透過多層變分量子電路緩解貧瘠平原問題

Abstract

本研究針對變分量子電路（Variational Quantum Circuits, VQC）在噪聲中尺度量子（NISQ）硬體上受「貧瘠平原」（barren plateaus）現象限制而難以訓練的問題，提出了一種結合分層正規化與殘差連結的多層變分量子電路（Multilayer VQC, MLVQC）架構，以提升深度電路的可訓練性。每一層 MLVQC 由資料編碼區塊（Encoding Block）與參數化量子區塊（Parameterized Quantum Block）組成，並比較了三種正規化方案（無正規化、L2 範數、Linf 範數）與兩種量測方式（Pauli‐$Z$ 期望值與測量機率）。 於 MNIST、FashionMNIST 及 CIFAR-10（各取二類、五類及十類子集）之分類任務中進行實驗。在大多數情境下，L2 正規化搭配 Pauli‐$Z$ 量測可提供最佳準確率：五層 MLVQC 在 CIFAR-10 十類任務上達到最高 32.7％，較未正規化基準提升逾 10％。引入殘差連結後，模型最高準確率進一步提升至 35.7％，且可穩定擴展至六層以上而無效能衰退。 理論分析基於 Haar 隨機電路證明，L2 正規化能有效阻止梯度方差的指數衰減。綜合實驗與理論結果，本研究所提出的設計原則為訓練深度 VQC 提供了一條可擴展且穩健的途徑，進而推動量子機器學習在 NISQ 時代的實際應用。

Variational Quantum Circuits (VQCs) hold promise for hybrid quantum–classical machine learning on noisy intermediate-scale quantum (NISQ) hardware, yet their practical utility is severely constrained by barren plateaus—regions of vanishing gradient that render parameter optimization infeasible as system size grows. In this work, we introduce a Multilayer Variational Quantum Circuit (MLVQC) architecture that integrates layer-wise normalization and residual connections to mitigate barren-plateau effects and enhance trainability. Each MLVQC layer comprises an input encoding block followed by a parameterized quantum block; we systematically evaluate three normalization schemes (none, L2, Linf) and two measurement protocols (Pauli-$Z$ expectation and outcome probabilities). We further incorporate a learnable residual gate—analogous to classical network shortcuts—that adaptively blends each layer’s output with its input, preserving gradient flow in deeper circuits. We benchmark our approach on classification tasks drawn from MNIST, FashionMNIST, and CIFAR-10—each reduced to two-, five-, and ten-class subsets—and observe that in the vast majority of scenarios, L2 normalization combined with Pauli-$Z$ measurements yields the highest accuracy. With a five-layer MLVQC, this setup achieves up to 32.7 \% accuracy on the ten-class CIFAR-10 task, representing an improvement of over 10 \% versus unnormalized baselines. Furthermore, the introduction of residual connections elevates peak performance to 35.7 \% and allows circuits to scale beyond six layers without any drop in accuracy. Our theoretical analysis, based on Haar-random circuit models, corroborates these findings by showing that L2 normalization arrests the exponential decay of gradient variance. Collectively, these design principles offer a scalable pathway for training deep quantum circuits, broadening the applicability of VQC in practical quantum machine learning.