A Family of Quantum Stabilizer Codes Based on the Weyl Commutation Relations over a Finite Field

Using the Weyl commutation relations over a finite field we introduce a family of error-correcting quantum stabilizer codes based on a class of symmetric matrices over the finite field satisfying certain natural conditions. When the field is GF(2) the existence of a rich class of such symmetric matrices is demonstrated by a simple probabilistic argument depending on the Chernoff bound for i.i.d symmetric Bernoulli trials. If, in addition, these symmetric matrices are assumed to be circulant it is possible to obtain concrete examples by a computer program. The quantum codes thus obtained admit elegant encoding circuits.Comment: 16 pages, 2 figure

Chess players’ fame versus their merit

We investigate a pool of international chess title holders born between 1901 and 1943. Using Elo ratings, we compute for every player his expected score in a game with a randomly selected player from the pool. We use this figure as the player's merit. We measure players' fame as the number of Google hits. The correlation between fame and merit is 0.38. At the same time, the correlation between the logarithm of fame and merit is 0.61. This suggests that fame grows exponentially with merit

A Resistant Strain: Revealing the Online Grassroots Rise of the Antivaccination Movement

An analysis of more than eight years of data from vaccination forums on mothering.com shows that the antivaccination movement is well-organized and widely dispersed, and that it emerged long before concerns about immunity were expressed. The findings are evidence of a formidable challenge to the social norms surrounding vaccination

The Quantum Entanglement of Binary and Bipolar Sequences

This paper highlights two partial entanglement measures, namely the 'Linear Entanglement' (LE) (Section 6, Definition 11), and 'Stubborness of Entanglement' (SE) (Section 7, Definition 16), which is a sequence of parameters. The paper is aimed at both coding theorists and sequence designers, and at quantum physicists, and argues that the best codes and/or sequences can be interpreted as describing multiparticle states with high entanglement. A binary linear error-correcting code (ECC), C, is often partially described by its parameters [n, k, d], where n is wordlength, k is code dimension, and d is minimum Hamming Distance [18], and more generally by its weight hierarchy. We show, by interpreting the length 2 n indicator for C as an n-particle quantum state that, for those states representing binary linear ECCs, the ECCs with optimal weight hierarchy also have optimal LE and optimal SE (Theorems 10 and 15). By action of local unitary transform on the indicator of C, we can also view the quantum state as a bipolar sequence. In this context a sequence is often partially described by its nonlinear order, N , and correlation immunity order, CI (Definitions 21, 22). We show that N and CI give a lower bound on 2 Matthew G. Parker and V. Rijmen LE (Theorem 16). LE is the n - log 2 of a spectral 'peak' measure of Peakto -Average Power Ratio (PAR l (Section 6, Definition 10)), which is also an important measure in telecommunications [7,21,20]. This paper refers both to PAR l and to LE, where the two parameters are trivially related (Definition 11). The quantum-mechanical rule of 'local unitary equivalence' is a generalisation of code duality. We now state the most important results of this paper. We emphasise quantum states s from the set # p , where # p is equivalent to t..

Two Geometric Optimization Problems

. We consider two optimization problems with geometric structures. The first one concerns the following minimization problem, termed as the rectilinear polygon cover problem: "Cover certain features of a given rectilinear polygon (possibly with rectilinear holes) with the minimum number of rectangles included in the polygon." Depending upon whether one wants to cover the interior, boundary or corners of the polygon, the problem is termed as the interior, boundary or corner cover problem, respectively. Most of these problems are known to be NP-complete. In this chapter we survey some of the important previous results for these problems and provide a proof of impossibility of a polynomial-time approximation scheme for the interior and boundary cover problems. The second problem concerns routing in a segmented routing channel. The related problems are fundamental to routing and design automation for Field Programmable Gate Arrays (FPGAs), a new type of electrically programmable VLSI. In ..