Stochastic Neural Networks for Quantum Devices

Bodo Rosenhahn; Tobias J. Osborne; Christoph Hirche

quant-phFeb 24, 2026

Stochastic Neural Networks for Quantum Devices

Authors: Bodo Rosenhahn, Tobias J. Osborne, Christoph Hirche

Organizations: Institute for Information Processing (tnt/L3S), Leibniz Universität Hannover, Germany · Institute for Theoretical Physics (ITP/L3S), Leibniz Universität Hannover, Germany

Abstract

This work presents a formulation to express and optimize stochastic neural networks as quantum circuits in gate-based quantum computing. Motivated by a classical perceptron, stochastic artificial neurons are introduced and combined into a quantum neural network. The Kiefer-Wolfowitz algorithm in combination with simulated annealing is used for training the network weights. Several topologies and models are presented, including shallow fully connected networks, Hopfield Networks, Restricted Boltzmann Machines, Autoencoders and convolutional neural networks. We also demonstrate the combination of our optimized neural networks as an oracle for the Grover algorithm to realize a quantum generative AI model.

Figures & tables

Figure 1: Realized network architectures for a quantum computer (from left to right): Shallow (fully connected) neural networks for classification, Hopfield networks for pattern memorization, Restricted Boltzmann Machines, autoencoders for representation learning and convolutional neural networks.

Figure 2: Examples for existing architectures to represent artificial neural networks on a quantum device (taken from [ 16 , 72 , 67 , 4 ] ).

Figure 3: Concept of a quantum perceptron with probabilistic activation. The bias is represented as an RX-Gate and the binary input qubits $a_{i}$ increment the probability of neuron excitation by their angular components $\omega_{i}$ using a controlled gate. To ensure a linear and additive behavior of the angles, directly mapping to the probabilities, the $asin$ -function is used. The right image visualizes the qubit activation on the Bloch sphere (just the real part is shown) and the effect of rotating the qubits, based on the bias and CRX-gates. Please note, that for complex input signals, the model can be extended with CRY- and CRZ-gates in a straightforward way.

Figure 4: Example activations for a perceptron with two binary inputs. The weights are set to [0.4 0.6]. The bottom histograms demonstrate the additive increase of the activation probability.

Panel of Figure 5 — Figure 5: Setup for measuring the impact of noisy gates for the implementation of an artificial neuron realizing a xor-decision.

Figure 6: Probability of activating $\left|q_{2}\right\rangle$ (Fig 5 ) for varying $\left|q_{0}\right\rangle$ and $\left|q_{1}\right\rangle$ . Thresholding the probabilities leads to the desired xor pattern for non-binary inputs (no noise on gates).

Figure 7: Trace distances of the quantum circuit from Fig. 5 when the gates are effected by noise $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ .

Figure 8: Implementation of a shallow neural network on a quantum device. This model consists of 4 qubits as input, three hidden artificial neurons and two output neurons. The (C)RX-Gates contain the learnable parameters.

Figure 9: TSNE-Plots of the used UCI datasets for classification, the iris dataset (left), the wine dataset (middle) and the zoo dataset (right).

Figure 10: Convergence of the optimization algorithm during training the iris dataset. Shown is the loss decrease in blue and the accuracy (of the test data) in red. In this run, the algorithm could perfectly classify all test samples.

Figure 11: Hopfield Network and its corresponding Quantum Circuit for four input qubits. The red boxes are visual marks to show that an input artificial neuron has no direct connection to its corresponding output neuron.

Figure 12: Pattern retrieval of a 9-dimensional input data. The patterns in the upper left (red box) are to be memorized from the Hopfield network. The first row shows the test pattern and the second row the retrieval pattern (after one iteration). Whereas the memorized patterns are successfully reconstructed, the noisy input patterns have activation probabilities leading to the closest memorized pattern.

Figure 13: Pattern retrieval of disturbed input data (left). As recurrent model, the optimized Hopfield network can recover the memorized pattern within a few model iterations (right).

