Fast-forwarding quantum algorithms for linear dissipative differential equations

Quantum 10, 1986 (2026).

https://doi.org/10.22331/q-2026-01-27-1986

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 $widetilde{mathcal{O}}(log(T) (log(1/epsilon))^2 )$, which is an exponential speedup over the best previous result. For final state preparation at time $T$, we show that its complexity is $widetilde{mathcal{O}}(sqrt{T} (log(1/epsilon))^2 )$, 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.