Neural Network Quantum Entanglement Computation

Abstract

This paper proposes a novel approach to computation by leveraging quantum entanglement within a simulated neural network. The core idea is to utilize the inherent parallelism of quantum mechanics, specifically quantum entanglement, to achieve computational speeds beyond those attainable by classical neural networks. We introduce a model where artificial neurons are interconnected to mimic quantum entanglement, employing quantum algorithms such as Grover's algorithm for distributed computation. The model incorporates neural network synaptic connections to simulate quantum state superposition and interference. The potential for significant computational advantages through this hybrid approach is explored. The key equations demonstrating the core model are presented below: Let $s_i$ represent the state of neuron $i$, where $i = 1, 2, ..., K$. $s_i \in \{0, 1\}$ The entanglement between neurons $i$ and $j$ is represented as: $E_{ij} = \sqrt{\frac{1}{2}} \cos(\theta_{ij})$ where $\theta_{ij}$ is the angle of entanglement between neurons $i$ and $j$. The update rule for neuron $i$ is given by: $s_i^{(t+1)} = \sum_{j=1}^{K} G_{ij} s_j^{(t)}$ where $G_{ij}$ is the interaction coefficient between neurons $i$ and $j$. Grover's Algorithm can be represented as: $O = \sum_{i=1}^{N} a_i \frac{|i\rangle}{\sqrt{|i\rangle\langle i|}}$ where $N$ is the number of qubits. The oracle operator $O$ flips the amplitude of the marked state. The Grover iteration is: $U = \mathcal{I} + 2O + \lambda \sigma_x$ where $\mathcal{I}$ is the identity operator, $\sigma_x$ is the Pauli-X operator, and $\lambda$ is the Grover iteration number.