Higgs Self-coupling Strategy at Linear e$^+$e$^-$ Colliders

The determination of the Higgs self-coupling is a key target for future colliders, in particular through di-Higgs production at $e^+e^-$ Linear Colliders with $\sqrt{s} > 450$\,GeV, e.g.\ ILC, C3 or CLIC. This contribution will discuss the roles and the interplay of di-Higgs production processes at various collider energies, including the case of non-SM values of the self-coupling. Previous studies, already based on Geant4-based detector simulation, established that the Higgs self-coupling can be extracted with $10-27\%$ precision and provided a solid understanding of the limiting factors. This provides a robust starting point to explore the potential of more modern and sophisticated reconstruction and analysis techniques. We summarize the impact of advanced, often machine-learning-based algorithms, including e.g.\ jet clustering, kinematic fitting and matrix element-inferred likelihoods on the reconstruction of $ZHH$ events and before discussing the dependence of the projected precision on the center-of-mass energy and on the actual value of the self-coupling.