26 research outputs found
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.
Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform
We enumerate the inequivalent self-dual additive codes over GF(4) of
blocklength n, thereby extending the sequence A090899 in The On-Line
Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a
well-known interpretation as quantum codes. They can also be represented by
graphs, where a simple graph operation generates the orbits of equivalent
codes. We highlight the regularity and structure of some graphs that correspond
to codes with high distance. The codes can also be interpreted as quadratic
Boolean functions, where inequivalence takes on a spectral meaning. In this
context we define PAR_IHN, peak-to-average power ratio with respect to the
{I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is
equivalent to the the size of the maximum independent set over the associated
orbit of graphs. Finally we propose a construction technique to generate
Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South
Korea, October 2004. 17 pages, 10 figure
A complete characterisation of All-versus-Nothing arguments for stabiliser states
An important class of contextuality arguments in quantum foundations are the
All-versus-Nothing (AvN) proofs, generalising a construction originally due to
Mermin. We present a general formulation of All-versus-Nothing arguments, and a
complete characterisation of all such arguments which arise from stabiliser
states. We show that every AvN argument for an n-qubit stabiliser state can be
reduced to an AvN proof for a three-qubit state which is local
Clifford-equivalent to the tripartite GHZ state. This is achieved through a
combinatorial characterisation of AvN arguments, the AvN triple Theorem, whose
proof makes use of the theory of graph states. This result enables the
development of a computational method to generate all the AvN arguments in
on n-qubit stabiliser states. We also present new insights into
the stabiliser formalism and its connections with logic.Comment: 18 pages, 6 figure
Partial Complementation of Graphs
A partial complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class G, is there a partial complement of G which is in G? We show that this problem can be solved in polynomial time for various choices of the graphs class G, such as bipartite, degenerate, or cographs. We complement these results by proving that the problem is NP-complete when G is the class of r-regular graphs