Compare sum-based and worst-case principal-angle objectives for larger systems

Investigate systematically whether the smallest-singular-value objective provides advantages over the sum-based principal-angle objective for larger analog quantum systems in which full algebraic alignment may be unattainable, particularly regarding worst-case alignment guarantees.

Background

The paper optimizes the total squared cosines of the principal angles between the target spectral algebra and the accessible Krylov space. This smooth sum-based objective aligns all algebraic directions collectively and succeeds for the small systems studied.

The paper also introduces a min-type objective based on the smallest singular value, which directly minimizes the largest principal angle and can therefore provide a worst-case guarantee. However, this objective is nonsmooth when the smallest singular value is degenerate. The authors leave a systematic comparison of the two objectives for larger systems—where complete alignment may not be achievable—as an unresolved direction for future work.

References

This min-type objective is however non-smooth at points where the smallest singular value is degenerate, which complicates gradient-based optimization on the unitary manifold and typically requires subgradient or smoothed-minimum techniques. In the applications of Sec.~\ref{sec:numerical}, the sum-based objective $F(U)$ already succeeds in driving every principal angle cosine to unity simultaneously, so the additional complexity of optimizing $F_{\rm min}(U)$ is not required for the cases considered here; we introduce it for completeness and leave a systematic comparison between the two objectives -- in particular for larger systems where full alignment of $F(U)$ may not be achievable and a worst-case guarantee could be preferable -- to future work.

— Optimizing quantum encodings for analog simulation through dynamical algebra reachability  (2609.11351 - Mangin-Brinet, 10 Sep 2026) in Section 2, Optimization over the unitary orbit