Bernstein-Vazirani Networks: Quantum Machine Learning by Interference

Natacha Kuete Meli; Tolga Birdal; Prayag Tiwari; Vladislav Golyanik; Michael Moeller

quant-phAug 19, 2026

Bernstein-Vazirani Networks: Quantum Machine Learning by Interference

Authors: Natacha Kuete Meli, Tolga Birdal, Prayag Tiwari, Vladislav Golyanik, Michael Moeller

Organizations: Department of Vision and Graphics, University of Siegen · Department of Computing, Imperial College London · School of Information Technology, Halmstad University · 44D and Quantum Vision, Max Planck Institute for Informatics

Abstract

We introduce Bernstein-Vazirani Networks (BVNs), a non-variational quantum machine learning framework that leverages quantum interference for supervised learning, demonstrated on vision and representation learning tasks. In their standard form, BVNs follow the principle of quantum Fourier sampling: labelled data are placed in superposition and interfered in the Fourier basis to extract globally informative features. We then define generalised BVNs that enable interference in problem-adapted bases, yielding more expressive models under the same measurement budget as in the standard setting. BVNs achieve universal function approximation through (over)complete interference bases, while training of BVNs is gradient-free. Experiments on synthetic and real-world classification tasks, as well as implicit image representation, show strong generalisation capabilities and competitive performance with classical and quantum baselines.

Figures & tables

Panel of Figure 1 — Figure 1: The proposed standard ( 1(a) ) and generalised ( 1(b) ) Bernstein–Vazirani Networks (BVNs). In the standard setting, binary inputs $x_{i}$ are linearly combined in a hidden layer with optimised binary weights $y^{j}$ , passed through a parity activation $\chi$ , and linearly combined at the output layer using Fourier coefficients $\widehat{f}(y^{j})$ . In the generalised setting, inputs pass through expressive optimised sub-models $\chi$ (cartooned in the red box) before the output combination, enabling richer representations. ( 1(c) ): Illustratively, we classify 2D points where the target function (left) is a sum of expressive sub-models $\chi$ of the generalised basis. The generalised model fits the boundaries exactly, while the standard model has not yet converged. “Acc.” denotes accuracy. See Secs. 4 and 5 for more details.

$\mathbb{Z}_{2}^{n}$	$n$ -dimensional Boolean space, $\mathbb{Z}_{2}^{n}=\{0,1\}^{n}$
$\odot$ , $\oplus$	inner-product and sum $\bmod 2$ , respectively
$\tilde{f}$ , $f$	standard and generalised target function
$\hat{f}(\cdot)$	Fourier coefficient of $f$
$H$	Hadamard transform
$I$	interference operator
$U_{f}$	oracle unitary for function $f$

Table 1: List of symbols used in this paper.

Figure 2: The Bernstein–Vazirani algorithm ( Bernstein and Vazirani, 1993 ) . The algorithm mainly consists of three steps: (i) initialise the system in a perfect superposition; (ii) call the function oracle; (iii) apply the Hadamard operator for interference. If

\tilde{f}

satisfies the promise

\tilde{f}(x)=s\odot x

, measuring reveals

s

with certainty.

Figure 3: Our generalised Bernstein–Vazirani algorithm. We use three quantum registers: An input register, a register that computes a weighted representation of the inputs, and a weight register controlling this representation. The algorithm has three stages: (i) Prepare a uniform superposition of all inputs (Prep.) and weight configurations, and compute the representations; (ii) Query the oracle encoding the ground‑truth labels across all samples in the superposition; (iii) Apply an interference operator so the labels interfere with the basis functions generated by the representation.

Panel of Figure 4 — Figure 4: Binary classification results on synthetic 2D datasets. The top and bottom rows show BVN results with Hadamard and Chebyshev interference, with columns specifying data fractions for training. Chebyshev outperforms Hadamard; the generalised BVN outperforms the standard BVN. BVNs outperform MLP, SVM, and SQC, particularly on high-frequency shapes (S2, S6, and S9).

Figure 5: Real‑world classification results on Iris and Penguins datasets, with filled vs. unfilled (original) comparison for BVNs. Filled data substantially improves the generalised BVN, allowing

{>}90\%

accuracy with only

25\%

data, comparable to the baselines.

Figure 6: Implicit image representation. The top row shows results on the coarse (64×64) training image, while the bottom row shows evaluations on a finer (128×128) test image. The generalised BVN outperforms the standard BVN and is competitive with the benchmarks.

