5 research outputs found
Homology Computation of Large Point Clouds using Quantum Annealing
Homology is a tool in topological data analysis which measures the shape of
the data. In many cases, these measurements translate into new insights which
are not available by other means. To compute homology, we rely on mathematical
constructions which scale exponentially with the size of the data. Therefore,
for large point clouds, the computation is infeasible using classical
computers. In this paper, we present a quantum annealing pipeline for
computation of homology of large point clouds. The pipeline takes as input a
graph approximating the given point cloud. It uses quantum annealing to compute
a clique covering of the graph and then uses this cover to construct a
Mayer-Vietoris complex. The pipeline terminates by performing a simplified
homology computation of the Mayer-Vietoris complex. We have introduced three
different clique coverings and their quantum annealing formulation. Our
pipeline scales polynomially in the size of the data, once the covering step is
solved. To prove correctness of our algorithm, we have also included tests
using D-Wave 2X quantum processor
Computing Wasserstein Distance for Persistence Diagrams on a Quantum Computer
Persistence diagrams are a useful tool from topological data analysis which
can be used to provide a concise description of a filtered topological space.
What makes them even more useful in practice is that they come with a notion of
a metric, the Wasserstein distance (closely related to but not the same as the
homonymous metric from probability theory). Further, this metric provides a
notion of stability; that is, small noise in the input causes at worst small
differences in the output. In this paper, we show that the Wasserstein distance
for persistence diagrams can be computed through quantum annealing. We provide
a formulation of the problem as a Quadratic Unconstrained Binary Optimization
problem, or QUBO, and prove correctness. Finally, we test our algorithm,
exploring parameter choices and problem size capabilities, using a D-Wave 2000Q
quantum annealing computer
The Topology of Mutated Driver Pathways
Much progress has been made, and continues to be made, towards identifying
candidate mutated driver pathways in cancer. However, no systematic approach to
understanding how candidate pathways relate to each other for a given cancer
(such as Acute myeloid leukemia), and how one type of cancer may be similar or
different from another with regard to their respective pathways (Acute myeloid
leukemia vs. Glioblastoma multiforme for instance), has emerged thus far. Our
work attempts to contribute to the understanding of {\em space of pathways}
through a novel topological framework. We illustrate our approach, using
mutation data (obtained from TCGA) of two types of tumors: Acute myeloid
leukemia (AML) and Glioblastoma multiforme (GBM). We find that the space of
pathways for AML is homotopy equivalent to a sphere, while that of GBM is
equivalent to a genus-2 surface. We hope to trigger new types of questions
(i.e., allow for novel kinds of hypotheses) towards a more comprehensive grasp
of cancer.Comment: Key words: topological data analysis, cancer genomics, mutation data,
acute myeloid leukemia, glioblastoma multiforme, persistent homology,
simplicial complex, Betti numbers, algebraic topolog
Quantum Computing: An Overview Across the System Stack
Quantum computers, if fully realized, promise to be a revolutionary
technology. As a result, quantum computing has become one of the hottest areas
of research in the last few years. Much effort is being applied at all levels
of the system stack, from the creation of quantum algorithms to the development
of hardware devices. The quantum age appears to be arriving sooner rather than
later as commercially useful small-to-medium sized machines have already been
built. However, full-scale quantum computers, and the full-scale algorithms
they would perform, remain out of reach for now. It is currently uncertain how
the first such computer will be built. Many different technologies are
competing to be the first scalable quantum computer
Finding universal structures in quantum many-body dynamics via persistent homology
Inspired by topological data analysis techniques, we introduce persistent
homology observables and apply them in a geometric analysis of the dynamics of
quantum field theories. As a prototype application, we consider simulated data
of a two-dimensional Bose gas far from equilibrium. We discover a continuous
spectrum of dynamical scaling exponents, which provides a refined
classification of nonequilibrium universal phenomena. A possible explanation of
the underlying processes is provided in terms of mixing wave turbulence and
vortex kinetics components in point clouds. We find that the persistent
homology scaling exponents are inherently linked to the geometry of the system,
as the derivation of a packing relation reveals. The approach opens new ways of
analyzing quantum many-body dynamics in terms of robust topological structures
beyond standard field theoretic techniques.Comment: 21 pages, 18 figure