Determine the exact logarithmic factors in bipartiteness and expansion testing

Determine the exact power of \(\log(N)\) in the quantum query complexities of bipartiteness testing and expansion testing in the bounded-degree graph model.

Background

The paper establishes near-optimal quantum query lower bounds for both bipartiteness testing and expansion testing in the bounded-degree graph model, matching known quantum algorithms up to polylogarithmic factors. Specifically, the lower bound for bipartiteness testing is Ω(N1/3/log⁡N)\Omega(N^{1/3}/\log N), while the expansion-testing lower bound is Ω(N1/3/(log⁡N)4/3)\Omega(N^{1/3}/(\log N)^{4/3}). Because the upper and lower bounds still differ in their logarithmic factors, the precise powers of log⁡(N)\log(N) governing the complexities remain unresolved.

References

One problem left open by this work is determining the exact power of $\log(N)$ in the complexities of both bipartiteness and expansion testing.

— Near-optimal quantum query lower bounds on bipartiteness and expansion testing in the bounded-degree graph model  (2610.01752 - Kayal et al., 1 Oct 2026) in Section 1, subsection “Open problems”