687 research outputs found
Quantum kernels for unattributed graphs using discrete-time quantum walks
In this paper, we develop a new family of graph kernels where the graph structure is probed by means of a discrete-time quantum walk. Given a pair of graphs, we let a quantum walk evolve on each graph and compute a density matrix with each walk. With the density matrices for the pair of graphs to hand, the kernel between the graphs is defined as the negative exponential of the quantum Jensen–Shannon divergence between their density matrices. In order to cope with large graph structures, we propose to construct a sparser version of the original graphs using the simplification method introduced in Qiu and Hancock (2007). To this end, we compute the minimum spanning tree over the commute time matrix of a graph. This spanning tree representation minimizes the number of edges of the original graph while preserving most of its structural information. The kernel between two graphs is then computed on their respective minimum spanning trees. We evaluate the performance of the proposed kernels on several standard graph datasets and we demonstrate their effectiveness and efficiency
A Quantum-inspired Similarity Measure for the Analysis of Complete Weighted Graphs
We develop a novel method for measuring the similarity between complete weighted graphs, which are probed by means of discrete-time quantum walks. Directly probing complete graphs using discrete-time quantum walks is intractable due to the cost of simulating the quantum walk. We overcome this problem by extracting a commute-time minimum spanning tree from the complete weighted graph. The spanning tree is probed by a discrete time quantum walk which is initialised using a weighted version of the Perron-Frobenius operator. This naturally encapsulates the edge weight information for the spanning tree extracted from the original graph. For each pair of complete weighted graphs to be compared, we simulate a discrete-time quantum walk on each of the corresponding commute time minimum spanning trees, and then compute the associated density matrices for the quantum walks. The probability of the walk visiting each edge of the spanning tree is given by the diagonal elements of the density matrices. The similarity between each pair of graphs is then computed using either a) the inner product or b) the negative exponential of the Jensen-Shannon divergence between the probability distributions. We show that in both cases the resulting similarity measure is positive definite and therefore corresponds to a kernel on the graphs. We perform a series of experiments on publicly available graph datasets from a variety of different domains, together with time-varying financial networks extracted from data for the New York Stock Exchange. Our experiments demonstrate the effectiveness of the proposed similarity measures
Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories
We define a new topological polynomial extending the Bollobas-Riordan one,
which obeys a four-term reduction relation of the deletion/contraction type and
has a natural behavior under partial duality. This allows to write down a
completely explicit combinatorial evaluation of the polynomials, occurring in
the parametric representation of the non-commutative Grosse-Wulkenhaar quantum
field theory. An explicit solution of the parametric representation for
commutative field theories based on the Mehler kernel is also provided.Comment: 58 pages, 23 figures, correction in the references and addition of
preprint number
A constructive commutative quantum Lovasz Local Lemma, and beyond
The recently proven Quantum Lovasz Local Lemma generalises the well-known
Lovasz Local Lemma. It states that, if a collection of subspace constraints are
"weakly dependent", there necessarily exists a state satisfying all
constraints. It implies e.g. that certain instances of the kQSAT quantum
satisfiability problem are necessarily satisfiable, or that many-body systems
with "not too many" interactions are always frustration-free.
However, the QLLL only asserts existence; it says nothing about how to find
the state. Inspired by Moser's breakthrough classical results, we present a
constructive version of the QLLL in the setting of commuting constraints,
proving that a simple quantum algorithm converges efficiently to the required
state. In fact, we provide two different proofs, one using a novel quantum
coupling argument, the other a more explicit combinatorial analysis. Both
proofs are independent of the QLLL. So these results also provide independent,
constructive proofs of the commutative QLLL itself, but strengthen it
significantly by giving an efficient algorithm for finding the state whose
existence is asserted by the QLLL. We give an application of the constructive
commutative QLLL to convergence of CP maps.
We also extend these results to the non-commutative setting. However, our
proof of the general constructive QLLL relies on a conjecture which we are only
able to prove in special cases.Comment: 43 pages, 2 conjectures, no figures; unresolved gap in the proof; see
arXiv:1311.6474 or arXiv:1310.7766 for correct proofs of the symmetric cas
Differential graded algebras for trivalent plane graphs and their representations
To any trivalent plane graph embedded in the sphere, Casals and Murphy
associate a differential graded algebra (dg-algebra), in which the underlying
graded algebra is free associative over a commutative ring. Our first result is
the construction of a generalization of the Casals--Murphy dg-algebra to
non-commutative coefficients, for which we prove various functoriality
properties not previously verified in the commutative setting. Our second
result is to prove that rank representations of this dg-algebra, over a
field , correspond to colorings of the faces of the graph by
elements of the Grassmannian so that
bordering faces are transverse, up to the natural action of
. Underlying the combinatorics, the
dg-algebra is a computation of the fully non-commutative Legendrian contact
dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not
prove as such in this paper. The graph coloring problem verifies that for
Legendrian 2-weaves, rank representations of the Legendrian contact
dg-algebra correspond to constructible sheaves of microlocal rank . This is
the first such verification of this conjecture for an infinite family of
Legendrian surfaces.Comment: Minor revision
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