Model	Num. of qubits	Params (PQC)	Params (class.)	Epochs $\equiv$ Basis states for BVNs	Shots (parameter-shift)	Runtime (back-prob.)
Classification (2D), Figure 4
MLP	–	–	151	30	–	2.70s
SVM	–	–	253	–	–	0.001s
SQC	1	120	–	30	$120\cdot 30(2c)$	37.5s
Std. BVN	8(+1)	–	–	45	100	0.027s
Gen. BVN	14(+1)	–	–	95	100	0.039s

Table 2: Resource comparison across all models (quantum methods in blue). We report the expected shot count for parameter‑shift training of PQCs and the runtime using back-propagation. Our BVNs are significantly more efficient in terms of shots and runtime.

		(0, 0.0)	(4, 0.25)	(4, 0.5)	(4, 0.75)	(4, 1)	(10, 1)	(100, 1)	(1000, 1)
Iris	Stad. BVN	0.84	0.78	0.65	0.39	0.33	0.33	0.33	0.33
Iris	Gen. BVN	0.77	0.76	0.74	0.99	0.88	0.99	0.93	0.85
Penguins	Std. BVN	0.60	0.38	0.40	0.23	0.20	0.20	0.20	0.20
	Gen. BVN	0.59	0.86	0.85	0.87	0.96	0.93	0.95	0.96

Table 3: Influence of the dummy filling on the performance of BVNs. Tuples (V, F) indicate dummy fill (value, fraction).

Figure 7: Ablation of interference-based sampling versus random uniform sampling. BVN’s performance indicates that interference effectively identifies informative basis functions.

	Noise-free	Noise	Random
Std. BVN	0.99 (0.005)	0.86 (13.610)	0.80 (0.005)
Gen. BVN	0.98 (0.028)	0.92 (13.525)	0.85 (0.028)
SQC	0.96 (112.066)	0.99 (11513.526)	-
MLP	0.94 (3.801)	-	-
SVM	0.98 (0.010)	-	-

Table 4: Accuracy and runtime comparison in noise simulation. The numbers in brackets are in seconds. The experimental setting does not apply for classical methods (denoted by “-”).

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: A quantum circuit realising an expressive representation of the input. It implements a fully connected two-layer network. The input is a two-dimensional register

\ket{x_{1}}\ket{x_{2}}

. The hidden layer consists of two units, each computing

\eta_{t_{h_{j}}}(x)=\sigma(t_{h_{j}^{1}}x_{1}+t_{h_{j}^{2}}x_{2})

and the output unit computes

\eta_{o}

in the same way, but acting on the hidden activations instead of the input. The circuit is implemented using quantum arithmetic with ancillary qubits (not shown).

Panel of Figure 9 — Figure 9: All distinct activation patterns produced by the 2-layer binary MLP for all 64 possible weight configurations. Top: thresholds (32, 1) for hidden and output units. Bottom: thresholds (64, 1). Despite the large number of weight settings, only a small number of unique input-space partitions are realisable, highlighting the limited expressivity of this bias-free architecture. Blue means $1$ and grey $0$ .

Figure 10: Visualising the fitting results of the standard (top row) and generalised (bottom row) BVNs for different numbers of shots, using the 2-layer MLP as the expressive representation. With few shots, we observe the effect of the MLP, which activates different regions of the input domain. However, the fixed region boundaries limit the model’s ability to capture fine details in the image.

Figure 11: Quantum circuit implementing the rectangle activation. We circularly shift each input dimension

x

by its corresponding rectangle parameter

t

, and activate the output qubit if all high-significance qubits of

x

are 0, which corresponds to activating the rectangle parametrised by

t

. To reduce artefacts, the superposition register allows for overlapping rectangles by further shifting the input conditionally.

Panel of Figure 12 — Figure 12: All distinct activation patterns produced by the rectangle activation for all $64$ possible weight configurations. Top: activations for $s=0$ . Bottom: activations for $s=1$ . Each weight configuration generates a distinct local activation, illustrating the expressive power of the rectangle activation without requiring biases or additional qubits. Blue indicates $1$ and grey indicates $0$ .

