The optimization landscape of peaked-circuit generation

Abstract: Peaked circuits are random quantum circuits whose measurement returns one bitstring far more often than chance. They are a candidate route to verifiable quantum advantage, and the bottleneck is classical generation. Aaronson and Zhang fix a random circuit, append a trainable brickwall of half that depth, and optimize it by gradient descent to raise the probability of one chosen output string. Their method plateaus at a size-dependent ceiling, which they attribute to a barren plateau. A dichotomy remains open: either the optimizer stops short, so a better algorithm would reach higher, or no efficient method exists. We map the landscape on which the answer depends. Across 18 instances per size at n = 8-16, at fixed and converged budgets, no fixed-base exponential, the law they fit, matches the optimizer's reach: at convergence the decay steepens from 1.16 to 1.295 per qubit through n = 16 (p = 0.011 frozen and 0.025 converged, on n = 8-14 alone), leaving their n = 50 estimate unsupported; at n = 16 they report more than we reach at any budget measured. One optimizer beats ours: L-BFGS-B ends 3.9 +/- 1.6% above converged Adam at n = 16, on three instances. That margin retires our hardness conjecture under its registered rule and leaves the rate untouched: every optimizer measured loses a factor 1.3 per qubit. The barren plateau is present and cannot explain that rate: the reach sits far above the Haar floor 2^-n, and the exact second-order amplitude data are depth-independent while the reach is not. Nor do solutions cluster at that floor: they are decorrelated yet path-connected by paths 10^2-10^3 above 2^-n whose floor falls from 0.73 to 0.23 of the endpoints. Path search is one-sided, so near-optimal clustering stays open. In the deep limit we prove no poly(n)-parameter family beats poly(n) 2^-n on average. What survives: a connected landscape and a shrinking reach.