Magic state distillation with low overhead

We propose a new family of error detecting stabilizer codes with an encoding rate 1/3 that permit a transversal implementation of the pi/8-rotation $T$ on all logical qubits. The new codes are used to construct protocols for distilling high-quality `magic' states $T|+>$ by Clifford group gates and Pauli measurements. The distillation overhead has a poly-logarithmic scaling as a function of the output accuracy, where the degree of the polynomial is $\log_2{3}\approx 1.6$. To construct the desired family of codes, we introduce the notion of a triorthogonal matrix --- a binary matrix in which any pair and any triple of rows have even overlap. Any triorthogonal matrix gives rise to a stabilizer code with a transversal $T$-gate on all logical qubits, possibly augmented by Clifford gates. A powerful numerical method for generating triorthogonal matrices is proposed. Our techniques lead to a two-fold overhead reduction for distilling magic states with output accuracy $10^{-12}$ compared with the best previously known protocol.Comment: 11 pages, 3 figure

Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes

We construct a new family of quantum MDS codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exists, then a quantum code of this type with the same parameters also exists. Thus as far as is known at present, they are the most important family of quantum MDS codes.Comment: 9 pages, no figure

Towards Collaborative Conceptual Exploration

In domains with high knowledge distribution a natural objective is to create principle foundations for collaborative interactive learning environments. We present a first mathematical characterization of a collaborative learning group, a consortium, based on closure systems of attribute sets and the well-known attribute exploration algorithm from formal concept analysis. To this end, we introduce (weak) local experts for subdomains of a given knowledge domain. These entities are able to refute and potentially accept a given (implicational) query for some closure system that is a restriction of the whole domain. On this we build up a consortial expert and show first insights about the ability of such an expert to answer queries. Furthermore, we depict techniques on how to cope with falsely accepted implications and on combining counterexamples. Using notions from combinatorial design theory we further expand those insights as far as providing first results on the decidability problem if a given consortium is able to explore some target domain. Applications in conceptual knowledge acquisition as well as in collaborative interactive ontology learning are at hand.Comment: 15 pages, 2 figure

Contextuality in Measurement-based Quantum Computation

We show, under natural assumptions for qubit systems, that measurement-based quantum computations (MBQCs) which compute a non-linear Boolean function with high probability are contextual. The class of contextual MBQCs includes an example which is of practical interest and has a super-polynomial speedup over the best known classical algorithm, namely the quantum algorithm that solves the Discrete Log problem.Comment: Version 3: probabilistic version of Theorem 1 adde

Guided growth for correction of knee flexion deformity: a series of four cases

Fixed knee flexion deformity can present as an insidious and significant problem in diverse etiologies, most commonly in cerebral palsy. Traditional surgical intervention has included posterior capsulotomy and supracondylar femoral osteotomy, both of which carry significant associated morbidity and risks. In the skeletally immature patient, guided growth may be used to correct or substantially diminish the deformity. We are presenting our early experience encompassing four subjects who completed instrumented gait analysis both prior to and after distal femoral anterior guided growth (hemiepiphysiodesis). Changes in gait and function resulting from surgery in each individual are reported. Outcomes indicate improved knee range of motion and alleviation of crouch at the knee with secondary improvements in the ankle, hip and pelvis. Three subjects with initially slow gait velocity improved to within normal limits by demonstrating increased stride length. A measure of overall gait kinematics showed improvements in all limbs. Anterior guided growth (hemiepiphysiodesis) of the distal femur resulted in positive quantitative changes in all four patients, though degree and types of changes were variable in this small series. Encouraged by these findings, we now prefer guided growth to extension supracondylar osteotomy for the skeletally immature patient with fixed knee flexion deformity

Neighbour transitivity on codes in Hamming graphs

We consider a \emph{code} to be a subset of the vertex set of a \emph{Hamming graph}. In this setting a \emph{neighbour} of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the property that some group of automorphisms acts transitively on the \emph{set of neighbours} of the code. We call these codes \emph{neighbour transitive}. We obtain sufficient conditions for a neighbour transitive group to fix the code setwise. Moreover, we construct an infinite family of neighbour transitive codes, with \emph{minimum distance} $\delta=4$, where this is not the case. That is to say, knowledge of even the complete set of code neighbours does not determine the code

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

Reed-Muller codes for random erasures and errors

This paper studies the parameters for which Reed-Muller (RM) codes over $GF(2)$ can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate $GF(2)$ polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about $E(m,r)$, the matrix whose rows are truth tables of all monomials of degree $\leq r$ in $m$ variables. What is the most (resp. least) number of random columns in $E(m,r)$ that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees $r$, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code $C$ of sufficiently high rate we construct a new code $C'$, also of very high rate, such that for every subset $S$ of coordinates, if $C$ can recover from erasures in $S$, then $C'$ can recover from errors in $S$. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent \cite{KLP} bounds from constant degree to linear degree polynomials

Quantum Error Correction in Spatially Correlated Quantum Noise

We consider quantum error correction of quantum-noise that is created by a local interaction of qubits with a common bosonic bath. The possible exchange of bath bosons between qubits gives rise to spatial and temporal correlations in the noise. We find that these kind of noise correlations have a strong negative impact on quantum error correction.Comment: 4 pages, 1 figure, final version with minor correction

Weighing matrices and spherical codes

Mutually unbiased weighing matrices (MUWM) are closely related to an antipodal spherical code with 4 angles. In the present paper, we clarify the relationship between MUWM and the spherical sets, and give the complete solution about the maximum size of a set of MUWM of weight 4 for any order. Moreover we describe some natural generalization of a set of MUWM from the viewpoint of spherical codes, and determine several maximum sizes of the generalized sets. They include an affirmative answer of the problem of Best, Kharaghani, and Ramp.Comment: Title is changed from "Association schemes related to weighing matrices