Re-uploading quantum data: a universal function approximator for quantum inputs

Hyunho Cha; Daniel K. Park; Jungwoo Lee

quant-phSep 23, 2025

Re-uploading quantum data: a universal function approximator for quantum inputs

Authors: Hyunho Cha, Daniel K. Park, Jungwoo Lee

Organizations: NextQuantum and Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Republic of Korea · Department of Statistics and Data Science, Yonsei University, Seoul 03722, Republic of Korea · Department of Applied Statistics, Yonsei University, Seoul 03722, Republic of Korea · Department of Quantum Information, Yonsei University, Incheon 21983, Republic of Korea

Abstract

Quantum data re-uploading has proved powerful for classical inputs, where repeatedly encoding features into a small circuit yields universal function approximation. Extending this idea to quantum inputs remains underexplored, as the information contained in a quantum state is not directly accessible in classical form. We propose and analyze a quantum data re-uploading architecture in which a qubit interacts sequentially with fresh copies of an arbitrary input state. The circuit can approximate any bounded continuous function using only one ancilla qubit and single-qubit measurements, provided that sufficiently many copies of the input state and, in general, sufficiently many layers are available. By alternating entangling unitaries with mid-circuit resets of the input register, the architecture realizes a discrete cascade of completely positive and trace-preserving maps, analogous to collision models in open quantum system dynamics. Our framework provides a space-efficient and expressive approach to designing quantum machine learning models that operate directly on quantum data.

Figures & tables

Figure 1: A single-qubit classical data re-uploading scheme.

U^{(l)}(\mathbf{x})

denotes an input-dependent data encoding gate, while

V^{(l)}(\boldsymbol{\theta})

denotes a trainable input-independent gate.

Panel of Figure 2 — Figure 2: (a) A general quantum data re-uploading scheme for universal function approximation. $\rho\in\mathcal{D}(\mathcal{H}_{d})$ denotes the quantum input and gray boxes indicate arbitrary $\text{SU}(2d)$ elements. (b) The Hadamard test circuit can be used to estimate $\mathrm{tr}(\rho U)$ by measuring the expectation value $\langle Z\rangle$ on system $A$ , using two separate configurations: $V=H$ to obtain $\text{Re}(\mathrm{tr}(\rho U))$ , and $V=S^{\dagger}H$ to obtain $\text{Im}(\mathrm{tr}(\rho U))$ .

Figure 3: A restricted single-qubit data re-uploading scheme for universal function approximation. $\rho\in\mathcal{D}(\mathcal{H}_{2})$ denotes the single-qubit input and gray boxes indicate arbitrary $\text{SU}(2)$ elements.

Panel of Figure 4 — Figure 4: Two equivalent representations of our re-uploading model.

Panel of Figure 5 — Figure 5: Polynomial fitting results for the state in Eq. ( 1 ), where $\lambda=2t^{2}-1$ . Solid green lines denote the target functions, and dotted blue lines denote the fitted results obtained with our re-uploading model. (a) Results for training the restricted re-uploading circuit in Figure 3 . The target function to be fitted is shown at the top of each plot. From left to right: plots with $L=1$ , $L=4$ , and $L=2$ layers, respectively. (b) Approximation without training, using Eq. ( 27 ) for different values of $\Delta$ .

Panel of Figure 6 — Figure 6: Test accuracy curves for (a) purity classification and (b) entanglement entropy classification with $L=1,2,3,4$ .

Panel of Figure 7 — Figure 7: Purity classification histograms for re-uploading depths $L=1$ to $L=4$ , where the corresponding test accuracies are 0.51, 0.61, 0.95, and 0.98, respectively. The $x$ -axis shows purity and the $y$ -axis the sample count. Blue bars indicate predictions of class 1 and red bars predictions of class 0.

Panel of Figure 8 — Figure 8: Entanglement entropy classification histograms for re-uploading depths $L=1$ to $L=4$ , where the corresponding test accuracies are 0.49, 0.84, 0.92, and 0.93, respectively. The $x$ -axis shows entanglement entropy and the $y$ -axis the sample count. Blue bars indicate predictions of class 1 and red bars predictions of class 0.

Panel of Figure 9 — Figure 9: Two classification datasets consisting of unit-norm Bloch vectors $(r_{1},r_{2},r_{3})$ . Red squares and blue circles correspond to classes 0 and 1, respectively.

Panel of Figure 10 — Figure 10: Bloch sphere classification histograms for (a) Figure 9(a) with $L=1$ , (b) Figure 9(a) with $L=2$ , (c) Figure 9(b) with $L=2$ , and (d) Figure 9(b) with $L=3$ , where the corresponding test accuracies are 0.47, 0.99, 0.53, and 0.99, respectively. The $x$ -axis shows $r_{3}$ and the $y$ -axis the sample count. Blue bars indicate predictions of class 1 and red bars predictions of class 0.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 11: The swap test circuit [ 1 , 8 ] , typically used to estimate inner products between pure states, also serves as a tool for estimating the purity of a quantum state

\rho

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Quantum machine learning uses quantum circuits and quantum data to build learning algorithms and study their computational and statistical limits.

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A Tutorial on Quantum Amplitude Embedding with Data Re-Uploading

Hyojun Ahn, Joongheon Kim

Department of Electrical & Computer Engineering, Korea University, Seoul, Republic of Korea

Quantum Circuit Expressibility Quantum Machine Learning Quantum Data Encoding Variational Quantum Circuits Quantum Feature Maps

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Amplitude embedding lets a variational quantum model read many classical features with few qubits, but the model output is then only a quadratic form in the normalized input. Data re-uploading, which encodes the input repeatedly between trainable blocks, can be used to enrich such a model. Because every additional upload of an amplitude embedding repeats a costly state preparation, the gain that an upload buys has to be known before the gates are spent. This tutorial thus addresses the problem of explaining what data re-uploading adds to an amplitude-embedded model and how large the gain is. We express sequential re-uploading and parallel re-uploading with two copies in one notation and show that each upload raises the reachable polynomial order by two. Under one trainable circuit family and one budget of trainable parameters, simulation results show that re-uploading raises the test coefficient of determination from 0.689 to between 0.883 and 0.975 on targets of order four and from 0.337 to between 0.642 and 0.783 on targets of order six, on all ten seeds. On a quadratic target and on handwritten digits, one upload remains the best choice, and every additional upload lowers the gate error that the classifier tolerates. These results help designers match the number of uploads to the polynomial order of the task at hand.

Abstract

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