arXiv:2607.11013quant-phcs.LG2026-07

通过频率渐进调度,解决量子分类器的傅里叶锁定问题

Overcoming Fourier Locking in Quantum Data Re-uploading Classifiers via Spectral Homotopy

  • 设计频率渐进协议,逐步提升目标频率以避免优化陷阱
  • 实验显示频率迁移率达-0.48(直接训练)和1.34(课程学习),逃逸率提升至18%
  • 适合研究量子机器学习优化与参数化量子电路的学者

数据重上传参数化量子电路(DRU-PQCs)虽具通用逼近能力,但其表达力导致非凸、振荡的损失景观,阻碍梯度优化。我们发现主要瓶颈并非容量不足,而是称为傅里叶锁定(FL)的结构性失败:编码权重与纠缠层非线性耦合,使高频率目标的随机初始化陷入虚假局部极小值。两个费雪诊断可刻画该现象:输入空间量子费雪信息 $F_x$ 衡量编码态的有效频率内容;特征的费雪判别比衡量其与标签对齐程度。在两组独立50次种子实验中,锁定表现为 $F_x$ 全程冻结,而逃逸电路则随课程学习迁移频率内容(直接训练:$r_{pb} = -0.48$;课程学习:$d = 1.34$;均 $p < 0.001$)。关键特征是频谱可动性而非 $F_x$ 终点值,且被困电路仍保持完全非退化的参数空间QFIM($r_{pb} o 0$):失败源于响应态的频谱错位,而非几何敏感性丧失。频率分阶段同伦协议($f: 1.0 \to 3.0$)使早期损失景观凸化;逃逸电路随课程提升 $F_x$,逃逸率提升至18%(原6%)。傅里叶锁定本质是频率对齐问题,解法在于频率调度。

原文摘要 · Abstract (English)

Data re-uploading parameterized quantum circuits (DRU-PQCs) are universal function approximators, yet their expressivity produces oscillatory, non-convex loss landscapes that resist gradient-based optimization. We show that the primary optimization bottleneck in DRU-PQCs is not insufficient capacity but a structural failure mode we term Fourier locking (FL): because encoding weights and entangling layers are nonlinearly coupled, random initialization on high-frequency targets collapses the encoding parameters into spurious local minima. Two Fisher diagnostics characterize FL. The input-space quantum Fisher information $F_x$ measures the effective frequency content of the encoded state; the Fisher discriminant ratio of the measured features measures their alignment with the class labels. In two independent 50-seed experiments, the locking is literal: trapped circuits hold $F_x$ frozen for the entire run, while escaping circuits migrate their frequency content (direct training: $r_{pb} = -0.48$; curriculum: $d = 1.34$; both $p < 0.001$). The replicated signature is this spectral mobility, not any endpoint value of $F_x$, and trapped circuits retain a fully non-degenerate parameter-space QFIM ($r_{pb} \approx 0$): the failure is spectral misalignment of a responsive state, not a loss of geometric sensitivity. A frequency-staged homotopy protocol that paces the target frequency ($f: 1.0 \to 3.0$) convexifies the early loss landscape; escaping circuits raise $F_x$ in step with the curriculum, and the escape rate triples (18% vs. 6%). Fourier locking is a frequency-alignment problem, and its remedy is frequency pacing.

量子机器学习优化方法参数化量子电路傅里叶锁定

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