Generating infinite monoids of cellular automata

For a group $G$ and a set $A$, let $\text{End}(A^G)$ be the monoid of all cellular automata over $A^G$, and let $\text{Aut}(A^G)$ be its group of units. By establishing a characterisation of surjunctuve groups in terms of the monoid $\text{End}(A^G)$, we prove that the rank of $\text{End}(A^G)$ (i.e. the smallest cardinality of a generating set) is equal to the rank of $\text{Aut}(A^G)$ plus the relative rank of $\text{Aut}(A^G)$ in $\text{End}(A^G)$, and that the latter is infinite when $G$ has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when $A=V$ is a vector space over a field $\mathbb{F}$, we study the monoid $\text{End}_{\mathbb{F}}(V^G)$ of all linear cellular automata over $V^G$ and its group of units $\text{Aut}_{\mathbb{F}}(V^G)$. We show that if $G$ is an indicable group and $V$ is finite-dimensional, then $\text{End}_{\mathbb{F}}(V^G)$ is not finitely generated; however, for any finitely generated indicable group $G$, the group $\text{Aut}_{\mathbb{F}}(\mathbb{F}^G)$ is finitely generated if and only if $\mathbb{F}$ is finite.Comment: 11 page

Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance

We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, distance scaling as nϵ for ϵ>0 and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate k/n of 13/72=0.180… and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime

Quantum memories based on engineered dissipation

Storing quantum information for long times without disruptions is a major requirement for most quantum information technologies. A very appealing approach is to use self-correcting Hamiltonians, i.e. tailoring local interactions among the qubits such that when the system is weakly coupled to a cold bath the thermalization process takes a long time. Here we propose an alternative but more powerful approach in which the coupling to a bath is engineered, so that dissipation protects the encoded qubit against more general kinds of errors. We show that the method can be implemented locally in four dimensional lattice geometries by means of a toric code, and propose a simple 2D set-up for proof of principle experiments.Comment: 6 +8 pages, 4 figures, Includes minor corrections updated references and aknowledgement

The braided Ptolemy-Thompson group is finitely presented

Pursueing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group $T^*$ (and its further generalizations) which is an extension of the Ptolemy-Thompson group $T$ by means of the full braid group $B_{\infty}$ on infinitely many strands. We prove that it is a finitely presented group with solvable word problem, and give an explicit presentation of it.Comment: 35

MaxSAT Evaluation 2018 : Solver and Benchmark Descriptions

Non peer reviewe