409 research outputs found
Codeword Stabilized Quantum Codes
We present a unifying approach to quantum error correcting code design that
encompasses additive (stabilizer) codes, as well as all known examples of
nonadditive codes with good parameters. We use this framework to generate new
codes with superior parameters to any previously known. In particular, we find
((10,18,3)) and ((10,20,3)) codes. We also show how to construct encoding
circuits for all codes within our framework.Comment: 5 pages, 1 eps figure, ((11,48,3)) code removed, encoding circuits
added, typos corrected in codewords and elsewher
Codeword Stabilized Quantum Codes for Asymmetric Channels
We discuss a method to adapt the codeword stabilized (CWS) quantum code
framework to the problem of finding asymmetric quantum codes. We focus on the
corresponding Pauli error models for amplitude damping noise and phase damping
noise. In particular, we look at codes for Pauli error models that correct one
or two amplitude damping errors. Applying local Clifford operations on graph
states, we are able to exhaustively search for all possible codes up to length
. With a similar method, we also look at codes for the Pauli error model
that detect a single amplitude error and detect multiple phase damping errors.
Many new codes with good parameters are found, including nonadditive codes and
degenerate codes.Comment: 5 page
Codeword stabilized quantum codes: algorithm and structure
The codeword stabilized ("CWS") quantum codes formalism presents a unifying
approach to both additive and nonadditive quantum error-correcting codes
(arXiv:0708.1021). This formalism reduces the problem of constructing such
quantum codes to finding a binary classical code correcting an error pattern
induced by a graph state. Finding such a classical code can be very difficult.
Here, we consider an algorithm which maps the search for CWS codes to a problem
of identifying maximum cliques in a graph. While solving this problem is in
general very hard, we prove three structure theorems which reduce the search
space, specifying certain admissible and optimal ((n,K,d)) additive codes. In
particular, we find there does not exist any ((7,3,3)) CWS code though the
linear programming bound does not rule it out. The complexity of the CWS search
algorithm is compared with the contrasting method introduced by Aggarwal and
Calderbank (arXiv:cs/0610159).Comment: 11 pages, 1 figur
Nonbinary Codeword Stabilized Quantum Codes
The codeword stabilized (CWS) quantum codes formalism presents a unifying
approach to both additive and nonadditive quantum error-correcting codes
(arXiv:0708.1021 [quant-ph]), but only for binary states. Here we generalize
the CWS framework to the nonbinary case (of both prime and nonprime dimension)
and map the search for nonbinary quantum codes to a corresponding search
problem for classical nonbinary codes with specific error patterns. We show
that while the additivity properties of nonbinary CWS codes are similar to the
binary case, the structural properties of the nonbinary codes differ
substantially from the binary case, even for prime dimensions. In particular,
we identify specific structure patterns of stabilizer groups, based on which
efficient constructions might be possible for codes that encode more dimensions
than any stabilizer codes of the same length and distance; similar methods
cannot be applied in the binary case. Understanding of these structural
properties can help prune the search space and facilitate the identification of
good nonbinary CWS codes.Comment: 7 pages, no figur
Entanglement-assisted codeword stabilized quantum codes
Entangled qubit can increase the capacity of quantum error correcting codes
based on stabilizer codes. In addition, by using entanglement quantum
stabilizer codes can be construct from classical linear codes that do not
satisfy the dual-containing constraint. We show that it is possible to
construct both additive and non-additive quantum codes using the codeword
stabilized quantum code framework. Nonadditive codes may offer improved
performance over the more common sta- bilizer codes. Like other
entanglement-assisted codes, the encoding procedure acts only the qubits on
Alice's side, and only these qubits are assumed to pass through the channel.
However, errors the codeword stabilized quantum code framework gives rise to
effective Z errors on Bob side. We use this scheme to construct new
entanglement-assisted non-additive quantum codes, in particular, ((5,16,2;1))
and ((7,4,5;4)) codes
Structured Error Recovery for Codeword-Stabilized Quantum Codes
Codeword stabilized (CWS) codes are, in general, non-additive quantum codes
that can correct errors by an exhaustive search of different error patterns,
similar to the way that we decode classical non-linear codes. For an n-qubit
quantum code correcting errors on up to t qubits, this brute-force approach
consecutively tests different errors of weight t or less, and employs a
separate n-qubit measurement in each test. In this paper, we suggest an error
grouping technique that allows to simultaneously test large groups of errors in
a single measurement. This structured error recovery technique exponentially
reduces the number of measurements by about 3^t times. While it still leaves
exponentially many measurements for a generic CWS code, the technique is
equivalent to syndrome-based recovery for the special case of additive CWS
codes.Comment: 13 pages, 9 eps figure
Clustered Error Correction of Codeword-Stabilized Quantum Codes
Codeword stabilized (CWS) codes are a general class of quantum codes that
includes stabilizer codes and many families of non-additive codes with good
parameters. For such a non-additive code correcting all t-qubit errors, we
propose an algorithm that employs a single measurement to test all errors
located on a given set of t qubits. Compared with exhaustive error screening,
this reduces the total number of measurements required for error recovery by a
factor of about 3^t.Comment: 4 pages, 2 figures, revtex4; number of editorial changes in v
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