Data Re-Uploading

A single layer of angle encoding loads each feature once, and a one-qubit model so built can only represent very simple functions. Data re-uploading breaks that ceiling: it interleaves the data encoding with trainable rotations repeatedly, so even one qubit becomes a universal function approximator (Pérez-Salinas et al., 2020).

The idea

Instead of encode-once-then-train, alternate the two:

|\psi(x,\boldsymbol{\theta})\rangle = \Bigl[\prod_{\ell=1}^{L} W(\boldsymbol{\theta}_\ell)\,S(x)\Bigr]|0\rangle,

where $S(x)$ is the data-encoding block (a Pauli rotation such as $R_Y(x)$ ) and each $W(\boldsymbol{\theta}_\ell)$ is an independent trainable block. The same feature $x$ is fed in $L$ times, once per layer. The repetition is what buys expressivity.

Why repetition helps: the Fourier view

Schuld, Sweke, and Meyer (2021) gave the clean explanation. A variational model with data entered through Pauli rotations computes a truncated Fourier series in the input:

f(x) = \sum_{\omega \in \Omega} c_\omega\, e^{i\omega x}.

The frequencies $\Omega$ that the model can access are fixed by how many times and how the data is encoded — not by the trainable weights. Encoding $x$ only once gives a single nonzero frequency, so the model is essentially a sinusoid. Re-uploading $L$ times enlarges the accessible frequency spectrum, letting the trainable coefficients $c_\omega$ build richer functions. The trainable layers set the coefficients; the encoding repetition sets the frequencies.

A two-layer single-qubit model

Take the encoding generator to be $R_Y(x)$ and a trainable $R_Z(\theta_\ell)$ per layer. Two layers give

|\psi\rangle = R_Z(\theta_2)\,R_Y(x)\,R_Z(\theta_1)\,R_Y(x)\,|0\rangle,

with the same $x$ uploaded in both $R_Y$ blocks. The two distinct trainable angles $\theta_1, \theta_2$ act between and after the uploads. Already this reaches a two-frequency Fourier model — far beyond what a single upload can express.

Try it

Build the two-layer re-uploading state for input $x = 0.7$ with trained angles $\theta_1 = 0.4$ and $\theta_2 = 1.1$ . Apply gates so the circuit realizes

R_Z(\theta_2)\,R_Y(x)\,R_Z(\theta_1)\,R_Y(x)\,|0\rangle,

i.e. (in call order) $R_Y(x)$ , $R_Z(\theta_1)$ , $R_Y(x)$ , $R_Z(\theta_2)$ . The grader checks the state vector. Note the same $x$ appears in both $R_Y$ gates — that repetition is the whole point.

Run your code to see the quantum state.