Breaking the One-Dimensional Expressibility-Trainability Tradeoff

2026-07-05

AI Summary

This work challenges the prevailing notion that enhanced expressibility of parameterized quantum circuits (PQCs) inevitably leads to vanishing gradients (i.e., barren plateaus), thereby forming a one-dimensional trade-off. The authors introduce a two-dimensional analytical framework based on entangling power (EP) and entangling power deviation (EPD), integrating global statistics from multiple circuit copies—specifically, two-copy averages and four-copy fluctuations—with local gradient moments constrained by parameter light cones. Their analysis reveals that expressibility and trainability are not mutually exclusive: highly expressive PQCs can retain trainable structures even after achieving Haar-like coverage. Moreover, the onset of such coverage and the emergence of barren plateaus correspond to distinct phase transitions. Guided by these insights, the proposed EP/EPD “dual-knob” design principle enables high expressibility without collapsing gradient variance, thereby transcending the conventional one-dimensional trade-off paradigm.

Abstract

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.

Problem

• expressibility
• trainability
• entangling power
• barren plateau
• parameterized quantum circuits

Innovation

• entangling power
• entangling power deviation
• barren plateau
• parameterized quantum circuits
• expressibility-trainability tradeoff