A Comparative Study of Encoding Strategies for Quantum Convolutional Neural Networks

Quantum convolutional neural networks (QCNNs) offer a promising architecture for near-term quantum machine learning by combining hierarchical feature extraction with modest parameter growth. However, any QCNN operating on classical data must rely on an encoding scheme to embed inputs into quantum states, and this choice can dominate both performance and resource requirements. This work presents an implementation-level comparison of three representative encodings – Angle, Amplitude, and a Hybrid phase/angle scheme – for QCNNs under depolarizing noise. We develop a fully differentiable PyTorch–Qiskit pipeline with a custom autograd bridge, batched parameter-shift gradients, and shot scheduling, and use it to train QCNNs on downsampled binary variants of MNIST and Fashion-MNIST at $4\times 4$ and $8\times 8$ resolutions. Our experiments reveal regime-dependent trade-offs. On aggressively downsampled $4\times 4$ inputs, Angle encoding attains higher accuracy and remains comparatively robust as noise increases, while the Hybrid encoder trails and exhibits non-monotonic trends. At $8\times 8$, the Hybrid scheme can overtake Angle under moderate noise, suggesting that mixed phase/angle encoders benefit from additional feature bandwidth. Amplitude-encoded QCNNs are sparsely represented in the downsampled grids but achieve strong performance in lightweight and full-resolution configurations, where training dynamics closely resemble classical convergence. Taken together, these results provide practical guidance for choosing QCNN encoders under joint constraints of resolution, noise strength, and simulation budget.

💡 Research Summary

This paper presents a systematic, implementation‑level comparison of three representative data‑encoding strategies for quantum convolutional neural networks (QCNNs) under depolarizing noise: Angle (rotation) encoding, Amplitude encoding, and a Hybrid phase/angle scheme. The authors develop a fully differentiable pipeline that couples PyTorch optimization with Qiskit simulation, introducing a custom autograd bridge that implements a batched parameter‑shift rule. By aggregating all ±π/2 parameter shifts for every trainable weight and every circuit in a mini‑batch into a single Estimator call, the backward pass becomes dramatically more efficient. In addition, a shot‑scheduling policy dynamically adjusts the number of measurement shots per epoch, using few shots in early training to accelerate convergence and many shots later for precise evaluation.

The experimental protocol uses binary classification tasks derived from MNIST and Fashion‑MNIST. To keep the computational load realistic, the images are aggressively down‑sampled to 4 × 4 (16 features) for the primary study and to 8 × 8 (64 features) for a replication. Class pairs are selected via a pairwise L2‑distance heuristic to ensure that the reduced data remain separable. Depolarizing noise is applied separately to single‑qubit and two‑qubit gates, reflecting the higher error rates of entangling operations on near‑term hardware. Training employs the Adam optimizer and cross‑entropy loss; the final readout is the Z‑expectation of the last remaining qubit, passed through a classical linear layer.

Results reveal regime‑dependent trade‑offs. In the highly compressed 4 × 4 regime, Angle encoding consistently achieves the highest test accuracy (≈92 %) and exhibits the most gradual degradation as noise probabilities increase. The Hybrid encoder lags behind and displays non‑monotonic performance curves, suggesting sensitivity to the interplay between added phase information and noise. Amplitude encoding, while qubit‑efficient (log₂ d qubits for d‑dimensional inputs), shows strong performance only when the ideal “Initialize” instruction is assumed; the authors acknowledge that realistic state‑preparation circuits would be deeper and more error‑prone, likely reducing its practical advantage.

When the resolution is increased to 8 × 8, the Hybrid scheme gains a noticeable edge under moderate noise levels (single‑qubit error ≈ 0.01, two‑qubit error ≈ 0.02), surpassing Angle encoding by 2–3 % in accuracy. This improvement is attributed to the hybrid’s ability to embed additional phase information, effectively expanding the feature bandwidth available to the QCNN. Amplitude‑encoded QCNNs continue to perform well (≈90 % accuracy) and display training dynamics similar to classical CNNs, but the authors caution that the omission of realistic preparation depth inflates these results.

The paper also discusses several practical contributions. The batched parameter‑shift technique reduces the number of required Estimator calls from O(P) to O(1) per batch, where P is the number of trainable parameters. The shot‑scheduling strategy reduces total wall‑clock time without sacrificing final model quality. All experiments are logged with full configuration details to ensure reproducibility.

Limitations are openly addressed. The depolarizing noise model does not capture all error sources (e.g., measurement errors, crosstalk). The Amplitude encoder’s reliance on an ideal Initialize instruction abstracts away the substantial circuit depth needed on real hardware. Moreover, the exact parameter‑shift rule scales linearly with the number of parameters, which may become a bottleneck for larger QCNNs. Future work is suggested to incorporate more realistic noise models, develop hardware‑friendly state‑preparation methods for amplitude encoding, and explore stochastic or gradient‑free alternatives to the parameter‑shift rule.

In summary, the study provides concrete guidance for practitioners: for low‑resolution, high‑noise scenarios, Angle encoding offers the most robust and hardware‑friendly choice; for moderate resolution and moderate noise, Hybrid encoding can leverage its richer feature map to outperform Angle; Amplitude encoding remains attractive when qubit resources are severely limited, but only if the overhead of state preparation can be mitigated. These insights help inform encoder selection when designing QCNNs under joint constraints of resolution, noise strength, and simulation or hardware budget.