One-Qubit Sampler-QNN Overview

• One-qubit Sampler-QNN is a parameterized quantum circuit that employs repeated data re-uploading blocks to encode classical inputs and generate probabilistic outputs via Fourier decomposition.
• The model achieves universal approximation for univariate functions through a constructive synthesis mapping circuit parameters to Fourier coefficients, ensuring expressive capabilities with moderate depth.
• Extensions to multivariate inputs via hybrid or multi-qubit architectures underline challenges in parameter scaling while opening avenues for high-dimensional generative modeling.

A one-qubit Sampler Quantum Neural Network (Sampler-QNN) is a parameterized quantum circuit designed to model conditional probability distributions over single-bit outputs by leveraging quantum data re-uploading architectures. These models utilize the full expressivity of the single-qubit Hilbert space and rigorous connections to partial Fourier series, yielding universality for univariate function approximation but succumbing to expressivity bottlenecks in the multivariate regime. The output, typically the probability of measuring '0' in the computational basis after quantum evolution determined by classical input and trainable gates, is treated as a sample from a learned distribution. The theoretical underpinnings and performance of these architectures have been comprehensively analyzed in Yu et al. (Yu et al., 2022) and Pérez-Salinas et al. (Pérez-Salinas et al., 2021).

1. Circuit Architecture and Data Re-uploading Principle

The circuit architecture of a one-qubit Sampler-QNN employs repeated "data re-uploading" blocks, each consisting of an encoding gate $R_Z(x)$ that injects the classical input $x$ into the qubit state, followed by a trainable single-qubit unitary. The most general form of the $L$-layer model can be written as

$$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$

where $U_3(\theta,\phi,\lambda)$ parametrizes an arbitrary single-qubit rotation:

$$U_3(\theta,\phi,\lambda) = \begin{pmatrix} \cos(\tfrac\theta2) & -e^{i\lambda}\sin(\tfrac\theta2) \ e^{i\phi}\sin(\tfrac\theta2) & e^{i(\phi+\lambda)}\cos(\tfrac\theta2) \end{pmatrix}$$

The initial state $|0\rangle$ is evolved according to $U_{\bm\theta,L}(x)$, after which a computational basis measurement is performed, yielding outcome $y \in \{0,1\}$ with probabilities

$$p(0|x) = |\langle 0|U(x)|0\rangle|^2, \quad p(1|x)=1-p(0|x).$$

The expectation value $x$0 may also be used as the quantum output signal. This data re-uploading principle ensures that the classical input $x$1 is injected at each layer, enabling increasingly expressive models as $x$2 increases (Yu et al., 2022, Pérez-Salinas et al., 2021).

2. Analytic Output Structure and Fourier Representation

The output of the $x$3-layer Sampler-QNN can be decomposed in terms of a partial Fourier series over the input $x$4. Repeated application of $x$5 and trainable unitaries yields an expectation value of the form

$x$6

A more general analytic form for the circuit unitary is

$x$7

where $x$8 are Laurent polynomials in $x$9 with degree at most $L$0 and $L$1. Consequently,

$L$2

This establishes an explicit isomorphism between the trainable parameters of the QNN and the Fourier coefficients of a real-valued function $L$3 (Yu et al., 2022).

3. Universal Approximation for Univariate Functions

A central theoretical result is the rigorous proof of universal approximation for univariate functions. For any square-integrable $L$4 and any $L$5, there exists circuit depth $L$6 and parameters such that the QNN expectation value uniformly approximates $L$7,

$L$8

The constructive procedure relies on: (i) approximating $L$9 via a truncated Fourier series $$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$0, (ii) defining $$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$1, a degree-$$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$2 Laurent polynomial, and (iii) synthesizing circuit angles for $$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$3 to realize $$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$4 as described in quantum signal processing algorithms. This mapping provides both necessary and sufficient conditions for one-qubit QNN expressivity in the univariate case (Yu et al., 2022, Pérez-Salinas et al., 2021).

4. Constructive Parameter Synthesis and Training Protocols

Constructing a QNN to match a target Fourier polynomial involves stepwise "peeling off" layers: given a target $$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$5, one iteratively identifies the leading Fourier term and chooses rotation angles $$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$6 to remove it, continuing until all terms are synthesized. This is precisely the approach in quantum signal processing, enabling control over the functional form of $$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$7.

Training is typically performed by stochastic gradient descent, using the parameter-shift rule to evaluate gradients. For a gate $$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$8,

$$U_{\bm\theta,L}(x) = U_3(\theta_0,\phi_0,\lambda_0) \prod_{j=1}^L [R_Z(x) U_3(\theta_j,\phi_j,\lambda_j)]$$9

Practical guidelines include random uniform angle initialization, using Adam optimization (learning rate $U_3(\theta,\phi,\lambda)$00.1, batch size 20), and 100 iterations to fit smooth univariate targets. Both mean-squared error (for regression) and negative log-likelihood (for sampling) losses are supported. In experimental demonstrations on superconducting transmons, layer depths up to $U_3(\theta,\phi,\lambda)$1 are attainable before hardware errors limit performance (Pérez-Salinas et al., 2021).

