Speaker
Description
Parameterized quantum circuits (PQCs) are often analysed through uniform, Haar-random, or approximate-design ensembles. These averages give exact barren-plateau statements, but they describe a particular random experiment rather than the process followed by a learning algorithm. A trained circuit is initialized, moved by an optimizer, perturbed by finite-shot or stochastic-gradient noise, and constrained by the geometry of its loss landscape. We introduce training-induced measures as an average-case framework that keeps this dynamical structure. In the zero-noise limit, gradient flow pushes the initialization law forward through parameter space, defining endpoint and path-local measures for losses, gradients, and observable statistics. Conserved or approximately conserved quantities can restrict this support to low-dimensional level sets, so the variance seen by training may differ sharply from a full-domain or Haar average. We then thicken the flow by Brownian noise, obtaining a one-parameter family of measures that interpolates between deterministic training paths, finite-shot or optimizer-noise regimes, and randomizing endpoints when the ansatz supports them. This also separates landscape variation from shot-noise precision floors. The resulting diagnostic asks whether the optimizer can statistically resolve descent directions at the points it actually reaches, rather than whether Haar-random circuits concentrate. This makes flatness a measure-relative notion, not landscape geometry alone.
| Topic | Theoretical foundations of quantum learning |
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