671 research outputs found
On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12
We consider additive codes over GF(4) that are self-dual with respect to the
Hermitian trace inner product. Such codes have a well-known interpretation as
quantum codes and correspond to isotropic systems. It has also been shown that
these codes can be represented as graphs, and that two codes are equivalent if
and only if the corresponding graphs are equivalent with respect to local
complementation and graph isomorphism. We use these facts to classify all codes
of length up to 12, where previously only all codes of length up to 9 were
known. We also classify all extremal Type II codes of length 14. Finally, we
find that the smallest Type I and Type II codes with trivial automorphism group
have length 9 and 12, respectively.Comment: 18 pages, 4 figure
An efficient algorithm to recognize local Clifford equivalence of graph states
In [Phys. Rev. A 69, 022316 (2004)] we presented a description of the action
of local Clifford operations on graph states in terms of a graph transformation
rule, known in graph theory as \emph{local complementation}. It was shown that
two graph states are equivalent under the local Clifford group if and only if
there exists a sequence of local complementations which relates their
associated graphs. In this short note we report the existence of a polynomial
time algorithm, published in [Combinatorica 11 (4), 315 (1991)], which decides
whether two given graphs are related by a sequence of local complementations.
Hence an efficient algorithm to detect local Clifford equivalence of graph
states is obtained.Comment: 3 pages. Accepted in Phys. Rev.
Edge local complementation for logical cluster states
A method is presented for the implementation of edge local complementation in
graph states, based on the application of two Hadamard operations and a single
controlled-phase (CZ) gate. As an application, we demonstrate an efficient
scheme to construct a one-dimensional logical cluster state based on the
five-qubit quantum error-correcting code, using a sequence of edge local
complementations. A single physical CZ operation, together with local
operations, is sufficient to create a logical CZ operation between two logical
qubits. The same construction can be used to generate any encoded graph state.
This approach in concatenation may allow one to create a hierarchical quantum
network for quantum information tasks.Comment: 15 pages, two figures, IOP styl
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
Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms
The local minimum degree of a graph is the minimum degree that can be reached
by means of local complementation. For any n, there exist graphs of order n
which have a local minimum degree at least 0.189n, or at least 0.110n when
restricted to bipartite graphs. Regarding the upper bound, we show that for any
graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n)
for bipartite graphs, improving the known n/2 upper bound. We also prove that
the local minimum degree is smaller than half of the vertex cover number (up to
a logarithmic term). The local minimum degree problem is NP-Complete and hard
to approximate. We show that this problem, even when restricted to bipartite
graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which
W[1]-hardness is a long standing open question. Finally, we show that the local
minimum degree is computed by a O*(1.938^n)-algorithm, and a
O*(1.466^n)-algorithm for the bipartite graphs
Graph-Based Classification of Self-Dual Additive Codes over Finite Fields
Quantum stabilizer states over GF(m) can be represented as self-dual additive
codes over GF(m^2). These codes can be represented as weighted graphs, and
orbits of graphs under the generalized local complementation operation
correspond to equivalence classes of codes. We have previously used this fact
to classify self-dual additive codes over GF(4). In this paper we classify
self-dual additive codes over GF(9), GF(16), and GF(25). Assuming that the
classical MDS conjecture holds, we are able to classify all self-dual additive
MDS codes over GF(9) by using an extension technique. We prove that the minimum
distance of a self-dual additive code is related to the minimum vertex degree
in the associated graph orbit. Circulant graph codes are introduced, and a
computer search reveals that this set contains many strong codes. We show that
some of these codes have highly regular graph representations.Comment: 20 pages, 13 figure
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