76 research outputs found
Quantum Goethals-Preparata Codes
We present a family of non-additive quantum codes based on Goethals and
Preparata codes with parameters ((2^m,2^{2^m-5m+1},8)). The dimension of these
codes is eight times higher than the dimension of the best known additive
quantum codes of equal length and minimum distance.Comment: Submitted to the 2008 IEEE International Symposium on Information
Theor
Non-Additive Quantum Codes from Goethals and Preparata Codes
We extend the stabilizer formalism to a class of non-additive quantum codes
which are constructed from non-linear classical codes. As an example, we
present infinite families of non-additive codes which are derived from Goethals
and Preparata codes.Comment: submitted to the 2008 IEEE Information Theory Workshop (ITW 2008
Non-Additive Quantum Codes from Goethals and Preparata Codes
We extend the stabilizer formalism to a class of non-additive quantum codes
which are constructed from non-linear classical codes. As an example, we
present infinite families of non-additive codes which are derived from Goethals
and Preparata codes.Comment: submitted to the 2008 IEEE Information Theory Workshop (ITW 2008
Kerdock Codes Determine Unitary 2-Designs
The non-linear binary Kerdock codes are known to be Gray images of certain
extended cyclic codes of length over . We show that
exponentiating these -valued codewords by produces stabilizer states, that are quantum states obtained using
only Clifford unitaries. These states are also the common eigenvectors of
commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the
Pauli group. We use this quantum description to simplify the derivation of the
classical weight distribution of Kerdock codes. Next, we organize the
stabilizer states to form mutually unbiased bases and prove that
automorphisms of the Kerdock code permute their corresponding MCS, thereby
forming a subgroup of the Clifford group. When represented as symplectic
matrices, this subgroup is isomorphic to the projective special linear group
PSL(). We show that this automorphism group acts transitively on the Pauli
matrices, which implies that the ensemble is Pauli mixing and hence forms a
unitary -design. The Kerdock design described here was originally discovered
by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is
new which simplifies its description and translation to circuits significantly.
Sampling from the design is straightforward, the translation to circuits uses
only Clifford gates, and the process does not require ancillary qubits.
Finally, we also develop algorithms for optimizing the synthesis of unitary
-designs on encoded qubits, i.e., to construct logical unitary -designs.
Software implementations are available at
https://github.com/nrenga/symplectic-arxiv18a, which we use to provide
empirical gate complexities for up to qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to
2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is
included in the arXiv packag
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
New Constant-Weight Codes from Propagation Rules
This paper proposes some simple propagation rules which give rise to new
binary constant-weight codes.Comment: 4 page
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
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
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