386 research outputs found
On optimal quantum codes
We present families of quantum error-correcting codes which are optimal in
the sense that the minimum distance is maximal. These maximum distance
separable (MDS) codes are defined over q-dimensional quantum systems, where q
is an arbitrary prime power. It is shown that codes with parameters
[[n,n-2d+2,d]]_q exist for all 3 <= n <= q and 1 <= d <= n/2+1. We also present
quantum MDS codes with parameters [[q^2,q^2-2d+2,d]]_q for 1 <= d <= q which
additionally give rise to shortened codes [[q^2-s,q^2-2d+2-s,d]]_q for some s.Comment: Accepted for publication in the International Journal of Quantum
Informatio
Asymptotic Bound on Binary Self-Orthogonal Codes
We present two constructions for binary self-orthogonal codes. It turns out
that our constructions yield a constructive bound on binary self-orthogonal
codes. In particular, when the information rate R=1/2, by our constructive
lower bound, the relative minimum distance \delta\approx 0.0595 (for GV bound,
\delta\approx 0.110). Moreover, we have proved that the binary self-orthogonal
codes asymptotically achieve the Gilbert-Varshamov bound.Comment: 4 pages 1 figur
Efficiently decoding Reed-Muller codes from random errors
Reed-Muller codes encode an -variate polynomial of degree by
evaluating it on all points in . We denote this code by .
The minimal distance of is and so it cannot correct more
than half that number of errors in the worst case. For random errors one may
hope for a better result.
In this work we give an efficient algorithm (in the block length ) for
decoding random errors in Reed-Muller codes far beyond the minimal distance.
Specifically, for low rate codes (of degree ) we can correct a
random set of errors with high probability. For high rate codes
(of degree for ), we can correct roughly
errors.
More generally, for any integer , our algorithm can correct any error
pattern in for which the same erasure pattern can be corrected
in . The results above are obtained by applying recent results
of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and
Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct
random erasures.
The algorithm is based on solving a carefully defined set of linear equations
and thus it is significantly different than other algorithms for decoding
Reed-Muller codes that are based on the recursive structure of the code. It can
be seen as a more explicit proof of a result of Abbe et al. that shows a
reduction from correcting erasures to correcting errors, and it also bares some
similarities with the famous Berlekamp-Welch algorithm for decoding
Reed-Solomon codes.Comment: 18 pages, 2 figure
Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes
We present two methods for the construction of quantum circuits for quantum
error-correcting codes (QECC). The underlying quantum systems are tensor
products of subsystems (qudits) of equal dimension which is a prime power. For
a QECC encoding k qudits into n qudits, the resulting quantum circuit has
O(n(n-k)) gates. The running time of the classical algorithm to compute the
quantum circuit is O(n(n-k)^2).Comment: 18 pages, submitted to special issue of IJFC
Decoding of Matrix-Product Codes
We propose a decoding algorithm for the -construction that
decodes up to half of the minimum distance of the linear code. We extend this
algorithm for a class of matrix-product codes in two different ways. In some
cases, one can decode beyond the error correction capability of the code
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
GHZ extraction yield for multipartite stabilizer states
Let be an arbitrary stabilizer state distributed between three
remote parties, such that each party holds several qubits. Let be a
stabilizer group of . We show that can be converted by local
unitaries into a collection of singlets, GHZ states, and local one-qubit
states. The numbers of singlets and GHZs are determined by dimensions of
certain subgroups of . For an arbitrary number of parties we find a
formula for the maximal number of -partite GHZ states that can be extracted
from by local unitaries. A connection with earlier introduced measures
of multipartite correlations is made. An example of an undecomposable
four-party stabilizer state with more than one qubit per party is given. These
results are derived from a general theoretical framework that allows one to
study interconversion of multipartite stabilizer states by local Clifford group
operators. As a simple application, we study three-party entanglement in
two-dimensional lattice models that can be exactly solved by the stabilizer
formalism.Comment: 12 pages, 1 figur
Pure Asymmetric Quantum MDS Codes from CSS Construction: A Complete Characterization
Using the Calderbank-Shor-Steane (CSS) construction, pure -ary asymmetric
quantum error-correcting codes attaining the quantum Singleton bound are
constructed. Such codes are called pure CSS asymmetric quantum maximum distance
separable (AQMDS) codes. Assuming the validity of the classical MDS Conjecture,
pure CSS AQMDS codes of all possible parameters are accounted for.Comment: Change in authors' list. Accepted for publication in Int. Journal of
Quantum Informatio
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
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