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 O ( log ( T ) ( log ( 1 / ) ) 2 ) , which is an exponential speedup over the best previous result. For final state preparation at time T , we show that its complexity is O ( T ( log ( 1 / ) ) 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 O ( 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.
An et al. (Tue,) studied this question.