Qudit Data Re-Uploading in Quantum ML

• Qudit data re-uploading is a quantum ML technique that repeatedly encodes classical or quantum data into higher-dimensional qudit systems, enabling universal function approximation.
• The architecture employs layered data embeddings and trainable unitaries—using Euler/product and combined exponential methods—to capture deep, nonlinear dependencies.
• Practical implementations balance resource efficiency and gradient behavior while navigating hardware constraints, making qudit re-uploading promising for quantum classifiers.

Qudit data re-uploading refers to a class of quantum machine learning architectures that achieve high expressivity and universal function approximation by interleaving repeated, parameter-dependent embeddings of classical or quantum data into a quantum circuit consisting of one or more $d$-level quantum systems (qudits), with interspersed trainable unitaries. Originally developed for qubit-based models, data re-uploading has been rigorously extended to qudit systems ($d>2$), allowing efficient learning in scenarios where higher local Hilbert space dimensions are required. This paradigm has direct implications for both efficient quantum information processing and the practical implementation of quantum neural networks and classifiers for classical and quantum input data.

1. Circuit Architectures for Qudit Data Re-Uploading

Qudit data re-uploading circuits comprise $L$ alternating layers, each consisting of a data-encoding unitary $E(\mathbf{x})$ and a trainable unitary $U(\boldsymbol{\theta})$ acting on a single qudit initialized in a reference state $|0\rangle$. The final quantum state after $L$ layers reads

$$|\psi(\mathbf{x};\{\boldsymbol{\theta},\boldsymbol{\omega}\})\rangle = \prod_{\ell=1}^L U\bigl(\boldsymbol{\theta}^{(\ell)}\bigr)\, E\bigl(\mathbf{x},\boldsymbol{\omega}^{(\ell)}\bigr)\, |0\rangle,$$

where $\mathbf{x}\in\mathbb{R}^M$ is the feature vector, and $E(\mathbf{x},\boldsymbol{\omega}^{(\ell)})$ encodes the $d>2$0 features into the qudit via a parameterized mapping (e.g., two-level rotations per transition) (Wach et al., 2023, Roca-Jerat et al., 2023).

Two equivalent forms are commonly used:

• Layered Euler/Product form: A sequence of elementary rotations generated by operators such as $d>2$1, $d>2$2, and $d>2$3, with angles affinely dependent on $d>2$4 and additional trainable parameters.
• Combined exponential form: A single exponential of a linear combination of generators with data-dependent and trainable coefficients.

The “re-uploading” property refers to applying $d>2$5 (possibly with distinct parameterizations per layer) multiple times, thereby attaining deep, nonlinear dependencies on $d>2$6 and circumventing the limited expressivity of “single shot” data encoding (Wach et al., 2023, Roca-Jerat et al., 2023).

2. Expressivity, Universality, and Approximation Power

Qudit data re-uploading circuits act as universal function approximators for bounded continuous functions of the input features or of quantum data parameters, provided sufficient layers and a generator set that spans the entire $d>2$7 algebra. The addition of non-commuting operators (e.g., squeezing $d>2$8 for $d>2$9) is essential for achieving full expressivity. The key mechanism is the ability to construct arbitrary polynomials in data parameters (e.g., generalized Bloch-vector components $L$0 for quantum data), by building up monomials across multiple layers and exploiting trainable unitary blocks for flexible coefficient control (Cha et al., 23 Sep 2025, Wach et al., 2023).

The Stone–Weierstrass theorem implies that any continuous target function can be approximated to arbitrary accuracy by increasing the number of re-uploading layers $L$1. Empirical results confirm that, for tasks such as multi-class classification or regression on benchmarks like MNIST or geometric synthetic data, qudit re-uploading attains or exceeds classical classifier performance when $L$2 and $L$3 are chosen appropriately. Notably, for aligned label structures, accuracies surpassing $L$4 have been observed (e.g., 7-class “stripes” task with $L$5, $L$6) (Wach et al., 2023).

3. Generalization to Quantum and Qudit Inputs

Re-uploading quantum data—where the register to be repeatedly encoded is itself a quantum state, not classical features—requires architectures capable of interacting sequentially with fresh copies of the input state $L$7. In the most general construction, an ancilla (either a qubit or qudit) interacts with the $L$8-dimensional data register via a controlled unitary: $L$9 with $E(\mathbf{x})$0, $E(\mathbf{x})$1 a basis for operators on $E(\mathbf{x})$2. Each round discards (traces out) and resets the data register to a fresh copy of $E(\mathbf{x})$3 before proceeding, resulting in a cascade of completely-positive trace-preserving maps acting on the ancilla (Cha et al., 23 Sep 2025).

This construction realizes a universal function approximator over functions of quantum states, with resource requirements independent of the data Hilbert space dimension: the signal register is always a single qudit (or qubit), the number of data qudits is fixed, and circuit depth scales as $E(\mathbf{x})$4 for $E(\mathbf{x})$5 data qudits and $E(\mathbf{x})$6 layers.

