Homological Error Correction: Classical and Quantum Codes

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.Comment: Revtex4 fil

LIPIcs

The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Stefankovic proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest

Permutation-partition pairs. III. Embedding distributions of linear families of graphs

AbstractFor any fixed graph H, and H-linear family of graphs is a sequence {Gn}n=1∞ of graphs in which Gn consists of n copies of H that have been linked in a consistent manner so as to form a chain. Generating functions for the region distribution of any such family are found. It is also shown that the minimum genus and the average genus of Gn are essentially linear functions of n

A Spin TQFT Related to the Ising Categories

Most interesting 3d Topological Quantum Field Theories (TQFTs) are constructed by starting with algebraic data, usually in the form of some kind of category. This category typically comes from an area of mathematics different from 3-manifold topology, and its topological nature can be hard to understand. This dissertation reverses the process, at least in one simple example, by constructing a Spin TQFT from pure topology and then uncovering some interesting categories.The topology used to construct the Spin TQFT is entirely classical. If $(M, s)$ is a closed Spin 3-manifold, then an embedded surface $\Sigma\subset M$ inherits a \Pin^- structure, $s|_{\Sigma}$, from $(M, s)$. If $\Sigma$ and $\Sigma'$ represent the same class in H_2(M;\bbZ/2), then $s|_{\Sigma}$ and $s|_{\Sigma'}$ are isomorphic. If $t$ is a \Pin^- structure on $\Sigma$, there is a classical invariant of the isomorphism type of $(\Sigma, t)$, denoted $\beta(\Sigma, t)$, that is an eight root of unity. One can therefore form the Spin 3-manifold invariant Z_3(M, s) := \frac{1}{2^{b_0(M)}} \sum_{[\Sigma]\in H_2(M;\bbZ/2)} \beta(\Sigma, s|_{\Sigma}).It turns out that this invariant fits into a Spin TQFT. The detailed construction of this TQFT is the subject of this dissertation.By a theorem of Kirby and Melvin, $Z_3$ is very much related to the Ising categories. In extending the TQFT for $Z_3$, one encounters a category (associated with the bounding Spin circle) which has most of the same properties as the Ising categories. One also encounters a category (associated with the interval) from which $Z_3$ can be reconstructed in the style of Turaev-Viro and Barrett-Westbury. Because $Z_3$ is a Spin TQFT, these categories are linear over super vector spaces. In fact, they are realized as the module categories of certain explicit super algebras