Vincent Gurgul of Humboldt-Universität zu Berlin and David Prinz of Rheinische Friedrich-Wilhelms-Universität Bonn have developed a new feature map that boosts the accuracy of quantum neural networks by up to 44% on benchmark datasets. Their work addresses a fundamental limitation of conventional amplitude encoding, a method for representing data on quantum computers, which cannot distinguish between inputs differing by nonzero real scaling.

The researchers demonstrate that this scale invariance restricts the types of functions a quantum neural network can learn, and their new feature map removes this ambiguity while maintaining efficient qubit scaling. This advancement establishes normalization-induced scale invariance as a concrete bottleneck and moves quantum models closer to capturing arbitrary relationships in classical data.

Amplitude Encoding’s Scale Invariance Limits Quantum Neural Networks

Conventional amplitude encoding, a technique for representing data within quantum neural networks, introduces a fundamental limitation by failing to differentiate between inputs that vary by a simple rescaling factor. This means a quantum neural network employing standard amplitude encoding would treat the inputs 2.0 and 4.0 as identical, regardless of any subsequent quantum processing. This discovery stems from the normalization inherent in the encoding process, which creates an invariance to non-zero real rescaling of the input data, up to a global phase.

No amount of further quantum computation can overcome this inability to distinguish scaled inputs, effectively limiting the model’s expressive power. Researchers addressed this constraint by developing a deterministic preprocessing step, a fixed nonlinear feature map, designed to remove the ambiguity before the data is even encoded into a quantum state.

This map operates without adding any trainable parameters, preserving the logarithmic qubit scaling that makes amplitude encoding attractive. The team’s solution involves a map, denoted as Φ, which transforms the original input data into a higher-dimensional space where scaled versions of the input become distinguishable. Evaluation on three canonical classification benchmarks and a nonlinear synthetic dataset demonstrated a consistent improvement in performance. Across these tasks, the proposed encoding consistently improves classification accuracy over both quantum encoding baselines by 21-44%.

Their results establish normalization-induced scale invariance as a concrete representational bottleneck and demonstrate that it can be removed while preserving a compact quantum representation. This work represents a step toward quantum models capable of capturing arbitrary relationships in classical data, addressing a key challenge in the development of practical quantum machine learning algorithms.

By pinpointing and mitigating this specific limitation of amplitude encoding, the team has opened the way to more expressive and accurate quantum neural networks, potentially broadening their applicability to complex real-world problems. The findings emphasise the importance of carefully considering the representational properties of data encoding methods in quantum machine learning, and the potential for clever preprocessing techniques to overcome inherent limitations.

Logarithmic Qubit Scaling with Amplitude vs. Angle Encoding

This improvement, achieved on datasets including Breast Cancer and Wine, narrows the performance gap with classical neural networks while retaining the efficiency of logarithmic qubit scaling. The work details a method for recovering scale information lost during conventional amplitude encoding, a critical step toward more expressive quantum models. The preservation of logarithmic qubit scaling, representing ‘d’ features using O(log₂d) qubits, is a key advantage for resource-constrained quantum computing. The synthetic dataset also showed a notable improvement, with the new approach reaching 0.7717 balanced accuracy, exceeding both 0.5996 (amplitude encoding) and 0.6458 (angle encoding).

The team’s analysis extends beyond simply improving accuracy; it identifies a fundamental constraint on the expressiveness of amplitude-encoded quantum neural networks. “Our geometric and empirical analyses show that this loss can limit the expressiveness of the subsequent Variational,” the researchers state, highlighting the importance of recovering lost radial information. While amplitude encoding offers logarithmic qubit scaling, normalization discards information about the magnitude of input features.

The introduced feature map addresses this by effectively recovering that lost information, allowing the quantum neural network to differentiate between scaled inputs. This contrasts with the logarithmic scaling of the new amplitude encoding method, which maintains efficiency even with the tripled dimensionality introduced by the feature map.

The researchers note that the new approach, while slightly less efficient than raw amplitude encoding, still requires significantly fewer qubits than angle encoding for larger datasets. The results demonstrate that the proposed feature map is not merely a performance boost but a fundamental improvement in how classical data is represented within a quantum framework.

Deterministic Feature Maps Resolve Ambiguity in Amplitude Encoding

This improvement stems from the map’s ability to distinguish between inputs that would otherwise appear identical to conventional amplitude encoding methods, a feat previously impossible regardless of subsequent quantum circuit complexity. The work details a deterministic preprocessing step that effectively recovers scale information lost during normalization, a critical factor in the performance of these networks. It cannot differentiate between a scaled input and the original, treating them as equivalent quantum states differing only by an unobservable global phase.

This means any downstream quantum circuit, no matter how detailed or trainable, is inherently unable to discern inputs differing by a non-zero real scaling factor. Their analysis establishes a general condition for removing this ambiguity, revealing that a simple nonlinear function is insufficient to achieve this distinction.

This approach requires O(log₂d) qubits to represent ‘d’ features, maintaining a significant advantage over angle encoding, which demands approximately one qubit per input feature. The performance gains were substantial. On the nonlinear synthetic dataset, the new feature map significantly surpassed angle encoding in terms of accuracy, while still retaining the favorable scaling properties of amplitude encoding.

They found that a feature map consisting only of a nonlinear function may still preserve scaling ambiguities for particular inputs, and can be undefined for inputs equal to zero. This highlights the need for a carefully designed preprocessing step to fully realize the potential of amplitude encoding.

Nonlinearity Insufficient for Removing Encoding Ambiguity

The team’s analysis establishes that simply adding nonlinearity to the encoding process is not enough to resolve this ambiguity. Their work demonstrates that a preprocessing map must be carefully constructed to eliminate the scaling issue before data is embedded into a quantum state. They 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.

This map is designed to provably distinguish all real inputs under subsequent amplitude encoding without introducing any trainable parameters, maintaining the logarithmic qubit scaling advantage of amplitude encoding over angle encoding. This ensures that the feature map eliminates the scaling ambiguity before the data is processed by the quantum circuit.

Similarly, on the wine dataset, the new approach yielded an accuracy of 0.9320 (0.0259), significantly surpassing the 0.6816 and 0.7600 achieved by angle and conventional amplitude encoding, respectively. The results, detailed in the paper, show that the proposed feature map consistently outperforms both angle and conventional amplitude encoding across a range of datasets. The team’s work extends beyond simply improving performance on existing benchmarks. This condition dictates that a deterministic preprocessing step is necessary to eliminate the scaling ambiguity, and that nonlinearity alone is insufficient to achieve this goal.

Performance Gains: Up to 44% Accuracy Boost on Benchmarks

The improvement was quantified using balanced accuracy, ROC-AUC and AUPR, metrics calculated from six cross-validation folds, with the best quantum neural network result per dataset shown in bold in the reported values. The team evaluated performance against a classical neural network, included for reference, which consistently achieved strong predictive performance but was not the primary focus of the study.

The objective, the researchers state, was not to demonstrate a quantum advantage, but to determine how much predictive capability is retained under different encoding schemes and whether the information loss of conventional amplitude encoding can be mitigated. The study’s findings contribute to a growing body of work focused on optimizing quantum data encoding strategies for enhanced machine learning capabilities.

👉 More information
🗞 Overcoming the Representational Limitations of Amplitude Encoding for Quantum Neural Networks
✍️ Vincent Gurgul (Chair of Information Systems); David Prinz (Lamarr Institute for Machine Learning and Artificial Intelligence)
🧠 ArXiv: https://arxiv.org/abs/2610.41824

More like this