89,569 research outputs found
Finite-Connectivity Spin-Glass Phase Diagrams and Low Density Parity Check Codes
We obtain phase diagrams of regular and irregular finite connectivity
spin-glasses. Contact is firstly established between properties of the phase
diagram and the performances of low density parity check codes (LDPC) within
the Replica Symmetric (RS) ansatz. We then study the location of the dynamical
and critical transition of these systems within the one step Replica Symmetry
Breaking theory (RSB), extending similar calculations that have been performed
in the past for the Bethe spin-glass problem. We observe that, away from the
Nishimori line, in the low temperature region, the location of the dynamical
transition line does change within the RSB theory, in comparison with the (RS)
case. For LDPC decoding over the binary erasure channel we find, at zero
temperature and rate R=1/4 an RS critical transition point located at p_c =
0.67 while the critical RSB transition point is located at p_c = 0.7450, to be
compared with the corresponding Shannon bound 1-R. For the binary symmetric
channel (BSC) we show that the low temperature reentrant behavior of the
dynamical transition line, observed within the RS ansatz, changes within the
RSB theory; the location of the dynamical transition point occurring at higher
values of the channel noise. Possible practical implications to improve the
performances of the state-of-the-art error correcting codes are discussed.Comment: 21 pages, 15 figure
X-code: MDS array codes with optimal encoding
We present a new class of MDS (maximum distance separable) array codes of size n×n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns
Quantum Error Correction and Fault-Tolerance
I give an overview of the basic concepts behind quantum error correction and
quantum fault tolerance. This includes the quantum error correction conditions,
stabilizer codes, CSS codes, transversal gates, fault-tolerant error
correction, and the threshold theorem.Comment: 8 pages, to appear in Encyclopaedia of Mathematical Physic
An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation
Quantum states are very delicate, so it is likely some sort of quantum error
correction will be necessary to build reliable quantum computers. The theory of
quantum error-correcting codes has some close ties to and some striking
differences from the theory of classical error-correcting codes. Many quantum
codes can be described in terms of the stabilizer of the codewords. The
stabilizer is a finite Abelian group, and allows a straightforward
characterization of the error-correcting properties of the code. The stabilizer
formalism for quantum codes also illustrates the relationships to classical
coding theory, particularly classical codes over GF(4), the finite field with
four elements. To build a quantum computer which behaves correctly in the
presence of errors, we also need a theory of fault-tolerant quantum
computation, instructing us how to perform quantum gates on qubits which are
encoded in a quantum error-correcting code. The threshold theorem states that
it is possible to create a quantum computer to perform an arbitrary quantum
computation provided the error rate per physical gate or time step is below
some constant threshold value.Comment: 46 pages, with large margins. Includes quant-ph/0004072 plus 30 pages
of new material, mostly on fault-toleranc
Phased burst error-correcting array codes
Various aspects of single-phased burst-error-correcting array codes are explored. These codes are composed of two-dimensional arrays with row and column parities with a diagonally cyclic readout order; they are capable of correcting a single burst error along one diagonal. Optimal codeword sizes are found to have dimensions n1×n2 such that n2 is the smallest prime number larger than n1. These codes are capable of reaching the Singleton bound. A new type of error, approximate errors, is defined; in q-ary applications, these errors cause data to be slightly corrupted and therefore still close to the true data level. Phased burst array codes can be tailored to correct these codes with even higher rates than befor
Experimental Implementation of a Codeword Stabilized Quantum Code
A five-qubit codeword stabilized quantum code is implemented in a seven-qubit
system using nuclear magnetic resonance (NMR). Our experiment implements a good
nonadditive quantum code which encodes a larger Hilbert space than any
stabilizer code with the same length and capable of correcting the same kind of
errors. The experimentally measured quantum coherence is shown to be robust
against artificially introduced errors, benchmarking the success in
implementing the quantum error correction code. Given the typical decoherence
time of the system, our experiment illustrates the ability of coherent control
to implement complex quantum circuits for demonstrating interesting results in
spin qubits for quantum computing
Coding for Racetrack Memories
Racetrack memory is a new technology which utilizes magnetic domains along a
nanoscopic wire in order to obtain extremely high storage density. In racetrack
memory, each magnetic domain can store a single bit of information, which can
be sensed by a reading port (head). The memory has a tape-like structure which
supports a shift operation that moves the domains to be read sequentially by
the head. In order to increase the memory's speed, prior work studied how to
minimize the latency of the shift operation, while the no less important
reliability of this operation has received only a little attention.
In this work we design codes which combat shift errors in racetrack memory,
called position errors. Namely, shifting the domains is not an error-free
operation and the domains may be over-shifted or are not shifted, which can be
modeled as deletions and sticky insertions. While it is possible to use
conventional deletion and insertion-correcting codes, we tackle this problem
with the special structure of racetrack memory, where the domains can be read
by multiple heads. Each head outputs a noisy version of the stored data and the
multiple outputs are combined in order to reconstruct the data. Under this
paradigm, we will show that it is possible to correct, with at most a single
bit of redundancy, deletions with heads if the heads are
well-separated. Similar results are provided for burst of deletions, sticky
insertions and combinations of both deletions and sticky insertions
Error suppression via complementary gauge choices in Reed-Muller codes
Concatenation of two quantum error correcting codes with complementary sets
of transversal gates can provide a means towards universal fault-tolerant
computation. We first show that it is generally preferable to choose the inner
code with the higher pseudo-threshold in order to achieve lower logical failure
rates. We then explore the threshold properties of a wide range of
concatenation schemes. Notably, we demonstrate that the concatenation of
complementary sets of Reed-Muller codes can increase the code capacity
threshold under depolarizing noise when compared to extensions of previously
proposed concatenation models. We also analyze the properties of logical errors
under circuit level noise, showing that smaller codes perform better for all
sampled physical error rates. Our work provides new insights into the
performance of universal concatenated quantum codes for both code capacity and
circuit level noise.Comment: 11 pages + 4 appendices, 6 figures. In v2, Fig.1 was added to conform
to journal specification
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