Breaking the One-Dimensional Expressibility-Trainability Tradeoff
2026-07-06 • Machine Learning
Machine Learning
AI summaryⓘ
The authors explain that designing powerful quantum circuits usually involves a tradeoff: making the circuits more expressive and entangled tends to make them harder to train. They show this tradeoff isn’t just one simple balance but actually involves different aspects of the circuit’s behavior that were previously combined into one idea. By separating how much the circuit covers the whole state space from how much its output varies, the authors provide a better way to understand when training becomes difficult. Their work suggests it’s possible to build circuits that are both expressive and still trainable by focusing on these two different properties separately.
Parameterized Quantum CircuitsExpressibilityEntangling PowerBarren PlateausGradient VarianceTrainabilityHaar MeasureQuantum AnsätzeCost ObservableEntangling-Power Deviation
Authors
Kyoungho Cho, Yu-Seong Jeon, Jinhyoung Lee, Jeongho Bang
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.