  Optimal Transport (OT) has fueled machine learning (ML) across many domains.
When paired data measurements $(\boldsymbol{\mu}, \boldsymbol{\nu})$ are
coupled to covariates, a challenging conditional distribution learning setting
arises. Existing approaches for learning a $\textit{global}$ transport map
parameterized through a potentially unseen context 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 unravelling a natural connection between OT
and quantum computation. We verify our method (QontOT) on synthetic and real
data by predicting variations in cell type distributions conditioned on drug
dosage. Importantly we conduct a 24-qubit hardware experiment on a task
challenging for classical computers and report a performance that cannot be
matched with our classical neural OT approach. In sum, this is a first step
toward learning to predict contextualized transportation plans through quantum
computing.
