Stochastic Neural Networks for Quantum Devices

TL;DR

This work presents a formulation to express and optimize stochastic neural networks as quantum circuits in gate-based quantum computing and demonstrates the combination of the optimized neural networks as an oracle for the Grover algorithm to realize a quantum generative AI model.

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 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 (19)

• 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 neurons on a quantum device (taken from Cong2019Tacchino2019SCHULD2015660Bai2023).
• 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 $\omega_i$. 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 controlled RX-gates.
• 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.
• Figure 5: Implementation of a shallow neural network on a quantum device. This model consists of 4 qubits as input, three hidden neurons and two output neurons. The (C)RX-Gates contain the learnable parameters.