4. Theoretical Analysis: Gradients, Frequency Profiles, and Trainability

The theoretical analysis of qudit data re-uploading architectures reveals several key features:

• Gradient behavior: The variance of gradients with respect to trainable parameters in data re-uploading circuits is bounded relative to data-less parameterized quantum circuits (PQCs) via the so-called “absorption witness,” a measure of how much data encoding can be “absorbed” into the parameter space. Properly designed generator sets (within the same Lie algebra) minimize absorption witnesses, avoiding excessive flattening (“barren plateaus”) (Barthe et al., 2023).
• Fourier spectrum: The output of a qudit data re-uploading model for integer-spectrum generators is a generalized Fourier series in the data, with high-frequency components vanishing exponentially fast as $E(\mathbf{x})$7 increases. The typical cut-off frequency increases only as $E(\mathbf{x})$8, so expressivity and model sensitivity to high-frequency structure can be managed by selecting $E(\mathbf{x})$9 (Barthe et al., 2023).
• Lipschitz constants: The effective Lipschitz constant (bounding the model's sensitivity to data changes) grows as $U(\boldsymbol{\theta})$0, reinforcing the smoothness of learned functions and limiting overfitting risk at large depth (Barthe et al., 2023).

5. Practical Considerations: Resource Efficiency, Inductive Bias, and Hardware

Qudit data re-uploading optimizes for compactness and resource efficiency:

• Parameter counts: Expressivity typically scales with the total number of parameters, which can be increased either by deeper circuits (higher $U(\boldsymbol{\theta})$1) or by using more operators per layer (increasing per-layer parameter count) (Wach et al., 2023).
• Label encoding and inductive bias: Embedding class labels directly as computational-basis states $U(\boldsymbol{\theta})$2 exploits the natural structure of the qudit Hilbert space and yields pronounced inductive bias when data geometry aligns with this basis. This can offer clear advantages over qubit encodings, which often require non-orthogonal label states (Wach et al., 2023, Roca-Jerat et al., 2023).
• Comparison with classical models: Empirical studies show that in regimes where the number of features and classes is not much larger than $U(\boldsymbol{\theta})$3, qudit re-uploading can match or outperform standard classical benchmarks (random forest, SVM, $U(\boldsymbol{\theta})$4-NN) for $U(\boldsymbol{\theta})$5 (Roca-Jerat et al., 2023).
• Hardware realization: On NISQ devices, especially when qudits are emulated on multiple qubits via Dicke-state encodings, performance is limited by the fidelity of entangling gates required for implementing squeeze operators (e.g., $U(\boldsymbol{\theta})$6). Without squeezing, hardware and noiseless simulation match up to moderate depths, while squeezing operations degrade with hardware error (Wach et al., 2023).

6. Extensions: Bosonic Systems and Alternative Quantum Platforms

Data re-uploading paradigms extend beyond finite-dimensional qudit systems. In bosonic implementations, such as integrated photonic classifiers, data is encoded into the parameters of interferometric circuits (e.g., phase shifters), re-uploaded across layers, and processed through multi-photon, multi-mode entangling unitaries (Ono et al., 2022). Experimental demonstrations with two-photon, two-mode silicon photonic circuits achieve approximately $U(\boldsymbol{\theta})$7 reproducibility, illustrating the flexibility and power of data re-uploading for non-qubit modalities.

These approaches naturally generalize to arbitrary $U(\boldsymbol{\theta})$8-photon, two-mode systems, with output probabilities taking the form of finite Fourier series determined by the re-uploaded phase parameters. Expressivity grows with the number of layers rather than directly with photon number, facilitating scalable learning on optical platforms (Ono et al., 2022).

7. Limitations, Trade-Offs, and Future Directions

Fundamental resource–expressivity trade-offs include the balance between the number of layers $U(\boldsymbol{\theta})$9 and gate parameter counts, the choice of generators (for covering all of $|0\rangle$0), and the dimension $|0\rangle$1 relative to feature and class count. Models face a bottleneck when input or class dimension substantially exceeds $|0\rangle$2, requiring hybrid quantum–classical strategies such as pre-embedding with classical neural networks before quantum processing (Roca-Jerat et al., 2023).

High-dimensional qudits theoretically provide superior label encoding and more compact Hilbert space representations, but their practical deployment is limited by hardware constraints and increased demands on precise multi-level control. The collision-model interpretation for quantum data re-uploading suggests deep connections to open-system simulation and interactive proof protocols, meriting further investigation (Cha et al., 23 Sep 2025).

A plausible implication is that future advances in native qudit platforms, improved gate fidelity for squeezing and multi-level couplings, and hybrid variational training methods will expand the practical power and scope of qudit data re-uploading in quantum machine learning and quantum information processing.

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Key references:

• Universal quantum data re-uploading: (Cha et al., 23 Sep 2025)
• Single-qudit models and comparison to qubit circuits: (Wach et al., 2023, Roca-Jerat et al., 2023)
• Theoretical analysis for gradients and expressivity: (Barthe et al., 2023)
• Bosonic classifier implementation: (Ono et al., 2022)