Near-optimal clustering of reachable solutions

Determine whether near-optimal solutions of the peaked-circuit variational landscape cluster at the near-optimal level, beyond the established connectivity of reachable solutions at the trap scale.

Background

The paper studies the peakedness objective obtained by applying a trainable brickwall circuit to a Haar-random circuit and optimizing the probability of a designated output string. Numerical string-method experiments find paths between decorrelated near-optimal solutions whose objective remains far above the Haar floor, ruling out fragmentation at the trap scale over the tested sizes.

Because the path search is one-sided and the registered near-optimal path criterion is not met, the results do not determine whether the near-optimal set itself forms clusters separated by an overlap gap. The authors explicitly identify this as an unresolved issue.

References

A path search is one-sided, so near-optimal clustering remains open.

— The optimization landscape of peaked-circuit generation  (2608.11890 - Jamoussi, 12 Aug 2026) in Section 1, Introduction; Section 5.3, The reachable ensemble: a single corrugating shelf

Over the sizes probed the corrugation floor $\varrho(n)$ along those paths deepens faster than any fixed power of $n$, an exponential fitting them adequately; persistence of that rate beyond $n = 16$ is the conjectural part.

— The optimization landscape of peaked-circuit generation  (2608.11890 - Jamoussi, 12 Aug 2026) in Conjecture C-shelf, Section 5.2, The conjecture