23 research outputs found
Hard limits on the postselectability of optical graph states
Coherent control of large entangled graph states enables a wide variety of
quantum information processing tasks, including error-corrected quantum
computation. The linear optical approach offers excellent control and
coherence, but today most photon sources and entangling gates---required for
the construction of large graph states---are probabilistic and rely on
postselection. In this work, we provide proofs and heuristics to aid
experimental design using postselection. We derive a fundamental limitation on
the generation of photonic qubit states using postselected entangling gates:
experiments which contain a cycle of postselected gates cannot be postselected.
Further, we analyse experiments that use photons from postselected photon pair
sources, and lower bound the number of classes of graph state entanglement that
are accessible in the non-degenerate case---graph state entanglement classes
that contain a tree are are always accessible. Numerical investigation up to
9-qubits shows that the proportion of graph states that are accessible using
postselection diminishes rapidly. We provide tables showing which classes are
accessible for a variety of up to nine qubit resource states and sources. We
also use our methods to evaluate near-term multi-photon experiments, and
provide our algorithms for doing so.Comment: Our manuscript comprises 4843 words, 6 figures, 1 table, 47
references, and a supplementary material of 1741 words, 2 figures, 1 table,
and a Mathematica code listin
Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm
Linear rank-width is a linearized variation of rank-width, and it is deeply
related to matroid path-width. In this paper, we show that the linear
rank-width of every -vertex distance-hereditary graph, equivalently a graph
of rank-width at most , can be computed in time , and a linear layout witnessing the linear rank-width can be computed with
the same time complexity. As a corollary, we show that the path-width of every
-element matroid of branch-width at most can be computed in time
, provided that the matroid is given by an
independent set oracle.
To establish this result, we present a characterization of the linear
rank-width of distance-hereditary graphs in terms of their canonical split
decompositions. This characterization is similar to the known characterization
of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex
separation and search number of a graph. Inf. Comput., 113(1):50--79, 1994].
However, different from forests, it is non-trivial to relate substructures of
the canonical split decomposition of a graph with some substructures of the
given graph. We introduce a notion of `limbs' of canonical split
decompositions, which correspond to certain vertex-minors of the original
graph, for the right characterization.Comment: 28 pages, 3 figures, 2 table. A preliminary version appeared in the
proceedings of WG'1
How to transform graph states using single-qubit operations: computational complexity and algorithms
Graph states are ubiquitous in quantum information with diverse applications ranging from quantum network protocols to measurement based quantum computing. Here we consider the question whether one graph (source) state can be transformed into another graph (target)state,using a specific set of quantum operations (LC+LPM+CC): single-qubit Clifford operations(LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. This question is of interest for effective routing or state preparation decisions in a quantum network or distributed quantum processor and also in the design of quantum repeater schemes and quantum error-correction codes. We first show that deciding whether a graph state|G〉can be transformed into another graph state|G′〉using LC+LPM+CC is NP-complete, which was previously not known. We also show that the problem remains NP-complete even if|G′〉is restricted to be the GHZ-state. However, we also provide efficient algorithms for two situations of practical interest. Our results make use of the insight that deciding whether a graph state|G〉can be transformed to another graph state|G′〉is equivalent to a known decision problem in graph theory, namely the problem of deciding whether a graph G′ is a vertex-minor of a graph G. The computational complexity of the vertex-minor problem was prior to this paper an open question in graph theory. We prove that the vertex-minor problem is NP-complete by relating it to a new decision problem on 4-regular graphs which we call the semi-ordered Eulerian tour problem
A Single-Exponential Fixed-Parameter Algorithm for Distance-Hereditary Vertex Deletion
Vertex deletion problems ask whether it is possible to delete at most
vertices from a graph so that the resulting graph belongs to a specified graph
class. Over the past years, the parameterized complexity of vertex deletion to
a plethora of graph classes has been systematically researched. Here we present
the first single-exponential fixed-parameter tractable algorithm for vertex
deletion to distance-hereditary graphs, a well-studied graph class which is
particularly important in the context of vertex deletion due to its connection
to the graph parameter rank-width. We complement our result with matching
asymptotic lower bounds based on the exponential time hypothesis. As an
application of our algorithm, we show that a vertex deletion set to
distance-hereditary graphs can be used as a parameter which allows
single-exponential fixed-parameter tractable algorithms for classical NP-hard
problems.Comment: 43 pages, 9 figures (revised journal version; an extended abstract
appeared in the proceedings of MFCS 2016