arXiv:2607.04598quant-phcs.LG2026-07

破解量子电路表达力与可训练性的单一权衡困局,提出双参数设计新范式。

Breaking the One-Dimensional Expressibility-Trainability Tradeoff

论文配图:Breaking the One-Dimensional Expressibility-Trainability Tradeoff
图 1 · 摘自论文原文
  • 区分表达力、纠缠方差与梯度方差,揭示三者不等价
  • 相同表达力下梯度方差差异显著,证明可训练性可独立优化
  • 适合设计高表达力且易训练的量子线路,尤其在变分量子算法中

参数化量子电路(PQC)常面临表达力与可训练性之间的困境:提升表达力和纠缠能力(EP)虽能扩大希尔伯特空间覆盖,却也导致波函数随机化并触发梯度消失(巴伦悬崖)。本文指出这一困境并非一维权衡。传统认知将参数分布覆盖、固定电路的纠缠响应和局部梯度矩三个不同概念混为一谈。对于哈尓乘积输入下的固定电路,EP是输出纠缠分布的全局二重平均,而纠缠能力方差(EPD)是四重波动描述;梯度方差则是由参数光锥与代价可观测量决定的局部二重收缩。该矩层级结构实现解析分离:同等EP未必等同可训练性,表现为具有相同EP但不同EPD与梯度方差的电路。因此,以EP和EPD构成双控制旋钮,引导电路设计:EP衡量覆盖程度,EPD监控输入依赖的波动性。研究发现,某些线路路径可在达到类似哈尓覆盖前,避免EPD与梯度方差崩溃,表明覆盖提升与巴伦悬崖激活是两个独立相变事件。该框架打破了看似一维的表达力-可训练性权衡,提供实用设计准则:在覆盖率高但梯度均质化尚未破坏可训练结构的区间内寻找高效量子电路。

原文摘要 · Abstract (English)

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.

量子电路可训练性表达力梯度消失

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