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This paper identifies a concrete limitation of amplitude encoding in quantum neural networks: normalization makes the encoded state invariant to nonzero real scaling, so no downstream circuit can distinguish such inputs. The proposed fixed nonlinear feature map removes that ambiguity without trainable parameters and keeps logarithmic qubit scaling. The reported 21-44% accuracy gains are against quantum encoding baselines on small, canonical benchmarks, so they do not establish practical or classical-beating performance.
Overcoming the Representational Limitations of Amplitude Encoding for Quantum Neural Networks
arXiv:2610.41824v1 Announce Type: cross Abstract: Quantum Neural Networks (QNNs) offer a framework for learning functions from classical data, but their expressive power depends critically on how inputs are encoded into quantum states. Angle encoding can require resources that scale with the number of features, whereas amplitude encoding represents $d$ features using only $\mathcal{O}(\log_2 d)$ qubits. This compactness comes at a representational cost: normalization makes conventional amplitude encoding invariant to nonzero real rescaling of the input, up to an unobservable global phase. We show that this invariance induces an exact, architecture-independent restriction on the resulting hypothesis class. No downstream quantum circuit can distinguish inputs differing by nonzero real scaling. We develop a general condition under which deterministic preprocessing can remove this ambiguity and show that nonlinearity alone is insufficient. Based on this analysis, we introduce a fixed nonlinear feature map that provably distinguishes all real inputs under subsequent amplitude encoding, without adding trainable parameters and while retaining logarithmic qubit scaling. We evaluate the resulting QNN on three canonical classification benchmarks and a nonlinear synthetic dataset against conventional amplitude and angle encoding, as well as a classical neural network. Across these tasks, the proposed encoding consistently improves classification accuracy over both quantum encoding baselines by 21$-$44 %. Our results establish normalization-induced scale invariance as a concrete representational bottleneck of amplitude encoding and demonstrate that it can be removed while preserving a compact quantum representation, marking a step toward quantum models capable of capturing arbitrary relationships in classical data.
Abstract:Quantum Neural Networks (QNNs) offer a framework for learning functions from classical data, but their expressive power depends critically on how inputs are encoded into quantum states. Angle encoding can require resources that scale with the number of features, whereas amplitude encoding represents $d$ features using only $\mathcal{O}(\log_2 d)$ qubits. This compactness comes at a representational cost: normalization makes conventional amplitude encoding invariant to nonzero real rescaling of the input, up to an unobservable global phase. We show that this invariance induces an exact, architecture-independent restriction on the resulting hypothesis class. No downstream quantum circuit can distinguish inputs differing by nonzero real scaling. We develop a general condition under which deterministic preprocessing can remove this ambiguity and show that nonlinearity alone is insufficient. Based on this analysis, we introduce a fixed nonlinear feature map that provably distinguishes all real inputs under subsequent amplitude encoding, without adding trainable parameters and while retaining logarithmic qubit scaling. We evaluate the resulting QNN on three canonical classification benchmarks and a nonlinear synthetic dataset against conventional amplitude and angle encoding, as well as a classical neural network. Across these tasks, the proposed encoding consistently improves classification accuracy over both quantum encoding baselines by 21$-$44 %. Our results establish normalization-induced scale invariance as a concrete representational bottleneck of amplitude encoding and demonstrate that it can be removed while preserving a compact quantum representation, marking a step toward quantum models capable of capturing arbitrary relationships in classical data.
Quantum Physics (quant-ph); Emerging Technologies (cs.ET); Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2610.41824 [quant-ph]
(or arXiv:2610.41824v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.41824 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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