Generalization of the lightcone-transition formula to k-local circuits

Verify whether the asymptotic lightcone-coverage transition formula derived for random all-to-all 2-local circuits generalizes to random all-to-all $k$-local circuits for every $k\geq 2$, with logarithms taken in base $k$.

Background

The paper derives the transition depth for full lightcone coverage in random all-to-all circuits with 2-qubit gates, obtaining a leading behavior of log⁡2n+log⁡2log⁡2n\log_2 n+\log_2\log_2 n plus lower-order terms. The derivation depends strongly on the comparatively simple edge-crossing statistics of random perfect matchings.

The authors suggest that an analogous expression may hold for higher locality, but the corresponding lightcone-growth process is more complicated because a single kk-local gate can connect multiple qubits already in the lightcone to multiple qubits outside it. The requested verification is therefore left unresolved.

References

We remark that a generalized expression plausibly holds for all $k\geq2$, with logarithms taken base $k$, but verification of this is left to future work.

— Proper Learning of Shallow All-to-All Quantum Circuits  (2608.20162 - Kordonowy et al., 20 Aug 2026) in Section 3, subsection “Analysis of lightcone growth”