Quantum State Assignment Flows
Jonathan Schwarz, Jonas Cassel, Bastian Boll, Martin G\"arttner, Peter, Albers, Christoph Schn\"orr

TL;DR
This paper introduces quantum state assignment flows on density matrices for data analysis on graphs, utilizing information geometry and Riemannian metrics to enable efficient computation and capture complex data correlations.
Contribution
It presents a novel class of quantum state assignment flows with geometric integration, extending previous categorial probability flows to quantum states, and demonstrates their potential for data representation.
Findings
Efficient computation via closed-form local expressions.
Flow convergence to pure quantum states.
Potential to represent data correlations through entanglement.
Abstract
This paper introduces assignment flows for density matrices as state spaces for representing and analyzing data associated with vertices of an underlying weighted graph. Determining an assignment flow by geometric integration of the defining dynamical system causes an interaction of the non-commuting states across the graph, and the assignment of a pure (rank-one) state to each vertex after convergence. Adopting the Riemannian Bogoliubov-Kubo-Mori metric from information geometry leads to closed-form local expressions which can be computed efficiently and implemented in a fine-grained parallel manner. Restriction to the submanifold of commuting density matrices recovers the assignment flows for categorial probability distributions, which merely assign labels from a finite set to each data point. As shown for these flows in our prior work, the novel class of quantum state assignment…
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Taxonomy
TopicsNeural Networks and Applications · Topological and Geometric Data Analysis · Stochastic Gradient Optimization Techniques
