69 research outputs found
Quantum Codes from Toric Surfaces
A theory for constructing quantum error correcting codes from Toric surfaces
by the Calderbank-Shor-Steane method is presented. In particular we study the
method on toric Hirzebruch surfaces. The results are obtained by constructing a
dualizing differential form for the toric surface and by using the cohomology
and the intersection theory of toric varieties. In earlier work the author
developed methods to construct linear error correcting codes from toric
varieties and derive the code parameters using the cohomology and the
intersection theory on toric varieties. This method is generalized in section
to construct linear codes suitable for constructing quantum codes by the
Calderbank-Shor-Steane method. Essential for the theory is the existence and
the application of a dualizing differential form on the toric surface. A.R.
Calderbank, P.W. Shor and A.M. Steane produced stabilizer codes from linear
codes containing their dual codes. These two constructions are merged to obtain
results for toric surfaces. Similar merging has been done for algebraic curves
with different methods by A. Ashikhmin, S. Litsyn and M.A. Tsfasman.Comment: IEEE copyrigh
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
Algebraic Quantum Error-Correction Codes
Based on the group structure of a unitary Lie algebra, a scheme is provided
to systematically and exhaustively generate quantum error correction codes,
including the additive and nonadditive codes. The syndromes in the process of
error-correction distinguished by different orthogonal vector subspaces, the
coset subspaces. Moreover, the generated codes can be classified into four
types with respect to the spinors in the unitary Lie algebra and a chosen
initial quantum state
Non-binary Unitary Error Bases and Quantum Codes
Error operator bases for systems of any dimension are defined and natural
generalizations of the bit/sign flip error basis for qubits are given. These
bases allow generalizing the construction of quantum codes based on eigenspaces
of Abelian groups. As a consequence, quantum codes can be constructed from
linear codes over \ints_n for any . The generalization of the punctured
code construction leads to many codes which permit transversal (i.e. fault
tolerant) implementations of certain operations compatible with the error
basis.Comment: 10 pages, preliminary repor
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