Use as a black-box component in quantum walk search

Determine whether the averaged-walk theorem is useful as a black-box classical component in quantum walk search constructions.

Background

The setup/update/check cost model is motivated by quantum walk search, and the paper presents the averaged-walk result as a classical mechanism that matches update and checking costs simultaneously for a single marked state.

Although the theorem is proved classically for arbitrary finite irreducible chains, its usefulness as a reusable black-box component within quantum walk search constructions is not established in the paper and is explicitly listed among the main open directions.

References

The main open directions are to understand the right analogue for multiple marked states, remove or reduce the need to know \pi(m)HT(m), and determine whether the theorem is useful as a black-box classical component in quantum walk search constructions.

— Optimizing Both Checking and Update Costs in Random Walk Search  (2609.18833 - Apers et al., 16 Sep 2026) in Section 7, “Conclusion”