Breaking the One-Dimensional Expressibility–Trainability Tradeoff

Parameterized quantum circuits (PQCs) are often designed under the assumption that increasing expressibility and entangling power (EP) inevitably leads to barren plateaus (BP). This paper challenges this scalar narrative by showing that expressibility, entangling power, and trainability are fundamentally different objects that do not necessarily collapse into a single tradeoff. The authors propose a moment hierarchy: EP is a global two-copy mean, EPD is a global four-copy fluctuation descriptor, and gradient variance is a local two-copy contraction.

The authors introduce a two-dial design rule to navigate the PQC landscape. The first dial, EP, measures how far an ansatz has moved toward Haar-like coverage. The second dial, EPD, monitors whether the output entanglement generated from product inputs remains input-dependent. By analyzing these two dials, the authors show that ansatz routes can reach high, Haar-like coverage before the EPD and gradient variance collapse. This identifies a sweet spot: a high-coverage window where the circuit is expressive enough for complex tasks but has not yet succumbed to the homogenization that triggers barren plateaus.

The paper proves that equal mean EP does not imply equal trainability. Through an explicit construction, the authors demonstrate that two entangling blocks can have identical EP but different EPDs and different gradient variances when inserted into the same architecture. This confirms that mean descriptors are structurally insufficient to characterize trainability. Instead, the authors provide a constructive screening principle: maximize coverage without crossing into the low-EPD regime where gradient variance is expected to vanish.

This framework provides a practical, task-independent screening tool for quantum machine learning and variational algorithms. Rather than blindly increasing circuit depth—which often drives a circuit toward the homogenized near-plateau sector—researchers can use the EP/EPD plane to select depths and architectures that balance coverage and variability. This approach helps identify high-performance candidates that retain finite-shot gradients, offering a more nuanced and constructive path for PQC design.