Homological Product Codes

Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their simple syndrome readout circuits and potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most $\sqrt{n}$ qubits, where $n$ is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight $n^a$ for any $a>0$. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.Comment: 49 page

Minimal kernels of Dirac operators along maps

Let $M$ be a closed spin manifold and let $N$ be a closed manifold. For maps $f\colon M\to N$ and Riemannian metrics $g$ on $M$ and $h$ on $N$, we consider the Dirac operator $D^f_{g,h}$ of the twisted Dirac bundle $\Sigma M\otimes_{\mathbb{R}} f^*TN$. To this Dirac operator one can associate an index in $KO^{-dim(M)}(pt)$. If $M$ is $2$-dimensional, one gets a lower bound for the dimension of the kernel of $D^f_{g,h}$ out of this index. We investigate the question whether this lower bound is obtained for generic tupels $(f,g,h)$

Electromagnetic mirror drive system

Oscillatory electromagnetic mirror drive system for horizon scanner

Euler Integration of Gaussian Random Fields and Persistent Homology

In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.Comment: 21 pages, 1 figur

B2 and G2 Toda systems on compact surfaces: a variational approach

We consider the B2 and G2 Toda systems on compact surfaces. We attack the problem using variational techniques. We get existence and multiplicity of solutions under a topological assumption on the surface and some generic conditions on the parameters. We also extend some of the results to the case of general systems.Comment: 28 pages, accepted on Journal of Mathematical Physic

More Torsion in the Homology of the Matching Complex

A matching on a set $X$ is a collection of pairwise disjoint subsets of $X$ of size two. Using computers, we analyze the integral homology of the matching complex $M_n$, which is the simplicial complex of matchings on the set $\{1, >..., n\}$. The main result is the detection of elements of order $p$ in the homology for $p \in \{5,7,11,13\}$. Specifically, we show that there are elements of order 5 in the homology of $M_n$ for $n \ge 18$ and for $n \in {14,16}$. The only previously known value was $n = 14$, and in this particular case we have a new computer-free proof. Moreover, we show that there are elements of order 7 in the homology of $M_n$ for all odd $n$ between 23 and 41 and for $n=30$. In addition, there are elements of order 11 in the homology of $M_{47}$ and elements of order 13 in the homology of $M_{62}$. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of $H_d(M_n;Z)$ for $13 \le n \le 16$; a complete description of the homology already exists for $n \le 12$. To prove the results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of $M_n$ obtained by letting certain groups act on the chain complex.Comment: 35 pages, 10 figure

The quantum algebra of superspace

We present the complete set of $N=1$, $D=4$ quantum algebras associated to massive superparticles. We obtain the explicit solution of these algebras realized in terms of unconstrained operators acting on the Hilbert space of superfields. These solutions are expressed using the chiral, anti-chiral and tensorial projectors which define the three irreducible representations of the supersymmetry on the superfields. In each case the space-time variables are non-commuting and their commutators are proportional to the internal angular momentum of the representation. The quantum algebra associated to the chiral or the anti-chiral projector is the one obtained by the quantization of the Casalbuoni-Brink-Schwarz (superspin 0) massive superparticle. We present a new superparticle action for the (superspin 1/2) case and show that their wave functions are the ones associated to the irreducible tensor multiplet.Comment: 20 pages;changes in the nomenclatur

Schrijver graphs and projective quadrangulations

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space $P^n$ is at least $(n+2)$-chromatic, unless it is bipartite. They conjectured that for any integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $SG(n,k)$ contains a spanning subgraph which is a quadrangulation of $P^{n-2k}$. The purpose of this paper is to prove the conjecture