Generalize the Overlap Gap Property beyond binary alphabets

Establish an appropriate definition of the Overlap Gap Property for optimization problems with $k$-ary labels, particularly Max-$k$-Cut, and determine how this generalized property affects the performance of QAOA beyond binary alphabets.

Background

The paper identifies the Overlap Gap Property as a structural obstruction to local algorithms, including QAOA, in binary optimization settings. For Max-kk-Cut, the labels form a larger alphabet and the paper states that even the appropriate formulation of the property is not yet established. Developing such a formulation would clarify whether analogous obstructions constrain QAOA and other local algorithms for qudit optimization.

References

Finally, another open problem is how the Overlap Gap Property which obstructs the performance of QAOA, can be generalized beyond binary alphabets. Even an appropriate definition of the overlap gap property is missing in this setting.

— Provable Classical and Quantum Local Algorithms for Max-$k$-Cut and Quantum Advantage at Moderate Girth  (2609.39042 - Apte et al., 30 Sep 2026) in Section 6, “Conclusion and Future Work”