Finite-size convergence rate for fixed-length localization

Determine whether the wrap-around boundary perturbation in the fixed-known-length quantum change-interval Gram matrix can be exploited to improve the proven square-root-measurement versus optimal-success-probability gap from O_{i,r}(N^{-1/2}) to O_{i,r}(N^{-1}), or to derive an asymptotic expansion with a nonzero N^{-1} coefficient.

Background

For a fixed anomalous interval length i and overlap parameter r=c2, the paper proves that the optimal collective-measurement success probability and the square-root-measurement success probability have the same asymptotic limit as the number N of admissible translations grows. The rigorous finite-size bound on their difference is O_{i,r}(N{-1/2}).

The authors note that numerical results are compatible with faster N{-1} decay, but do not establish such a rate or a corresponding asymptotic expansion. The unresolved issue concerns whether the boundary perturbation arising in the Toeplitz/circulant comparison contains exploitable structure that yields sharper finite-size control.

References

For fixed known length, Theorem~\ref{thm:fixed-length} proves the rigorous rate P_{\rm opt}-P_{\rm SRM}=O_{i,r}(N{-1/2}). Over the explored fixed-length parameter ranges and computed sizes, the numerical gaps are compatible with an N{-1} decay, but they do not determine a uniform exponent. Determining whether the wrap-around boundary perturbation can be exploited to prove O_{i,r}(N{-1}), or an asymptotic expansion with a nonzero N{-1} coefficient, remains an open problem.

— Quantum Change Intervals: Exact Asymptotic Localization with Collective Measurements  (2608.24543 - Chen et al., 25 Aug 2026) in Discussion and limitations, Section 7