36,250 research outputs found
Universal fault-tolerant gates on concatenated stabilizer codes
It is an oft-cited fact that no quantum code can support a set of
fault-tolerant logical gates that is both universal and transversal. This no-go
theorem is generally responsible for the interest in alternative universality
constructions including magic state distillation. Widely overlooked, however,
is the possibility of non-transversal, yet still fault-tolerant, gates that
work directly on small quantum codes. Here we demonstrate precisely the
existence of such gates. In particular, we show how the limits of
non-transversality can be overcome by performing rounds of intermediate
error-correction to create logical gates on stabilizer codes that use no
ancillas other than those required for syndrome measurement. Moreover, the
logical gates we construct, the most prominent examples being Toffoli and
controlled-controlled-Z, often complete universal gate sets on their codes. We
detail such universal constructions for the smallest quantum codes, the 5-qubit
and 7-qubit codes, and then proceed to generalize the approach. One remarkable
result of this generalization is that any nondegenerate stabilizer code with a
complete set of fault-tolerant single-qubit Clifford gates has a universal set
of fault-tolerant gates. Another is the interaction of logical qubits across
different stabilizer codes, which, for instance, implies a broadly applicable
method of code switching.Comment: 18 pages + 5 pages appendix, 12 figure
The Error-Pattern-Correcting Turbo Equalizer
The error-pattern correcting code (EPCC) is incorporated in the design of a
turbo equalizer (TE) with aim to correct dominant error events of the
inter-symbol interference (ISI) channel at the output of its matching Viterbi
detector. By targeting the low Hamming-weight interleaved errors of the outer
convolutional code, which are responsible for low Euclidean-weight errors in
the Viterbi trellis, the turbo equalizer with an error-pattern correcting code
(TE-EPCC) exhibits a much lower bit-error rate (BER) floor compared to the
conventional non-precoded TE, especially for high rate applications. A
maximum-likelihood upper bound is developed on the BER floor of the TE-EPCC for
a generalized two-tap ISI channel, in order to study TE-EPCC's signal-to-noise
ratio (SNR) gain for various channel conditions and design parameters. In
addition, the SNR gain of the TE-EPCC relative to an existing precoded TE is
compared to demonstrate the present TE's superiority for short interleaver
lengths and high coding rates.Comment: This work has been submitted to the special issue of the IEEE
Transactions on Information Theory titled: "Facets of Coding Theory: from
Algorithms to Networks". This work was supported in part by the NSF
Theoretical Foundation Grant 0728676
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
Density Evolution for Asymmetric Memoryless Channels
Density evolution is one of the most powerful analytical tools for
low-density parity-check (LDPC) codes and graph codes with message passing
decoding algorithms. With channel symmetry as one of its fundamental
assumptions, density evolution (DE) has been widely and successfully applied to
different channels, including binary erasure channels, binary symmetric
channels, binary additive white Gaussian noise channels, etc. This paper
generalizes density evolution for non-symmetric memoryless channels, which in
turn broadens the applications to general memoryless channels, e.g. z-channels,
composite white Gaussian noise channels, etc. The central theorem underpinning
this generalization is the convergence to perfect projection for any fixed size
supporting tree. A new iterative formula of the same complexity is then
presented and the necessary theorems for the performance concentration theorems
are developed. Several properties of the new density evolution method are
explored, including stability results for general asymmetric memoryless
channels. Simulations, code optimizations, and possible new applications
suggested by this new density evolution method are also provided. This result
is also used to prove the typicality of linear LDPC codes among the coset code
ensemble when the minimum check node degree is sufficiently large. It is shown
that the convergence to perfect projection is essential to the belief
propagation algorithm even when only symmetric channels are considered. Hence
the proof of the convergence to perfect projection serves also as a completion
of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor
Improving the efficiency of the LDPC code-based McEliece cryptosystem through irregular codes
We consider the framework of the McEliece cryptosystem based on LDPC codes,
which is a promising post-quantum alternative to classical public key
cryptosystems. The use of LDPC codes in this context allows to achieve good
security levels with very compact keys, which is an important advantage over
the classical McEliece cryptosystem based on Goppa codes. However, only regular
LDPC codes have been considered up to now, while some further improvement can
be achieved by using irregular LDPC codes, which are known to achieve better
error correction performance than regular LDPC codes. This is shown in this
paper, for the first time at our knowledge. The possible use of irregular
transformation matrices is also investigated, which further increases the
efficiency of the system, especially in regard to the public key size.Comment: 6 pages, 3 figures, presented at ISCC 201
Low-complexity quantum codes designed via codeword-stabilized framework
We consider design of the quantum stabilizer codes via a two-step,
low-complexity approach based on the framework of codeword-stabilized (CWS)
codes. In this framework, each quantum CWS code can be specified by a graph and
a binary code. For codes that can be obtained from a given graph, we give
several upper bounds on the distance of a generic (additive or non-additive)
CWS code, and the lower Gilbert-Varshamov bound for the existence of additive
CWS codes. We also consider additive cyclic CWS codes and show that these codes
correspond to a previously unexplored class of single-generator cyclic
stabilizer codes. We present several families of simple stabilizer codes with
relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl
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