244 research outputs found
A Lex-BFS-based recognition algorithm for Robinsonian matrices
Robinsonian matrices arise in the classical seriation problem and play an important role
in many applications where unsorted similarity (or dissimilarity) information must be re-
ordered. We present a new polynomial time algorithm to recognize Robinsonian matrices
based on a new characterization of Robinsonian matrices in terms of straight enumerations
of unit interval graphs. The algorithm is simple and is based essentially on lexicographic
breadth-first search (Lex-BFS), using a divide-and-conquer strategy. When applied to a non-
negative symmetric n Ă— n matrix with m nonzero entries and given as a weighted adjacency
list, it runs in O(d(n + m)) time, where d is the depth of the recursion tree, which is at most
the number of distinct nonzero entries of A
A quantum Bose-Hubbard model with evolving graph as toy model for emergent spacetime
We present a toy model for interacting matter and geometry that explores
quantum dynamics in a spin system as a precursor to a quantum theory of
gravity. The model has no a priori geometric properties, instead, locality is
inferred from the more fundamental notion of interaction between the matter
degrees of freedom. The interaction terms are themselves quantum degrees of
freedom so that the structure of interactions and hence the resulting local and
causal structures are dynamical. The system is a Hubbard model where the graph
of the interactions is a set of quantum evolving variables. We show
entanglement between spatial and matter degrees of freedom. We study
numerically the quantum system and analyze its entanglement dynamics. We
analyze the asymptotic behavior of the classical model. Finally, we discuss
analogues of trapped surfaces and gravitational attraction in this simple
model.Comment: 23 pages, 6 figures; updated to published versio
Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs
We examine the existence and structure of particular sets of mutually
unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known
power-of-prime MUB constructions, we restrict ourselves to using maximally
entangled stabilizer states as MUB vectors. Consequently, these bipartite
entangled stabilizer MUBs (BES MUBs) provide no local information, but are
sufficient and minimal for decomposing a wide variety of interesting operators
including (mixtures of) Jamiolkowski states, entanglement witnesses and more.
The problem of finding such BES MUBs can be mapped, in a natural way, to that
of finding maximum cliques in a family of Cayley graphs. Some relationships
with known power-of-prime MUB constructions are discussed, and observables for
BES MUBs are given explicitly in terms of Pauli operators.Comment: 8 pages, 1 figur
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
New Approaches to Complexity via Quantum Graphs
Problems based on the structure of graphs -- for example finding cliques,
independent sets, or colourings -- are of fundamental importance in classical
complexity. It is well motivated to consider similar problems about quantum
graphs, which are an operator system generalisation of graphs. Defining
well-formulated decision problems for quantum graphs faces several technical
challenges, and consequently the connections between quantum graphs and
complexity have been underexplored.
In this work, we introduce and study the clique problem for quantum graphs.
Our approach utilizes a well-known connection between quantum graphs and
quantum channels. The inputs for our problems are presented as quantum channels
induced by circuits, which implicitly determine a corresponding quantum graph.
We also use this approach to reimagine the clique and independent set problems
for classical graphs, by taking the inputs to be circuits of deterministic or
noisy channels which implicitly determine confusability graphs. We show that,
by varying the collection of channels in the language, these give rise to
complete problems for the classes , , ,
and . In this way, we exhibit a classical complexity problem
whose natural quantisation is , rather than ,
which is commonly assumed.
To prove the results in the quantum case, we make use of methods inspired by
self-testing. To illustrate the utility of our techniques, we include a new
proof of the reduction of to via cliques
for quantum graphs. We also study the complexity of a version of the
independent set problem for quantum graphs, and provide preliminary evidence
that it may be in general weaker in complexity, contrasting to the classical
case where the clique and independent set problems are equivalent.Comment: 45 pages, 3 figure
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