Fast-forwarding quantum algorithms for linear dissipative differential equations

TL;DR

This work establishes fast-forwarding results for quantum algorithms solving linear dissipative ODEs $\frac{du}{dt}=A(t)u+b(t)$ under the condition $A(t)+A(t)^{\dagger}\le -2\eta$. By framing the dynamics with an all-at-once linear system and leveraging a truncated Dyson series, the authors achieve an exponential speedup in evolution time $T$ for history-state preparation, with complexity $\widetilde{O}(\log(T)(\log(1/\epsilon))^2)$, and a quadratic speedup for final-state preparation, with complexity $\widetilde{O}(\sqrt{T}(\log(1/\epsilon))^2)$. They further show that simpler lower-order discretizations such as forward Euler and trapezoidal rule still yield sublinear time dependence for final-state preparation, specifically $\widetilde{O}(\sqrt{T})$ up to polylogarithmic factors, while detailing padding strategies to boost success probabilities. The results extend to dissipative non-Hermitian quantum dynamics and heat processes, offering fast-forwarded quantum simulations in these contexts and guiding how dissipation enables improved time-scaling over prior linear-in-$T$ approaches.

Abstract

We establish improved complexity estimates of quantum algorithms for linear dissipative ordinary differential equations (ODEs) and show that the time dependence can be fast-forwarded to be sub-linear. Specifically, we show that a quantum algorithm based on truncated Dyson series can prepare history states of dissipative ODEs up to time $T$ with cost $\widetilde{\mathcal{O}}(\log(T) (\log(1/ε))^2 )$, which is an exponential speedup over the best previous result. For final state preparation at time $T$, we show that its complexity is $\widetilde{\mathcal{O}}(\sqrt{T} (\log(1/ε))^2 )$, achieving a polynomial speedup in $T$. We also analyze the complexity of simpler lower-order quantum algorithms, such as the forward Euler method and the trapezoidal rule, and find that even lower-order methods can still achieve $\widetilde{\mathcal{O}}(\sqrt{T})$ cost with respect to time $T$ for preparing final states of dissipative ODEs. As applications, we show that quantum algorithms can simulate dissipative non-Hermitian quantum dynamics and heat processes with fast-forwarded complexity sub-linear in time.