Computing Wasserstein Distance for Persistence Diagrams on a Quantum Computer

Abstract

Persistence diagrams are a useful tool from topological data analysis that provide a concise description of a filtered topological space. They are even more useful in practice because 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 computer.

Finding self-similar behavior in quantum many-body dynamics via persistent homology

“…The latter provide further evidence for the suitability of the self-similar scaling ansatz for the asymptotic persistence pair distribution, as given by Eq. (20).…”

“…Besides the mathematical investigations on persistent homology, very fruitful applications to physical systems include studies in astrophysics and cosmology [9][10][11][12], physical chemistry [13], amorphous materials [14], the vacua landscape in string theories [15] and the theory of quantum phase space [16]. Moreover, it has been shown that quantum algorithms can speed up persistent homology computations [17][18][19][20][21].…”

Quantum Bridge Analytics I: a tutorial on formulating and using QUBO models

“…Recent QUBO quantum computing applications, complementing earlier applications on classical computing systems, include those for graph partitioning problems in Mniszewski et al (2016) and Ushijima-Mwesigwa et al (2017); graph clustering (quantum community detection problems) in Negre et al (2018Negre et al ( , 2019; traffic-flow optimization in Neukart et al (2017); vehicle routing problems in Feld et al (2018), Clark et al (2019) and Ohzeki et al(2018); maximum clique problems in Chapuis et al (2018); cybersecurity problems in Munch et al (2018) and Reinhardt et al(2018); predictive health analytics problems in De Oliveira et al (2018) and Sahner et al (2018); and financial portfolio management problems in Elsokkary et al (2017) and Kalra et al (2018). In another recent development, QUBO models are being studied using the IBM neuromorphic computer at as reported in Alom et al (2017) and Aimone et al (2018).…”

Topological Machine Learning for Multivariate Time Series

“…The 1-Wasserstein distance is widely used in computer science to compare discrete distributions (Rabin, Delon, & Gousseau, 2009;Rubner, Tomasi, & Guibas, 2000). The Wasserstein distance is defined as follows (Berwald, Gottlieb, & Munch, 2018;Cohen-Steiner et al, 2010;Mileyko, Mukherjee, & Harer, 2011).…”