Fast-forwarding quantum algorithms for linear dissipative differential equations

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.

💡 Research Summary

The paper investigates quantum algorithms for solving linear ordinary differential equations (ODEs) of the form du/dt = A(t)u(t) + b(t) under the assumption that the matrix A(t) is uniformly dissipative, i.e., its logarithmic norm satisfies A(t)+A†(t) ≤ –2η < 0 for some constant η > 0. This condition guarantees exponential decay of perturbations and enables a substantial reduction of the condition number of the linear system that arises after time discretization.

The authors first present a general framework: discretize the time interval