Breaking the One-Dimensional Expressibility-Trainability Tradeoff

Expressive parameterized quantum circuits (PQCs) are often designed under a dilemma: the growth of expressibility and entangling power (EP) that improves Hilbert-space coverage is also expected to randomize an ansatz and activate barren-plateau (BP) conditions. We show that this dilemma is not a one-dimensional tradeoff. The usual picture collapses three inequivalent objects -- parameter-ensemble coverage, fixed-circuit entangling response, and local gradient moments -- into one scalar narrative. For a fixed circuit probed by Haar-product inputs, EP is a global two-copy mean of the output-entanglement distribution, whereas entangling-power deviation (EPD) is a global four-copy fluctuation descriptor. Gradient variance, however, is a local two-copy contraction selected by a parameter light cone and a cost observable. This moment hierarchy yields an analytic separation: equal EP need not imply equal trainability, as witnessed by equal-EP circuits with different EPDs and different gradient variances. These separations turn EP and EPD into a two-dial design rule for PQC ansatzes: EP measures how far the circuit has moved along the coverage dial, while EPD monitors whether input-dependent variability remains. We find that ansatz routes can reach high, Haar-like coverage before EPD and gradient variance collapse, showing that coverage and BP activation are distinct crossover events. The EP/EPD framework thus breaks the apparent one-dimensional expressibility-trainability tradeoff into a practical design rule: search for highly expressive PQCs in the window where coverage is high but BP-like homogenization has not yet erased trainable structure.