DARUAN: Quantum Data Re-Uploading Activation

• DARUAN is a quantum-inspired activation mechanism utilizing repeated data encoding to achieve universal function approximation with low parameter counts.
• It employs variational quantum circuits where input features are re-uploaded through shallow single-qubit rotations, enriching the trigonometric frequency spectrum.
• DARUAN integrates seamlessly into quantum and classical neural architectures, offering exponential expressivity and robust generalization with efficient parameterization.

Data Re-Uploading Activation (DARUAN) is a quantum and quantum-inspired nonlinear activation mechanism realized via repeated encoding of input features into a variational quantum circuit, typically a shallow single-qubit circuit. Originally introduced to address the expressivity bottlenecks of parametric quantum circuits and later formalized for integration in Kolmogorov-Arnold Networks (KANs), DARUAN achieves universal function approximation with dramatically lower parameter counts than classical Fourier-based or spline-based activations, while offering strong generalization and trainability guarantees on both quantum hardware and classical neural architectures (Jiang et al., 17 Sep 2025).

1. Circuit Architecture and Mathematical Foundations

DARUAN is built by composing multiple "re-uploading" layers, where each layer interleaves a feature-dependent unitary (encoding the input variable) and a trainable single-qubit (or multi-qubit, but typically single-qubit) unitary. For a scalar input $x$ (or vector $x \in \mathbb{R}^d$), the canonical single-qubit DARUAN circuit with $r$ repetitions takes the form: $$U_\omega(x) = W^{(r+1)} \prod_{\ell=1}^r \left[ S(w_\ell x + b_\ell) W^{(\ell)} \right]$$ where $$S(w_\ell x + b_\ell) = \exp[-i (w_\ell x + b_\ell) H]$$ with $H = \sigma_j/2$, and $W^{(\ell)}$ are arbitrary trainable unitaries (e.g., parameterized Euler rotations). The input feature(s) is re-uploaded $r$ times, each time with its own pre-processing weight $w_\ell$ and bias $b_\ell$.

Measurement of an observable $M$ (typically $M = \sigma_z$) on the evolved state yields the activation output: $$\varphi_\theta(x) = \langle 0 | U_\omega(x)^\dagger M U_\omega(x) | 0 \rangle$$ This gives a smooth, bounded, real-valued nonlinearity, and serves as the activation in downstream neural architectures.

2. Frequency Spectrum Expansion and Parameter Complexity

DARUAN circuits are distinguished by an exponentially enriched trigonometric spectrum as a function of re-upload repetitions $r$. With constant weights $w_\ell \equiv 1$, the set of accessible frequencies is $\{-r, ..., r\}$ (linear growth). When the $w_\ell$ are trainable, the spectrum comprises all integer linear combinations $\sum_{\ell=1}^r m_\ell w_\ell$ with $m_\ell \in \{-1,0,1\}$, admitting up to $3^r-1$ distinct frequencies. Notably, choosing $w_\ell = 2^{\ell-1}$ yields a maximum frequency $K_B = 2^r-1$ and $|\Omega_B| \sim 2^r$ accessible frequencies—a doubly-exponential expressive palette for only $r$ trainable weights.

Classical Fourier-based activations require $M = \Theta(\epsilon^{-1/(k+1-m)})$ parameters for error $\epsilon$ in the $C^m$ norm for target functions in $C^{k+1}$. By contrast, DARUAN attains the same error with merely $r = \Theta(\log(1/\epsilon))$ repetitions, constituting an exponential reduction in parameter count (Jiang et al., 17 Sep 2025).

