Fast-forwarding quantum algorithms for linear dissipative differential equations

TL;DR

This paper demonstrates that quantum algorithms can efficiently simulate linear dissipative differential equations with complexity that scales sub-linearly with time, representing a significant speedup over previous methods.

Contribution

The authors introduce improved quantum algorithms for dissipative ODEs that achieve exponential and polynomial speedups, including fast-forwarding the time dependence to sub-linear complexity.

Findings

Quantum algorithms can prepare history states with cost $ ilde{O}( ext{log}(T))$ for dissipative ODEs.

Final state preparation complexity is $ ilde{O}( ext{sqrt}(T))$, showing polynomial speedup.

Lower-order quantum methods also achieve $ ilde{O}( ext{sqrt}(T))$ complexity for dissipative systems.

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 $\widetilde{\mathcal{O}}(\log(T) (\log(1/\epsilon))^2 )$, 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.