Determine the Potts–Parisi optimum for Max-$k$-Cut

Compute the large-degree Potts–Parisi limit for each number of labels $k$ and thereby determine the optimal leading $1/\sqrt d$ coefficient $P_k^{\star}$ for Max-$k$-Cut on random $d$-regular graphs, in order to quantify the gap between the optimum and the guarantees of the Local Vector algorithm and QAOA.

Background

The paper compares the leading 1/d1/\sqrt d improvements achieved by the Local Vector algorithm and QAOA against the random-labeling baseline (k−1)/k(k-1)/k. For Max-Cut, the corresponding optimal coefficient is known through the Parisi value of the Sherrington–Kirkpatrick model. For general Max-kk-Cut, the authors identify the relevant asymptotic benchmark with the ground-state energy of the mean-field Potts antiferromagnet, but do not compute it numerically for each kk. Establishing this benchmark would provide an absolute reference for the algorithms studied in the paper.

References

Three natural questions remain open. The first concerns the ceiling against which all of these coefficients should be measured. For Max-Cut, Dembo, Montanari, and Sen showed that the optimal cut fraction on random $d$-regular graphs is $1/2 + P_/\sqrt d + o(1/\sqrt d)$ with $P_ \approx 0.7632$ the Parisi value of the Sherrington--Kirkpatrick model, and Sen extended this Gaussian-optimization picture to Max-$k$-Cut, where the large-degree limit is governed by the ground-state energy of the corresponding mean-field Potts antiferromagnet, a Potts--Parisi variational problem. Computing this limit numerically for each $k$ would fix the optimal $1/\sqrt d$ coefficient $P{\star}_{k}$ and turn our tables into an absolute comparison, quantifying how far both the Local Vector algorithm and QAOA sit below the optimum.

— 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”