{I}{ˆ}_1⟩$ and $|\ddot{I}{ˆ}_2⟩$.

39 pages