Figure 14: RBM Network and its corresponding Quantum Circuit for four input qubits and two latent space artificial neurons. The red connections indicate a weight sharing. The RBM is bidirectional, thus the weights are used for both, encoding and decoding.

Figure 15: Example features from the Iris dataset and its encoding and decoding. Since the RBM is a probabilistic model, the gray values indicate the probability for a perceptron activation. Thus, already two hidden perceptrons are sufficient to represent the 12-dimensional input space with reasonable quality.

Figure 16: Example features from the Iris dataset and its encoding and decoding using an autoencoder. Since the autoencoder can carry more information than an RBM, the overall reconstruction quality is higher, compared to the former model.

Figure 17: Input/Output to learn all 3-dimensional binary patterns. The subtitle shows the input data, which can be observed with a high likelihood when executing the quantum circuit. To represent this dataset, a three dimensional latent space is at least required.

Figure 18: Training data: All $2^{3}$ 3-dimensional binary digits from $000$ to $111$ . Left: Optimization of a RBM with a latent space of 1-4 dimensions. Right: Optimization of an autoencoder with a latent space of 1-4 dimensions. Both models can capture the data with 3 and more latent space dimensions.

Figure 19: The training data (see text) consists of five 6-dimensional (asymmetric) patterns. Left: Optimization of a RBM with a latent space of 1-4 dimensions. Right: Optimization of an autoencoder with a latent space of 1-4 dimensions. The asymmetry of the training data is very hard to capture for a RBM, whereas an AE can represent the data. This example shows that an AE has in general a higher capacity over a RBM.

Figure 21: The Bars and Stripes patterns.

Figure 22: Binary neural network for classifying a simple pattern

Figure 23: Architecture of the Grover circuit for generating patterns.

Figure 24: Sampled bins when executing Grovers algorithm

Figure 25: Histogram after executing Grovers algorithm for a neural network which detects patterns with two dots.

Quantum Neural Networks

Quantum neural networks use quantum circuits and measurements to learn representations, make predictions, and model complex data.

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Stochastic Quantum Spiking Neural Networks with Quantum Memory and Local Learning

Jiechen Chen, Bipin Rajendran, Osvaldo Simeone

Department of Engineering, King’s College London, London, WC2R 2LS, UK · Institute for Intelligent Networked Systems, Northeastern University London, One Portsoken Street, London, E1 8PH, UK

Quantum Machine Learning Spiking Neural Networks Local Learning Quantum Neural Networks

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Neuromorphic and quantum computing have recently emerged as promising paradigms for advancing artificial intelligence, each offering complementary strengths. Neuromorphic systems built on spiking neurons excel at processing time series data efficiently through sparse, event-driven computation, consuming energy only upon input events. Quantum computing, on the other hand, operates on state spaces that grow exponentially in dimension with the number of qubits -- as a consequence of tensor-product composition -- with quantum states admitting superposition across basis states and entanglement between subsystems. Hybrid approaches combining these paradigms have begun to show potential, but existing quantum spiking models have important limitations. Notably, they implement classical memory mechanisms on single qubits, requiring repeated measurements to estimate firing probabilities, while relying on conventional backpropagation for training. In this paper, we propose a novel stochastic quantum spiking (SQS) neuron model that addresses these challenges. The SQS neuron uses multi-qubit quantum circuits to realize a spiking unit with internal quantum memory, enabling event-driven probabilistic spike generation in a single shot during inference. Furthermore, we study networks of SQS neurons, dubbed SQS neural networks (SQSNN), and demonstrate that they can be trained via a hardware-friendly local learning rule, eliminating the need for global classical backpropagation. The proposed SQSNN model is shown via experiments with both conventional and neuromorphic datasets to improve over previous quantum spiking neural networks, as well as over classical counterparts, when fixing the overall number of trainable parameters, highlighting its potential for event-driven applications such as neuromorphic integrated sensing and communications (N-ISAC).

Abstract

Quantum Neural Networks

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