Modelling microtubules in the brain asn-qudit quantum Hopfield network and beyond

Abstract

The scientific approach to understand the nature of consciousness revolves around the study of human brain. Neurobiological studies that compare the nervous system of different species have accorded highest place to the humans on account of various factors that include a highly developed cortical area comprising of approximately 100 billion neurons, that are intrinsically connected to form a highly complex network. Quantum theories of consciousness are based on mathematical abstraction and Penrose-Hameroff Orch-OR Theory is one of the most promising ones. Inspired by Penrose-Hameroff Orch-OR Theory, Behrman et. al. (Behrman, 2006) have simulated a quantum Hopfield neural network with the structure of a microtubule. They have used an extremely simplified model of the tubulin dimers with each dimer represented simply as a qubit, a single quantum two-state system. The extension of this model to n-dimensional quantum states, or n-qudits presented in this work holds considerable promise for even higher mathematical abstraction in modelling consciousness systems.

A Normal Form for Single-Qudit Clifford+$T$ Operators

“…Here, we address the problem of exact and approximate synthesis for single-qudit gates for qudits of generic odd prime dimension p. Our motivation for focusing our attention on qudits of odd prime dimension [7] includes interesting results in qudit magic state distillation [8][9][10][11][12][13][14][15][16][17], and the fact that magic, as a resource, may be easier to quantify when one restricts to qudits of odd dimension. [18][19][20][21][22].…”

Normal form for single-qutrit Clifford+Toperators and synthesis of single-qutrit gates

“…At a practical level, high thresholds for magic state distillation have been observed for magic state distillation schemes in odd prime dimension [3][4][5]. See also, e.g., [6][7][8] for additional results and motivation for studying qudits of odd prime dimension.…”