Entangled Quantum Networks

TL;DR

This paper explores the simulation of entangled quantum networks using a unitary form of c-NOT gates, demonstrating complex entanglement dynamics, the possibility of state recovery, and the limitations on periodicity in quantum evolutions.

Contribution

It introduces a unitary transition matrix for quantum networks with c-NOT gates, analyzes entanglement evolution, and discusses the potential for creating periodic orbits for pattern recognition.

Findings

Complete entanglement of the network achieved

Original input states can be recovered via logical back projection

Quantum networks cannot naturally transition from aperiodic to periodic regimes

Abstract

We present some results from simulation of a network of nodes connected by c-NOT gates with nearest neighbors. Though initially we begin with pure states of varying boundary conditions, the updating with time quickly involves a complicated entanglement involving all or most nodes. As a normal c-NOT gate, though unitary for a single pair of nodes, seems to be not so when used in a network in a naive way, we use a manifestly unitary form of the transition matrix with c?-NOT gates, which invert the phase as well as flipping the qubit. This leads to complete entanglement of the net, but with variable coefficients for the different components of the superposition. It is interesting to note that by a simple logical back projection the original input state can be recovered in most cases. We also prove that it is not possible for a sequence of unitary operators working on a net to make it move from an aperiodic regime to a periodic one, unlike some classical cases where phase-locking happens in course of evolution. However, we show that it is possible to introduce by hand periodic orbits to sets of initial states, which may be useful in forming dynamic pattern recognition systems.