3. Embedding DARUAN in Neural and Sequential Architectures

The modularity of DARUAN enables it to serve as a drop-in activation for univariate nodes in mesh architectures such as KAN and its quantum-inspired generalizations (QKAN, QuIRK, HQKAN).

a) KAN Integration:

In classical KANs, each edge $(i \to j)$ applies a learnable univariate nonlinearity. DARUAN replaces $\varphi_{l,j,i}(x)$ by a single-qubit quantum circuit expectation, reducing the required parameter set from grid-size-dependent splines to $O(r)$ per edge. The layer computation is then: $$x_{l+1,j} = \sum_{i=1}^{n_l} \varphi_{l,j,i}(x_{l,i}) \rightarrow \sum_{i=1}^{n_l} \langle 0 | U_{l,j,i}(x_{l,i}; \theta_{l,j,i})^\dagger M U_{l,j,i}(x_{l,i}; \theta_{l,j,i}) | 0 \rangle$$ (Jiang et al., 17 Sep 2025, Sharma et al., 9 Oct 2025).

b) LSTM and RNN Structures:

DARUAN modules have been incorporated into gating mechanisms of LSTM variants, notably the QKAN-LSTM. Each gate computes: $$\Phi_g(v_t) = \sum_{p=1}^\alpha \varphi_{g,p}(w \cdot v_t; \theta_{g,p}, \{w_{g,p}^{(\ell)}\})$$ where $\varphi_{g,p}$ is a single-qubit DARUAN activation, $w$ are classical projection weights, and $\alpha \ll m$ is the number of parallel subunits per gate. This structure matches or surpasses classical LSTM performance with up to 79% reduction in parameter count, retaining full expressivity for both short- and long-memory components (Hsu et al., 4 Dec 2025).

c) Reinforcement Learning and Q-Function Approximation:

DARUAN is effective in VQC-based RL agents, as shown in both deep Q-learning and batch-constrained Q-learning settings (BCQQ). Repeated data re-uploading augments network capacity and expands the functional class, enabling rapid convergence and competitive or superior accuracy relative to classical MLP or shallow NNs, especially in low-data regimes (Coelho et al., 21 Jan 2024, Periyasamy et al., 2023).

4. Theoretical Expressivity, Universality, and Generalization

DARUAN’s theoretical guarantees stem from a multivariate Fourier analysis. For activations implemented by DARUAN with suitably chosen $w_\ell$, any $f \in C^{k+1}$ is approximated within $C^m$ error $O(2^{-r(k+1-m)})$ by a depth-$r$ circuit. This exponential convergence equates to the rates obtained by spline-based KANs with exponentially more parameters (Jiang et al., 17 Sep 2025).

Universal function approximation for classical inputs is established for a single-qubit DARUAN activation by repeated data re-uploading, leveraging the Stone–Weierstrass theorem: the functional class grows with the number of layers/repetitions and covers any bounded continuous function on compact domains (Pérez-Salinas et al., 2019, Mauser et al., 7 Jul 2025). The generalization capacity is quantified via a finite VC dimension for the separated (data-encode + rotation) circuit ansatz, with $VCdim = 2L + 1$ for $L$ layers, balancing expressivity and generalization on finite data (Mauser et al., 7 Jul 2025).

For quantum inputs, the re-uploading paradigm extends to interaction between a signal qubit and sequentially reset quantum data registers. Universality still holds: $L$ re-uploadings suffice to approximate any bounded continuous function on quantum state space with error set by the degree of the polynomial approximation induced by circuit depth (Cha et al., 23 Sep 2025).

5. Physical Implementations and Empirical Evaluation

DARUAN has been deployed in various hardware and simulation environments, leveraging the simplicity and coherence of its one-qubit structure.

