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Fast-forwarding quantum algorithms for linear dissipative differential equations

Quantum Journal

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Researchers Dong An, Akwum Onwunta, and Gengzhi Yang developed a quantum algorithm that exponentially accelerates solving linear dissipative differential equations, reducing the cost to prepare history states to near-logarithmic time dependence. Their truncated Dyson series method achieves Õ(log(T)(log(1/ε))²) complexity for history states up to time T, surpassing previous best results by exponential factors while maintaining precision ε. For final state preparation at time T, the algorithm reaches Õ(√T(log(1/ε))²) complexity, marking the first polynomial speedup in time dependence for dissipative systems. Even low-order methods like forward Euler and trapezoidal rules demonstrate Õ(√T) scaling, proving simpler quantum algorithms can still outperform classical approaches for dissipative ODEs. Applications include simulating non-Hermitian quantum dynamics and heat processes, enabling sub-linear time complexity for previously intractable dissipative systems.

Fast-forwarding quantum algorithms for linear dissipative differential equations

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AbstractWe 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.► BibTeX data@article{An2026fastforwarding, doi = {10.22331/q-2026-01-27-1986}, url = {https://doi.org/10.22331/q-2026-01-27-1986}, title = {Fast-forwarding quantum algorithms for linear dissipative differential equations}, author = {An, Dong and Onwunta, Akwum and Yang, Gengzhi}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1986}, month = jan, year = {2026} }► References [1] Dominic W. Berry. ``High-order quantum algorithm for solving linear differential equations''. Journal of Physics A: Mathematical and Theoretical 47, 105301 (2014). https://doi.org/10.1088/1751-8113/47/10/105301 [2] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. ``Quantum algorithm for linear systems of equations''.

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Physical Review Letters 125, 260601 (2020). https://doi.org/10.1103/physrevlett.125.260601 [29] Kun Ding, Chen Fang, and Guancong Ma. ``Non-Hermitian topology and exceptional-point geometries''.

Physical Review Letters 133, 086301 (2024). https://doi.org/10.1103/physrevlett.133.086301 [33] Lawrence C. Evans. ``Partial differential equations''.

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Fast-forwarding quantum algorithms for linear dissipative differential equations

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