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

Topological Quantum Error Correction with Optimal Encoding Rate

We prove the existence of topological quantum error correcting codes with encoding rates $k/n$ asymptotically approaching the maximum possible value. Explicit constructions of these topological codes are presented using surfaces of arbitrary genus. We find a class of regular toric codes that are optimal. For physical implementations, we present planar topological codes.Comment: REVTEX4 file, 5 figure

MUBs inequivalence and affine planes

There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes "old" results that do not appear to be well-known concerning known families of complete sets of MUBs and their associated planes.Comment: This is the version of this paper appearing in J. Mathematical Physics 53, 032204 (2012) except for format changes due to the journal's style policie

An improved lower bound for (1,<=2)-identifying codes in the king grid

We call a subset $C$ of vertices of a graph $G$ a $(1,\leq \ell)$-identifying code if for all subsets $X$ of vertices with size at most $\ell$, the sets $\{c\in C |\exists u \in X, d(u,c)\leq 1\}$ are distinct. The concept of identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin. Identifying codes have been studied in various grids. In particular, it has been shown that there exists a $(1,\leq 2)$-identifying code in the king grid with density 3/7 and that there are no such identifying codes with density smaller than 5/12. Using a suitable frame and a discharging procedure, we improve the lower bound by showing that any $(1,\leq 2)$-identifying code of the king grid has density at least 47/111

Query-Efficient Locally Decodable Codes of Subexponential Length

We develop the algebraic theory behind the constructions of Yekhanin (2008) and Efremenko (2009), in an attempt to understand the ``algebraic niceness'' phenomenon in $\mathbb{Z}_m$. We show that every integer $m = pq = 2^t -1$, where $p$, $q$ and $t$ are prime, possesses the same good algebraic property as $m=511$ that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki's composition method. More precisely, we construct a $3^{\lceil r/2\rceil}$-query LDC for every positive integer $r<104$ and a $\left\lfloor (3/4)^{51}\cdot 2^{r}\right\rfloor$-query LDC for every integer $r\geq 104$, both of length $N_{r}$, improving the $2^r$ queries used by Efremenko (2009) and $3\cdot 2^{r-2}$ queries used by Itoh and Suzuki (2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.Comment: to appear in Computational Complexit

Topological Color Codes and Two-Body Quantum Lattice Hamiltonians

Topological color codes are among the stabilizer codes with remarkable properties from quantum information perspective. In this paper we construct a four-valent lattice, the so called ruby lattice, governed by a 2-body Hamiltonian. In a particular regime of coupling constants, degenerate perturbation theory implies that the low energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. The gauge symmetry $\mathbf{Z}_{2}\times\mathbf{Z}_{2}$ of color code could already be realized by identifying three distinct plaquette operators on the lattice. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other giving rise to exact topological degeneracy of the model. Connection to 2-colexes can be established at the non-perturbative level. The particular structure of the 2-body Hamiltonian provides a fruitful interpretation in terms of mapping to bosons coupled to effective spins. We show that high energy excitations of the model have fermionic statistics. They form three families of high energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. Also, we use Jordan-Wigner transformation in order to test the integrability of the model via introducing of Majorana fermions. The four-valent structure of the lattice prevents to reduce the fermionized Hamiltonian into a quadratic form due to interacting gauge fields. We also propose another construction for 2-body Hamiltonian based on the connection between color codes and cluster states. We discuss this latter approach along the construction based on the ruby lattice.Comment: 56 pages, 16 figures, published version

Code Construction and Decoding Algorithms for Semi-Quantitative Group Testing with Nonuniform Thresholds

We analyze a new group testing scheme, termed semi-quantitative group testing, which may be viewed as a concatenation of an adder channel and a discrete quantizer. Our focus is on non-uniform quantizers with arbitrary thresholds. For the most general semi-quantitative group testing model, we define three new families of sequences capturing the constraints on the code design imposed by the choice of the thresholds. The sequences represent extensions and generalizations of Bh and certain types of super-increasing and lexicographically ordered sequences, and they lead to code structures amenable for efficient recursive decoding. We describe the decoding methods and provide an accompanying computational complexity and performance analysis