Beyond the Classical Ceiling: Multi-Layer Fully-Connected Variational Quantum Circuits

TL;DR

The paper tackles the scalability barrier of variational quantum circuits for high-dimensional data by introducing Multi-layer Fully-Connected VQCs (FC-VQCs), a modular architecture composed of small quantum blocks with structured inter-block mixing that enables end-to-end quantum learning without classical encoders. By restricting local Hilbert space dimensions and using block mixing to expand the receptive field, FC-VQCs achieve linear scalability $O(d)$ and demonstrate the ability to process 300-dimensional industrial data, breaking the Classical Ceiling observed for monolithic VQCs. Empirically, FC-VQCs match or exceed state-of-the-art gradient-boosting methods on high-dimensional tasks while achieving substantial parameter efficiency (often ~15–17× fewer parameters). The work is supported by theoretical results on noise accumulation, receptive-field expansion, and irreducible error across mixing regimes, and points toward practical deployment on NISQ hardware with further future exploration of hardware-noise resilience and scalability.

Abstract

Standard Variational Quantum Circuits (VQCs) struggle to scale to high-dimensional data due to the ``curse of dimensionality,'' which manifests as exponential simulation costs ($\mathcal{O}(2^d)$) and untrainable Barren Plateaus. Existing solutions often bypass this by relying on classical neural networks for feature compression, obscuring the true quantum capability. In this work, we propose the \textbf{Multi-Layer Fully-Connected VQC (FC-VQC)}, a modular architecture that performs \textbf{end-to-end quantum learning} without trainable classical encoders. By restricting local Hilbert space dimensions while enabling global feature interaction via structured block mixing, our framework achieves \textbf{linear scalability $\mathcal{O}(d)$}. We empirically validate this approach on standard benchmarks and a high-dimensional industrial task: \textbf{300-asset Option Portfolio Pricing}. In this regime, the FC-VQC breaks the ``Classical Ceiling,'' outperforming state-of-the-art Gradient Boosting baselines (XGBoost/CatBoost) while exhibiting \textbf{$\approx 17\times$ greater parameter efficiency} than Deep Neural Networks. These results provide concrete evidence that pure, modular quantum architectures can effectively learn industrial-scale feature spaces that are intractable for monolithic ansatzes.

Figures (8)

• Figure 1: Overview of Scalable VQC Architectures.
• Figure 2: Block Mixing Rules.
• Figure 3: Schematic plot of 9t3t1
• Figure 4: Gradient Dynamics on Concrete (SingleVQC_8). The grid displays gradient variance over training epochs. Rows correspond to Stacking Layers $L \in \{1, 3, 5, 7, 9\}$ and Columns to Internal Depth $K \in \{1, 3, 5, 7, 9\}$. Row 1 ($L=1$) represents the monolithic Type 1 baseline, which immediately succumbs to barren plateaus (vanishing variance). Rows 2--5 (Type 2) show that simply stacking monolithic blocks fails to stabilize the gradients.
• Figure 5: Gradient Dynamics on Concrete (Type 3: 8t3t1). Grid layout: Rows $L \in \{1..9\}$, Columns $K \in \{1..9\}$. Unlike the SingleVQC baseline, this modular fully-connected architecture begins to show signs of trainability. While extremely shallow blocks ($K=1$, Column 1) still face barren plateaus, deeper configurations ($K \ge 3$) combined with stacking ($L \ge 3$) exhibit non-zero, healthy gradient variance.

Theorems & Definitions (9)

• Theorem 4.1: Type 2 error propagation bound
• Theorem 4.2: Receptive-field growth under sliding-window mixing
• Theorem 4.3: One-step global receptive field under fully-connected mixing
• Theorem 4.4: Support mismatch bounds and monotone improvement with mixing
• Theorem 3.1: Type 2 error propagation bound
• Lemma 3.2: Block separability without exchange
• Theorem 3.3: Receptive-field growth under sliding-window mixing
• Theorem 3.4: One-step global receptive field under fully-connected mixing
• Theorem 3.5: Support mismatch bounds and monotone improvement with mixing