Quantum Theory and Application of Contextual Optimal Transport

Optimal Transport (OT) has fueled 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.