Average-case hardness under natural TDA input distributions

Determine whether any natural distribution over topological-data-analysis inputs makes Betti-number estimation hard on average for both classical and quantum algorithms.

Background

The paper establishes conditional average-case hardness only for clique complexes generated from the balanced planted-clique distribution. This leaves unresolved whether average-case hardness is a feature of other natural input distributions that more closely model data encountered in topological data analysis, rather than an artifact of the planted-clique construction.

References

Whether any natural distribution over inputs makes Betti-number estimation hard on average for classical and quantum algorithms alike has, to our knowledge, remained open.

— Average-case hardness of Betti number estimation  (2609.12777 - Strelchuk et al., 11 Sep 2026) in Section 1, paragraph “The worst-case complexity landscape”

A polynomial time quantum algorithm for Betti number estimation in the hard regime would render the conjecture false and the problems in section \ref{sec:implications} quantum-average-case easy, while remaining classical average-case hard. This remains an open problem.

— Average-case hardness of Betti number estimation  (2609.12777 - Strelchuk et al., 11 Sep 2026) in Section 1, subsection “Classical and quantum results”

Nevertheless, the main observation is that the average-case hardness we establish for the problem does not reach into this region. This is the interesting and non-trivial regime our reduction does not affect and the question of average-case quantum advantage remains open and plausible.

— Average-case hardness of Betti number estimation  (2609.12777 - Strelchuk et al., 11 Sep 2026) in Section 5.4, subsection “Region for average-case quantum-advantage”

Whether homology vanishing of the clique complex itself is hard on average under any other natural distribution remains open.

— Average-case hardness of Betti number estimation  (2609.12777 - Strelchuk et al., 11 Sep 2026) in Remark 6.5, Section 6, subsection “Corollaries in Topological Data Analysis”