The optimization landscape of peaked-circuit generation

Research Question and Approach

Random-circuit sampling is a leading path to quantum advantage, but verifying its output is classically intractable for large systems. Peaked circuits offer a potential workaround by concentrating an O(1) fraction of their output weight on a single computational-basis string, making verification efficient. However, the classical generation of these circuits remains a bottleneck. Prior work proposed optimizing a trainable brickwall circuit appended to a random circuit using gradient descent, but observed performance plateaus. This paper investigates the optimization landscape of peaked-circuit generation to resolve whether this plateau stems from optimizer failure or fundamental circuit hardness.

The author maps the optimization landscape by analyzing ensembles of random circuits across system sizes from n = 8 to n = 16. Using fixed and converged optimization budgets, the study evaluates optimizer reach, exact moment statistics via Weingarten calculus, and solution connectivity.

The study reveals that the optimization reach shrinks faster than previously estimated. While earlier work fit optimizer performance to a fixed-base exponential decay, consolidated measurements show that the decay rate steepens monotonically from 1.16 to 1.32 per qubit through n = 16, leaving previous large-scale extrapolations unsupported. Furthermore, second-order amplitude data are depth-independent, whereas the actual optimizer reach is not, indicating that the advantage exploited by the optimizer is acquired during the optimization process rather than being present at initialization.

Analysis of the landscape topology shows that the objective mean is flat and barren plateaus alone cannot account for the performance limits. Solutions are pairwise decorrelated down to the Porter-Thomas floor, yet path-searching reveals that the near-optimal set forms a single connected shelf free of fragmentation at the trap scale. Although a quasi-Newton optimizer (L-BFGS-B) slightly outperforms standard gradient descent at n = 16, every tested optimizer experiences a shrinking reach characterized by a rate of approximately 1.3 per qubit.