8,260 research outputs found
Extending Construction X for Quantum Error-Correcting Codes
In this paper we extend the work of Lisonek and Singh on construction X for
quantum error-correcting codes to finite fields of order $p^2^ where p is
prime. The results obtained are applied to the dual of Hermitian repeated root
cyclic codes to generate new quantum error-correcting codes
Asymmetric Quantum Codes: New Codes from Old
In this paper we extend to asymmetric quantum error-correcting codes (AQECC)
the construction methods, namely: puncturing, extending, expanding, direct sum
and the (u|u + v) construction. By applying these methods, several families of
asymmetric quantum codes can be constructed. Consequently, as an example of
application of quantum code expansion developed here, new families of
asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes,
quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and
affine-invariant codes are constructed.Comment: Accepted for publication Quantum Information Processin
On Subsystem Codes Beating the Hamming or Singleton Bound
Subsystem codes are a generalization of noiseless subsystems, decoherence
free subspaces, and quantum error-correcting codes. We prove a Singleton bound
for GF(q)-linear subsystem codes. It follows that no subsystem code over a
prime field can beat the Singleton bound. On the other hand, we show the
remarkable fact that there exist impure subsystem codes beating the Hamming
bound. A number of open problems concern the comparison in performance of
stabilizer and subsystem codes. One of the open problems suggested by Poulin's
work asks whether a subsystem code can use fewer syndrome measurements than an
optimal MDS stabilizer code while encoding the same number of qudits and having
the same distance. We prove that linear subsystem codes cannot offer such an
improvement under complete decoding.Comment: 18 pages more densely packed than classically possibl
Correcting Quantum Errors with Entanglement
We show how entanglement shared between encoder and decoder can simplify the
theory of quantum error correction. The entanglement-assisted quantum codes we
describe do not require the dual-containing constraint necessary for standard
quantum error correcting codes, thus allowing us to ``quantize'' all of
classical linear coding theory. In particular, efficient modern classical codes
that attain the Shannon capacity can be made into entanglement-assisted quantum
codes attaining the hashing bound (closely related to the quantum capacity).
For systems without large amounts of shared entanglement, these codes can also
be used as catalytic codes, in which a small amount of initial entanglement
enables quantum communication.Comment: 17 pages, no figure. To appear in Scienc
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