## Empires: The Nonlocal Properties of Quasicrystals

In quasicrystals, any given local patch—called an emperor—forces at all distances the existence of accompanying tiles—called the empire—revealing thus their inherent nonlocality. In this chapter, we review and compare the methods currently used for generating the empires, with a focus on the cut-and-project method, which can be generalized to calculate empires for any quasicrystals that […]

## Holographic Code Rate

Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inflation rule protect quantum information stored in the bulk from errors on the boundary provided the code rate is less than one. Hyperbolic geometry bounds the holographic code rate and guarantees quantum error correction for codes grown with any inflation rule on all […]

## Group geometrical axioms for magic states of quantum computing

Let H be a non trivial subgroup of index d of a free group G and N the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of MIC states associated to minimal informationally […]

## Quantum Gravity at the Fifth Root of Unity

We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within $SU(2)$ quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering fermionic cycles through the foam we couple this $SU(2)$ quantum group with the same deformation of $SU(3)$, so that we have quantum […]

## Constructing numbers in quantum gravity: infinions

Based on the Cayley-Dickson process, a sequence of multidimensional structured natural numbers (infinions) creates a path from quantum information to quantum gravity. Octonionic structure, exceptional Jordan algebra, and E 8 Lie algebra are encoded on a graph with E 9 connectivity, decorated by integral matrices. With the magic star, a toy model for a quantum gravity is presented with its naturally […]

## Quantum computing, Seifert surfaces and singular fibers

The fundamental group $π1(L)$ of a knot or link $L$ may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the $L$ of such a quantum computer model and computes their Seifert […]

## Emergence Theory Conceptual Overview

Emergence theory is a code-theoretic first-principles based discretized quantum field theoretic approach to quantum gravity and particle physics. This overview covers the primary set of ideas being assembled by Quantum Gravity Research.

## Assessment of discrete breather in the metallic hydrides

Computational assessment of the discrete breathers (also known as intrinsic localised modes) is performed in nickel and palladium hydrides with an even stoichiometry by means of molecular dynamics simulations. The breathers consisting of hydrogen and metallic atoms were excited following the experience obtained earlier by modelling the breathers in pure metallic systems. Stable breathers were […]