6,941 research outputs found
Quantum error-correcting codes associated with graphs
We present a construction scheme for quantum error correcting codes. The
basic ingredients are a graph and a finite abelian group, from which the code
can explicitly be obtained. We prove necessary and sufficient conditions for
the graph such that the resulting code corrects a certain number of errors.
This allows a simple verification of the 1-error correcting property of
fivefold codes in any dimension. As new examples we construct a large class of
codes saturating the singleton bound, as well as a tenfold code detecting 3
errors.Comment: 8 pages revtex, 5 figure
Sparse Graph Codes for Quantum Error-Correction
We present sparse graph codes appropriate for use in quantum
error-correction. Quantum error-correcting codes based on sparse graphs are of
interest for three reasons. First, the best codes currently known for classical
channels are based on sparse graphs. Second, sparse graph codes keep the number
of quantum interactions associated with the quantum error correction process
small: a constant number per quantum bit, independent of the blocklength.
Third, sparse graph codes often offer great flexibility with respect to
blocklength and rate. We believe some of the codes we present are unsurpassed
by previously published quantum error-correcting codes.Comment: Version 7.3e: 42 pages. Extended version, Feb 2004. A shortened
version was resubmitted to IEEE Transactions on Information Theory Jan 20,
200
Scheme for constructing graphs associated with stabilizer quantum codes
We propose a systematic scheme for the construction of graphs associated with
binary stabilizer codes. The scheme is characterized by three main steps:
first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum
code; second, the canonical form of the CWS code is uncovered; third, the input
vertices are attached to the graphs. To check the effectiveness of the scheme,
we discuss several graphical constructions of various useful stabilizer codes
characterized by single and multi-qubit encoding operators. In particular, the
error-correcting capabilities of such quantum codes are verified in
graph-theoretic terms as originally advocated by Schlingemann and Werner.
Finally, possible generalizations of our scheme for the graphical construction
of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum
codes are briefly addressed.Comment: 42 pages, 12 figure
On the Minimum Degree up to Local Complementation: Bounds and Complexity
The local minimum degree of a graph is the minimum degree reached by means of
a series of local complementations. In this paper, we investigate on this
quantity which plays an important role in quantum computation and quantum error
correcting codes. First, we show that the local minimum degree of the Paley
graph of order p is greater than sqrt{p} - 3/2, which is, up to our knowledge,
the highest known bound on an explicit family of graphs. Probabilistic methods
allows us to derive the existence of an infinite number of graphs whose local
minimum degree is linear in their order with constant 0.189 for graphs in
general and 0.110 for bipartite graphs. As regards the computational complexity
of the decision problem associated with the local minimum degree, we show that
it is NP-complete and that there exists no k-approximation algorithm for this
problem for any constant k unless P = NP.Comment: 11 page
Quantum computation with Turaev-Viro codes
The Turaev-Viro invariant for a closed 3-manifold is defined as the
contraction of a certain tensor network. The tensors correspond to tetrahedra
in a triangulation of the manifold, with values determined by a fixed spherical
category. For a manifold with boundary, the tensor network has free indices
that can be associated to qudits, and its contraction gives the coefficients of
a quantum error-correcting code. The code has local stabilizers determined by
Levin and Wen. For example, applied to the genus-one handlebody using the Z_2
category, this construction yields the well-known toric code.
For other categories, such as the Fibonacci category, the construction
realizes a non-abelian anyon model over a discrete lattice. By studying braid
group representations acting on equivalence classes of colored ribbon graphs
embedded in a punctured sphere, we identify the anyons, and give a simple
recipe for mapping fusion basis states of the doubled category to ribbon
graphs. We explain how suitable initial states can be prepared efficiently, how
to implement braids, by successively changing the triangulation using a fixed
five-qudit local unitary gate, and how to measure the topological charge.
Combined with known universality results for anyonic systems, this provides a
large family of schemes for quantum computation based on local deformations of
stabilizer codes. These schemes may serve as a starting point for developing
fault-tolerance schemes using continuous stabilizer measurements and active
error-correction.Comment: 53 pages, LaTeX + 199 eps figure
Quantum Stabilizer Codes, Lattices, and CFTs
There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples
Relation between spectra of Narain CFTs and properties of associated boolean functions
Recently, the construction of Narain CFT from a certain class of quantum
error correcting codes has been discovered. In particular, the spectral gap of
Narain CFT corresponds to the binary distance of the code, not the genuine
Hamming distance. In this paper, we show that the binary distance is identical
to the so-called EPC distance of the boolean function uniquely associated with
the quantum code. Therefore, seeking Narain CFT with high spectral gap is
equivalent to getting a boolean function with high EPC distance. Furthermore,
this problem can be addressed by finding lower Peak-to-Average Power ratio
(PAR) with respect to the binary truth table of the boolean function. Though
this is neither sufficient nor necessary condition for high EPC distance, we
construct some examples of relatively high EPC distances referring to the
constructions for lower PAR. We also see that codes with high distance are
related to induced graphs with low independence numbers.Comment: 27 pages, 1 figur
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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