Programmable Quantum-Like bits from Signed Regular Graphs

Abstract

Extending upon observations of the emergence of quantum-like (QL) states from classical complex synchronized networks, this work adds mathematical rigor to the analysis of single QL bits constructed from adjacency-matrix eigenvectors. First, we rigorously show that symmetric construction of such networks (regular undirected/symmetric bipartite graph $G_C$ connecting two regular undirected subgraphs $G_A,\,G_B$) leads to an equal superposition of the $|+\rangle, |-\rangle$ Hadamard states (with basis $|0\rangle,\,|1\rangle$ set from eigenvectors of the subgraphs), and provide an analysis of sufficient conditions on the network for construction of such states. Second, we prove two methods to construct arbitrary single qubit states $|\psi\rangle = a|0\rangle + b|1\rangle,\, |a|^2+|b|^2=1$, and give switching lemmas for their boundaries: (i) by detuning the two subgraphs regularities and (ii) by asymmetrically allowing the bipartite connection matrix $C$ to be directed and detuning those regularities. Although motivated by using complex synchronized networks for quantum information storage and computation, the proofs for these methods rely only on the structure of the graph embedded in the adjacency matrix. Thus, synchronization is unnecessary; QL bits arise when edge weights are unit (or near-unit) and subgraphs are regular. Results on combinations of random k-regular graphs (more precisely Erd\H{o}s-R\'enyi graphs) may be independently interesting.