10,997 research outputs found
CSS-like Constructions of Asymmetric Quantum Codes
Asymmetric quantum error-correcting codes (AQCs) may offer some advantage
over their symmetric counterparts by providing better error-correction for the
more frequent error types. The well-known CSS construction of -ary AQCs is
extended by removing the \F_{q}-linearity requirement as well as the
limitation on the type of inner product used. The proposed constructions are
called CSS-like constructions and utilize pairs of nested subfield linear codes
under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian
inner products.
After establishing some theoretical foundations, best-performing CSS-like
AQCs are constructed. Combining some constructions of nested pairs of classical
codes and linear programming, many optimal and good pure -ary CSS-like codes
for up to reasonable lengths are found. In many
instances, removing the \F_{q}-linearity and using alternative inner products
give us pure AQCs with improved parameters than relying solely on the standard
CSS construction.Comment: Accepted by IEEE Trans. Information Theory in June 2013, to appea
Additive Asymmetric Quantum Codes
We present a general construction of asymmetric quantum codes based on
additive codes under the trace Hermitian inner product. Various families of
additive codes over \F_{4} are used in the construction of many asymmetric
quantum codes over \F_{4}.Comment: Accepted for publication March 2, 2011, IEEE Transactions on
Information Theory, to appea
Xing-Ling Codes, Duals of their Subcodes, and Good Asymmetric Quantum Codes
A class of powerful -ary linear polynomial codes originally proposed by
Xing and Ling is deployed to construct good asymmetric quantum codes via the
standard CSS construction. Our quantum codes are -ary block codes that
encode qudits of quantum information into qudits and correct up to
\flr{(d_{x}-1)/2} bit-flip errors and up to \flr{(d_{z}-1)/2} phase-flip
errors.. In many cases where the length
and the field size are fixed and for chosen values of and , where is the designed distance of
the Xing-Ling (XL) codes, the derived pure -ary asymmetric quantum CSS codes
possess the best possible size given the current state of the art knowledge on
the best classical linear block codes.Comment: To appear in Designs, Codes and Cryptography (accepted Sep. 27, 2013
On privacy amplification, lossy compression, and their duality to channel coding
We examine the task of privacy amplification from information-theoretic and
coding-theoretic points of view. In the former, we give a one-shot
characterization of the optimal rate of privacy amplification against classical
adversaries in terms of the optimal type-II error in asymmetric hypothesis
testing. This formulation can be easily computed to give finite-blocklength
bounds and turns out to be equivalent to smooth min-entropy bounds by Renner
and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a
bound in terms of the divergence by Yang, Schaefer, and Poor
[arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy
amplification based on linear codes can be easily repurposed for channel
simulation. Combined with known relations between channel simulation and lossy
source coding, this implies that privacy amplification can be understood as a
basic primitive for both channel simulation and lossy compression. Applied to
symmetric channels or lossy compression settings, our construction leads to
proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing
to the notion of channel duality recently detailed by us in [IEEE Trans. Info.
Theory 64, 577 (2018)], we show that linear error-correcting codes for
symmetric channels with quantum output can be transformed into linear lossy
source coding schemes for classical variables arising from the dual channel.
This explains a "curious duality" in these problems for the (self-dual) erasure
channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and
partly anticipates recent results on optimal lossy compression by polar and
low-density generator matrix codes.Comment: v3: updated to include equivalence of the converse bound with smooth
entropy formulations. v2: updated to include comparison with the one-shot
bounds of arXiv:1706.03866. v1: 11 pages, 4 figure
Concatenated Codes for Amplitude Damping
We discuss a method to construct quantum codes correcting amplitude damping
errors via code concatenation. The inner codes are chosen as asymmetric
Calderbank-Shor-Steane (CSS) codes. By concatenating with outer codes
correcting symmetric errors, many new codes with good parameters are found,
which are better than the amplitude damping codes obtained by any previously
known construction.Comment: 5 page
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