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

Hyunho Cha, Daniel K. Park, Jungwoo Lee

TL;DR

This work proposes and analyzes a quantum data re-uploading architecture in which a qubit interacts sequentially with fresh copies of an arbitrary input state, and provides a qubit-efficient and expressive approach to designing quantum machine learning models that operate directly on quantum data.

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. 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 qubit-efficient and expressive approach to designing quantum machine learning models that operate directly on quantum data.

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

TL;DR

This work proposes and analyzes a quantum data re-uploading architecture in which a qubit interacts sequentially with fresh copies of an arbitrary input state, and provides a qubit-efficient and expressive approach to designing quantum machine learning models that operate directly on quantum data.

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. 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 qubit-efficient and expressive approach to designing quantum machine learning models that operate directly on quantum data.

Paper Structure

This paper contains 16 sections, 1 theorem, 94 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Main results
  4. Re-uploading a single-parameter single-qubit state
  5. Re-uploading an arbitrary single-qubit state
  6. Re-uploading an arbitrary multi-qubit state
  7. Interpretation as an entangling measurement
  8. Experiments
  9. Polynomial fitting
  10. Purity classification
  11. Entanglement entropy classification
  12. Classification on the Bloch Sphere
  13. Discussion and outlook
  14. Derivation of Eq. \ref{['equation:evolution_formula']}
  15. Details of the purity test
  16. ...and 1 more sections

Key Result

Theorem 1

If $f$ is a continuously differentiable function from an open subset $\Tilde{S} \subset \mathbb{R}^n$ into $\mathbb{R}^n$, and the derivative $f'(p)$ is invertible at a point $p$, then there exist neighborhoods $S$ of $p$ in $\Tilde{S}$ and $T$ of $q = f(p)$ such that $f(S) \subset T$ and $f : S \to

Figures (11)

  • 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.
  • 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.
  • Figure 4: Two equivalent representations of our re-uploading model.
  • Figure 5: Polynomial fitting results for the state in Eq. \ref{['equation:psi_t_vector']}, 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 \ref{['fig:reupload_restricted']}. 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. \ref{['equation:delta_parameters']} for different values of $\Delta$.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Theorem 1: hormander2015analysis, Theorem 1.1.7
  • proof
  • proof