Panel of Figure 13 — Figure 13: Visualising the fitting results of the standard and generalised BVNs for different numbers of shots and two rectangle resolutions, 3x3 (top) and 5x5 (bottom). With a few shots, we observe the effect of the rectangle activation functions. Each rectangle locally modifies the standard BV basis, allowing the model to better fit the target in that region. With a sufficient number of shots, the rectangles are no longer visible, and the generalised model becomes more accurate than the standard one. The finer rectangle resolution outperforms the coarse.

Figure 14: Interference operators in comparison. We visualise the basis functions

\xi_{y}(\cdot)

for the Hadamard, the Fourier and the Chebyshev operators. The Hadamard basis functions are parity activations that yield

\pm 1

depending on whether the weighted sum of the input is odd or even. The Fourier and Chebyshev basis functions, in contrast, allow more variations than

\pm 1

and smoother activations.

Figure 15: Regression of random 1D functions with different frequencies using the proposed interference-based quantum networks. The top and bottom rows show results with the Hadamard and Chebyshev interference operators, respectively. We compare the standard Bernstein–Vazirani model (blue) with the generalised model (red). Within each model, we contrast the Gram inversion regularisation factors

\lambda

. The Chebyshev interference produces smooth functions as it acts in the real domain, while Hadamard, due to rounding, produces step functions as it acts in the binary domain. The generalised method with

\lambda=0.1

fits the target function better than that of the standard model (see step function in column 1). Increasing

\lambda

regularises the approximation.

Panel of Figure 16 — Figure 16: Histograms of sampled coefficients for the 2D synthetic datasets using (a) the standard BVN and (b) the generalised BVN.

Figure 17: Histograms of the sampled coefficients obtained with the standard (top row) and generalised BV (bottom row) methods on the Iris and Penguins datasets. The coefficients are not concentrated and exhibit dispersed magnitudes.

Panel of Figure 18 — Figure 18: Classification results for unfilled (a) and filled (b) training data on the Iris and Penguins datasets for different $\lambda$ . Filling the data strongly affects the performance of the BVNs: Unfilled data lead to poor generalisation, whereas filled data, while degrading the standard model, substantially improve the generalised model, allowing $>90\%$ accuracy with only $25\%$ training data, comparable to the baselines.

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Classical $\mathrm{SU}(2)$ Models Match or Exceed Shallow Variational Quantum Circuits on Vision Benchmarks

Christopher Fulton, Irene Tsapara, Lawrence Fulton

United States Air Force Test Pilot School, Edwards Air Force Base, California, 93524, USA · Department of Engineering, Data, and Computer Science; National University, San Diego, California, 92123, USA · Applied Analytics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA, 02467, USA

Variational Quantum Circuits Image Classification

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Quaternion-valued neural networks and variational quantum circuits (VQCs) both derive local transformations from

\mathrm{SU}(2)

geometry, yet their performance on classical supervised learning remains poorly understood. We compare real-valued, quaternion-valued, and quantum classification heads on identical frozen features across MNIST, FashionMNIST, and CIFAR-10. CIFAR-10 uses a learned 16-dimensional bottleneck and frozen ImageNet-pretrained ResNet18 features to separate architecture from representation quality. Quaternion classifiers match or approach real-valued baselines while outperforming shallow VQCs. On MNIST and FashionMNIST, quaternion networks nearly equal real-valued MLPs, whereas product-state VQCs show lower accuracy and higher cost. On CIFAR-10, quaternion networks retain 94--97% of real-valued performance and remain stable under a 32-fold increase in dimensionality. Product-state circuits underperform quaternion classifiers, while entanglement gives modest grayscale gains but reverses under pretrained CNN features (9.25 pp degradation vs.\ product-state). Fubini--Study/QFI natural gradients improve geometric alignment but not short-horizon loss reduction vs.\ Adam. A Friedman test on five-seed MNIST detects model differences (

χ^2=12.796

p=0.0051

n=5

), with Wilcoxon tests yielding large effect sizes (

d>5

) for QuatNet vs.\ quantum comparisons. For FashionMNIST and CIFAR-10, large effects (

d>2.0

) are the primary statistic given

n=3

. These results indicate that quaternion networks provide efficient, stable

\mathrm{SU}(2)

alternatives to shallow VQCs on tasks lacking intrinsic quantum structure. Shared local

\mathrm{SU}(2)

geometry and shallow entanglement are insufficient, within the regime studied, to confer practical quantum advantage. Conclusions are limited to shallow, measurement-limited circuits on such tasks.

Abstract

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