410 research outputs found
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 on
all logical qubits. The new codes are used to construct protocols for
distilling high-quality `magic' states 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
. 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 -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 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} , 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
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . 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
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 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 , the matrix whose rows are truth tables of all
monomials of degree in variables. What is the most (resp. least)
number of random columns in 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 , 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 of sufficiently high rate we construct a new code
, also of very high rate, such that for every subset of coordinates, if
can recover from erasures in , then can recover from errors in .
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
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