5. Empirical Performance and Benchmarking

Benchmark studies have confirmed that one-qubit Sampler-QNNs with modest depth (e.g., $U_3(\theta,\phi,\lambda)$2) can accurately fit varied real and complex univariate targets (e.g., ReLU, $U_3(\theta,\phi,\lambda)$3, sign, absolute cubic polynomials, damped sinc, square waves), often matching or exceeding classical Fourier-series and shallow neural-network approximators for equivalent parameter budgets.

Notably, deeper circuits capture high-frequency structure (e.g., $U_3(\theta,\phi,\lambda)$4 learning a 20-periodic square wave), though with Gibbs phenomena at discontinuities. Error scales with the frequency cutoff in the Fourier expansion; for functions with $U_3(\theta,\phi,\lambda)$5 derivatives, the truncation error decays as $U_3(\theta,\phi,\lambda)$6. Diminishing returns are observed for very large $U_3(\theta,\phi,\lambda)$7, often due to implementation noise for devices with gate errors $U_3(\theta,\phi,\lambda)$8 per layer (Yu et al., 2022, Pérez-Salinas et al., 2021).

The table below summarizes representative applications and key outcomes:

Target Function	Circuit Depth (L)	Empirical Behavior
Damped sinc	5	Captures main lobes
Square wave (20-per)	45	Extrapolates, shows Gibbs ringing
Classical regression / classification	5–6	Matches classical nets / Fourier series
Superconducting qubit experiment	1–6	Quantum/classical parity, limited by hardware error

6. Multivariate Input Expressivity and Extensions

Native one-qubit Sampler-QNNs, when extended to $U_3(\theta,\phi,\lambda)$9-dimensional inputs by inserting $$U_3(\theta,\phi,\lambda) = \begin{pmatrix} \cos(\tfrac\theta2) & -e^{i\lambda}\sin(\tfrac\theta2) \ e^{i\phi}\sin(\tfrac\theta2) & e^{i(\phi+\lambda)}\cos(\tfrac\theta2) \end{pmatrix}$$0 for each $$U_3(\theta,\phi,\lambda) = \begin{pmatrix} \cos(\tfrac\theta2) & -e^{i\lambda}\sin(\tfrac\theta2) \ e^{i\phi}\sin(\tfrac\theta2) & e^{i(\phi+\lambda)}\cos(\tfrac\theta2) \end{pmatrix}$$1, exhibit a severe expressivity constraint. The model output is

$$U_3(\theta,\phi,\lambda) = \begin{pmatrix} \cos(\tfrac\theta2) & -e^{i\lambda}\sin(\tfrac\theta2) \ e^{i\phi}\sin(\tfrac\theta2) & e^{i(\phi+\lambda)}\cos(\tfrac\theta2) \end{pmatrix}$$2

with parameter count scaling as $$U_3(\theta,\phi,\lambda) = \begin{pmatrix} \cos(\tfrac\theta2) & -e^{i\lambda}\sin(\tfrac\theta2) \ e^{i\phi}\sin(\tfrac\theta2) & e^{i(\phi+\lambda)}\cos(\tfrac\theta2) \end{pmatrix}$$3 and $$U_3(\theta,\phi,\lambda) = \begin{pmatrix} \cos(\tfrac\theta2) & -e^{i\lambda}\sin(\tfrac\theta2) \ e^{i\phi}\sin(\tfrac\theta2) & e^{i(\phi+\lambda)}\cos(\tfrac\theta2) \end{pmatrix}$$4, much smaller than the full expansion which would require $$U_3(\theta,\phi,\lambda) = \begin{pmatrix} \cos(\tfrac\theta2) & -e^{i\lambda}\sin(\tfrac\theta2) \ e^{i\phi}\sin(\tfrac\theta2) & e^{i(\phi+\lambda)}\cos(\tfrac\theta2) \end{pmatrix}$$5 coefficients. This leads to a curse of dimensionality: most possible multivariate Fourier coefficients cannot be independently controlled. Hybrid architectures that perform classical preprocessing with trainable weights can mitigate this but shift universality to classical operations. Multi-qubit extensions that enable parallelization and entanglement via CNOT or universal two-qubit gates restore expressivity, allowing for the successful learning of non-trivial multivariate targets absent in the single-qubit regime. Empirical studies on bivariate polynomial regression have verified this principle, with two-qubit, ten-layer circuits capturing 2D targets that one-qubit QNNs cannot (Yu et al., 2022).

7. Applications and Scope of One-Qubit Sampler-QNNs

Sampler-QNNs provide a complete, analytic, and constructive blueprint for modeling any univariate probability density $$U_3(\theta,\phi,\lambda) = \begin{pmatrix} \cos(\tfrac\theta2) & -e^{i\lambda}\sin(\tfrac\theta2) \ e^{i\phi}\sin(\tfrac\theta2) & e^{i(\phi+\lambda)}\cos(\tfrac\theta2) \end{pmatrix}$$6, making them suitable as conditional generative bit-samplers. This is achieved via the explicit map between circuit parameters and Fourier coefficients, and the guarantee that shallow-to-moderate circuit depths suffice for smooth target distributions. Extensions to multi-qubit settings and the inclusion of entangling layers enable controlled sampling of joint bit distributions and move towards true high-dimensional generative modeling.

The theoretical machinery of data re-uploading models—rooted in quantum signal processing synthesis and partial Fourier series representation—has clarified the boundaries between classical and quantum expressiveness within VQAs, provided constructive methods for implementation, and suggested practical training guidelines for real hardware (Yu et al., 2022, Pérez-Salinas et al., 2021).