Lower bounds for short-term-memory VIDE simulation

Establish lower bounds in the simulation time T and target error ε for quantum algorithms simulating linear Volterra integro-differential equations with general kernels in the short-term-memory regime \(\mathcal M<1\), including determining whether logarithmic dependence on \(1/\varepsilon\) is achievable.

Background

The paper develops efficient quantum algorithms for general convolution linear Volterra integro-differential equations (VIDEs) under the short-term-memory condition M<1\mathcal M<1, but the resulting dependence on the simulation time and precision is not shown to be optimal. The authors specifically identify the need to understand whether the non-Markovian memory term fundamentally prevents near-linear-in-time or logarithmic-in-precision complexity.

References

Related to this, developing lower bounds in terms of $T$ and $\varepsilon$ for $VIDE$s with $\mathcal M<1$ would be interesting. In particular, can quantum algorithms achieve $O(T)$ complexity, or does the presence of non-Markovian behavior not allow this? Furthermore, is it possible to achieve $O(\log(1/\epsilon))$ scaling for general kernels?

— Quantum simulation of non-Markovian dynamical systems  (2608.13533 - Ameri et al., 13 Aug 2026) in Section 1, subsection “Outlook,” paragraph “Higher-order methods”

Lastly, would it be possible to develop efficient algorithms for dynamical systems that have no Markovian term?

— Quantum simulation of non-Markovian dynamical systems  (2608.13533 - Ameri et al., 13 Aug 2026) in Section 1, subsection “Outlook,” paragraph “Higher-order methods”

While we have efficient quantum algorithms in the $\mathcal M<1$ regime, it is far from optimal. However, what lower bounds can one prove in this regime? For instance, is it possible to have algorithms with $(\log(1/\epsilon))$ complexity?

— Quantum simulation of non-Markovian dynamical systems  (2608.13533 - Ameri et al., 13 Aug 2026) in Section 1, subsection “Outlook,” paragraph “Lower bounds in the short-term memory regime”