Achieving O(sqrt(N)) spatial-search time in two dimensions

Determine whether quantum spatial search on a two-dimensional grid can be performed in O(sqrt(N)) time, improving on the O(sqrt(N)(log N)^{3/2}) bound achieved by the recursive construction and prior recursive-search analyses.

Background

The paper reviews spatial search on a d-dimensional grid and notes that recursive quantum-search methods achieve O(sqrt(N)) time for d greater than or equal to 3, but only O(sqrt(N)(log N){3/2}) time for d=2. Quantum-walk methods achieve O(sqrt(N log N)) in two dimensions, while the paper's own two-dimensional construction also retains a polylogarithmic overhead. The authors explicitly state that the optimal O(sqrt(N)) two-dimensional scaling is unresolved.

References

To our knowledge, whether $O(\sqrt{N})$ is achievable at $d=2$ remains an open question.

— Exact and Optimal Recursive Quantum Search via Hilbert-Space Decomposition  (2608.23002 - Burke et al., 24 Aug 2026) in Section 1, Related Work, paragraph 'Spatial search'