Asymptotic validity of the cube-root conjecture

Determine whether the cube-root relation $\delta^*\simeq b_{\rm triv}^{1/3}$ holds asymptotically for peaked circuits at fixed peaking-to-random-depth ratio, despite the observed depth dependence of the fitted exponent at the tested sizes.

Background

The paper examines the cube-root conjecture proposed for peaked-circuit generation, comparing the optimal peakedness δ∗\delta^* with the peakedness btrivb_{\rm triv} of a truncated random circuit. Its finite-size, fixed-depth tests find that the fitted exponent is not depth-independent over the tested grid.

Those results do not resolve the original asymptotic, mean-based conjecture at a fixed ratio of peaking depth to random depth. The authors explicitly state that the available two sizes and four depths cannot determine the asymptotic claim.

References

The shallow-depth computation of Sec.~\ref{sec:atom} bears on the cube-root conjecture of Ref. (their Conj.~3.2). We test the instance-exact form, $\delta* \simeq b_{\rm triv}{\gamma}$ with $\gamma = 1/3$ independent of the peaking depth, where $b_{\rm triv}$ is the peakedness of the same instance truncated to its first $\tau_r - \tau_p$ random layers. Ref. states the conjecture asymptotically, in the mean, and at fixed ratio $\tau_p = k\tau_r$, a ratio the depth scan varies, so what follows constrains this per-instance, fixed-depth reading rather than the original claim.

— The optimization landscape of peaked-circuit generation  (2608.11890 - Jamoussi, 12 Aug 2026) in Appendix B, The cube-root conjecture at fixed depth