• Photonic Processors: Experimental demonstrations on integrated photonic chips, using dual-rail single-qubit MZIs, achieve universal image classification and match best-in-class accuracy (e.g., $>95\%$ on MNIST-car-vs-ship with only $L=4$ layers) (Mauser et al., 7 Jul 2025).
• Simulation on NISQ and Classical Hardware: Classical GPU-accelerated simulators tractably run DARUAN-based QuIRK and QKANs with up to $\sim$ 50 qubits and moderate depth, while NISQ hardware supports high-fidelity realization for small depths. The forward pass in classical emulation reduces to sine/cosine evaluations and sparse matrix-vector multiplications, yielding substantial training speedups compared to classical spline- or MLP-based models (Sharma et al., 9 Oct 2025, Hsu et al., 4 Dec 2025).
• Application Domains:
  • Symbolic regression: QKANs outperform classical KANs in accuracy and parameter efficiency on Feynman benchmark tasks (Jiang et al., 17 Sep 2025).
  • Image classification: Hybrid CNN-QKAN models halve the parameter count versus KAN or MLP backends at fixed accuracy on MNIST and CIFAR datasets (Jiang et al., 17 Sep 2025).
  • Time series: Hybrid classical-quantum networks with multiple re-uploaded qubits achieve lower mean squared error than LSTMs on urban traffic and telecommunications datasets (Schetakis et al., 22 Jan 2025, Hsu et al., 4 Dec 2025).
  • Language modeling: Transformer architectures with all MLP components replaced by HQKAN (using DARUAN) reduce parameter counts and memory usage by $35\%$ or more with no loss in perplexity (Jiang et al., 17 Sep 2025).

Empirical evidence further demonstrates that gradient magnitudes and variances in DARUAN-based circuits resist barren-plateau effects, supporting trainability even with moderate circuit depths (Coelho et al., 21 Jan 2024, Mauser et al., 7 Jul 2025).

6. Extensions, Variations, and Optimization

Variants of the basic DARUAN ansatz include:

• Cyclic data re-uploading, where the order of input features is shifted between each layer, enhancing the effective circuit dimension and mitigating feature-starvation in VQCs (Periyasamy et al., 2023).
• Entangled multi-qubit re-uploaded circuits, which accelerate convergence and capture cross-feature correlations for complex, non-convex decision boundaries (Pérez-Salinas et al., 2019, Aminpour et al., 15 May 2024).
• Bosonic and photonic realizations, where multi-photon interference in MZI and phase-shifter networks implement data re-uploading in continuous-variable systems, with theoretical generalization to arbitrary $N$-photon states (Ono et al., 2022).

Classic optimization strategies for training DARUAN layers include L-BFGS-B, COBYLA, Nelder-Mead, SLSQP, and autodiff with parameter-shift rules, with empirical accuracy approaching $98\%$ on benchmark tasks (Aminpour et al., 15 May 2024). Design guidelines emphasize inclusion of trainable input and output scaling, use of regularization (for noise robustness), and judicious selection of repetition depth and qubit count for the targeted functional complexity and available hardware (Jiang et al., 17 Sep 2025, Sharma et al., 9 Oct 2025).

7. Interpretability, Analytic Extraction, and Knowledge Distillation

DARUAN modules, due to their single-variable nature and trigonometric kernel, admit closed-form analytic expressions for activations, supporting full symbolic regression and functional transparency. For smooth regression tasks, DARUAN can be distilled into spline or polynomial coefficients for initializing or regularizing classical networks—a process shown to yield up to 70% lower test loss post-distillation compared to direct spline fitting (Jiang et al., 17 Sep 2025, Sharma et al., 9 Oct 2025). Plotting the learned activation curves for each unit provides both interpretability and insight into frequency adaptation during training. The additive, univariate structure ensures that analysis and visualization of individual components remains as transparent as in classical KANs.

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In summary, Data Re-Uploading Activation (DARUAN) constitutes a foundational primitive for quantum and quantum-inspired neural architectures, enabling exponentially rich nonlinearities, universal function approximation, and highly efficient parameterization, with tractable analytic, computational, and experimental properties across classical and quantum computational substrates (Jiang et al., 17 Sep 2025, Hsu et al., 4 Dec 2025, Mauser et al., 7 Jul 2025, Sharma et al., 9 Oct 2025, Coelho et al., 21 Jan 2024, Cha et al., 23 Sep 2025, Schetakis et al., 22 Jan 2025, Periyasamy et al., 2023, Pérez-Salinas et al., 2019).