Determine tight bounds for unitary property-testing problems

Determine tight query-complexity bounds for the recurrence-time and entropy-estimation problems, and identify further nontrivial unitary property-testing problems invariant under conjugation by specified groups.

Background

The paper’s construction uses local-unitary invariance and symmetric/antisymmetric subspace projectors to derive a quantum oracle separation. The authors suggest extending these ideas to other unitary property-testing problems.

They specifically identify recurrence-time and entropy-estimation problems as having unresolved quantitative complexity, and also point to the search for interesting invariance classes under group conjugation.

References

Known upper and lower bounds for these problems agree up to polylogarithmic factors, but the sharp rates remain unresolved.

— Optimal spectrum estimation  (2609.30171 - Bakshi et al., 24 Sep 2026) in Section 4, paragraph “Applications to learning other symmetric properties”

Concretely, leaves open the problem of finding tight bounds for the recurrence time and entropy estimation problems, as well as finding interesting problems related to unitaries that are invariant under conjugation by various groups (discussed in ).

— A quantum oracle separation between QMA(2) and QMA  (2609.02865 - Bostanci et al., 2 Sep 2026) in Section 6, Conclusion, “Future directions,” item 2

Other open problems include uniform qutrit phase-space separation beyond nine queries, sufficient conditions for a locked marginal to be realizable by an optimal protocol, constraints on optimal measurement effects, and stability relative to the actual optimal-marginal~set.

— Resource Compatibility in Optimal Quantum Symmetry Testing  (2609.38134 - Cao et al., 29 Sep 2026) in Discussion; Section “Wigner-positive qutrit-Clifford testing”

Other open problems include uniform qutrit phase-space separation beyond nine queries, sufficient conditions for a locked marginal to be realizable by an optimal protocol, constraints on optimal measurement effects, and stability relative to the actual optimal-marginal~set.

— Resource Compatibility in Optimal Quantum Symmetry Testing  (2609.38134 - Cao et al., 29 Sep 2026) in Discussion