Bernstein-Vazirani Networks

Bernstein-Vazirani Networks:
Quantum Machine Learning by Interference

¹University of Siegen ²Imperial College London ³Halmstad University ⁴MPI für Informatik

^\(\dagger\) denotes equal advising

ArXiv 2026

Overview

BNVs in action

Bernstein- Vazirani Networks introduce a non-variational quantum machine learning (QML) framework driven by quantum interference for supervised learning. 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. Generalised BVNs enable interference in problem-adapted bases and yield more expressive models. BVNs achieve universal function approximation through (over)complete interference bases, while training of BVNs is gradient-free.

Motivation

SoTA QML Pradigm

Current predominant QML paradigms attempt to project classical machine learning paradigms onto the quantum domain. They use parameterised quantum circuits (PQCs) and an outer classical loop to optimise the circuit parameters.

Challenges of variational QML

Classical–quantum mismatch: classical neural networks rely on nonlinear activations, whereas closed quantum systems evolve through linear, unitary dynamics.
Barren plateaus: gradients can vanish exponentially with the number of qubits.
Gradient evaluation: certified methods such as the parameter-shift rule require many circuit evaluations.

→

Zoom-in: gradient-based training cost

The expected circuit-evaluation cost scales as

Cost of partial derivative evaluation
\(\cdot \)
number of trainable parameters
\(\cdot \)
number of training epochs

Consequence: Vanishing gradients + parameter-shift cost + measurement randomness → substantial sampling overhead.

But What are Quantum Computers really good at?

Superposition: Explore many candidate solutions of a problem simultaneously.
Interference: Amplify best solutions and suppress irrelevant ones.

Idea: Define a quantum model (architecture) for the unknown learning target:

explore all possible configurations of this model;
identify the configurations that align with the target;

→ Interference-based learning.

(Generalised) Bernstein-Vazirani Networks

The Bernstein-Vazirani Algorithm as a Learning Model

The Bernstein-Vazirani (BV) algorithm provides a canonical example of quantum interference extracting global structure from data. Given a Boolean function \(f:\{0,1\}^n\rightarrow\{0,1\}\) with the promise \(f(x)=s^Tx \mod 2\), the goal is to recover the unknown parameter vector \(s\) from observed pairs \((x,f(x))\).

The algorithm performs three steps:

(i) Input encoding: Encode all possible inputs simultaneously: \(\frac{1}{\sqrt{2^n}}\sum_x |x\rangle\).
(ii) Oracle encoding: Label each input according to the function value: \(\frac{1}{\sqrt{2^n}}\sum_x(-1)^{f(x)}|x\rangle\).
(iii) Interference: Apply a basis (Hadamard) transform that concentrates the amplitude on the hidden string \(s\).

This is a genuine learning algorithm. BV implement the parametric model \[ N(s,x)=s^Tx \mod 2, \] an identifies the parameter \(s\) out of \(2^n\) many that explains the observed training pairs \((x,f(x))\) via \(f(x) = N(s,x)\) completely by interference.

Standard BVNs

Implicit model: Assume \(f(x) = \chi_y(x), \quad \chi_y(x) = (-1)^{y\odot x}\).
Test the hypothesis: The quantum computer probes \(\widehat f(y) = \langle f, \chi_y\rangle\) for all \(y\)'s simultanously and returns the \(y\)'s with probability \(|\widehat f(y)|^2\).
Effective model: \(f(x) = \sum_y \widehat f(y) \chi_y(x)\).

Generalised BVNs

Assume a more expressive implicit model \(f(x) = \chi_{yzt}(x)\), e.g., by encoding an expressive representations \(\eta_t(x)\) controlled by parameters \(t\).
Test the hypothesis: The quantum computer probes \(\widehat f(yzt) = \langle f, \chi_{yzt}\rangle\) and returns the \(yzt\)'s with probability \(|\widehat f(yzt)|^2\).
Effective model: \(f(x) \approx \sum_{yzt} \widehat f(yzt) \chi_{yzt}(x)\).

Architecture of BVNs

BVNs are two-layer networks with one hidden layer composed of units \(\chi_y\) or \(\chi_{yzt}\), which are already fully optimised, and an output layer that aggregates the hidden activations with weights \(\widehat f(y)\) or \(\widehat f(yzt)\) for the standard and generalised models, respectively. Given the sampled basis states \(y\) or \(yzt\), the output weights \(\widehat f(y)\) or \(\widehat f(yzt)\) are reconstructed classically to find the best approximation of \(f\) in the sampled basis \(\chi_y\) or \(\chi_{yzt}\).

Results

We validate our BVN paradigm on several representative supervised learning tasks of adequate size for classical simulation:

Binary Classification on Synthetic 2D Shapes

BVNs succesfully perform binary-class classification on the the different shapes S1-S10 and challenge the quantum and calssical baselines. On high-frequency shapes such as S2, S6 and S9, the basilines are subject to the spectral bias and perform badly, whereas BVNs succeed. We use a total of 8 and 14 qubits for the standard and generalised BVN, with a total of 100 measurement shots in both cases.

Three-Class Classification on the Real-World Iris and Pinguins Datasets

BVNs succesfully perform multi-class classification on the 4-dimensional real-world Iris and Penguins datasets and stand competitevely along the quantum and calssical baselines. We use a total of 16 and 22 qubits for the standard and generalised BVN, with a total of 100 measurement shots in both cases.

Implicit Image Representation

BVNs are excellent in implicit image representations. The top row shows results on the coarse (64×64) training image, while the bottom evaluates the optimised models on a finer (128×128) testing image. The caption high-variations, and can even interpolate on unseen pixels coordinates. We use a total of 12 and 24 qubits for the standard and generalised BVN, with a total of 10.000 measurement shots in both cases.

Ablations

We ablate the value of quantum interference by replacing the quantum-selected basis states functions by uniformly random basis states of identical cardinality. Clearly, randomly selecting basis functions from the exponentially large pool do not lean to an accurate approximation of our unkown learning target function. Quantum interference truly finds informative basis function that aling with the target.

Citation

@article{meli2026bvns,
  title={Bernstein-Vazirani Networks: Quantum Machine Learning by Interference},
  author={Kuete Meli, Natacha and Birdal, Tolga and Tiwari, Prayag and Golyanik, Vladislav and Moeller, Michael},
  journal={#},
  year={2026}
}

Contact

Natacha Kuete Meli
natacha.kuetemeli@uni-siegen.de
Vladislav Golyanik
golyanik@mpi-inf.mpg.de