Quantum Theory and Application of Contextual Optimal Transport
Nicola Mariella Albert Akhriev Francesco Tacchino Christa Zoufal Juan Carlos Gonzalez-Espitia Benedek Harsanyi Eugene Koskin Ivano Tavernelli Stefan Woerner Marianna Rapsomaniki Sergiy Zhuk Jannis Born
Optimal Transport (OT) has fueled machine learning (ML) applications across
many domains. In cases where paired data measurements ($\mu$, $\nu$) are
coupled to a context variable $p_i$ , one may aspire to learn a global
transportation map that can be parameterized through a potentially unseen
con-text. Existing approaches utilize Neural OT and largely rely on Brenier's
theorem. Here, we propose a first-of-its-kind quantum computing formulation for
amortized optimization of contextualized transportation plans. We exploit a
direct link between doubly stochastic matrices and unitary operators thus
finding a natural connection between OT and quantum computation. We verify our
method on synthetic and real data, by predicting variations in cell type
distributions parameterized through drug dosage as context. Our comparisons to
several baselines reveal that our method can capture dose-induced variations in
cell distributions, even to some extent when dosages are extrapolated and
sometimes with performance similar to the best classical models. In summary,
this is a first step toward learning to predict contextualized transportation
plans through quantum.