Quantum Theory and Application of Contextual Optimal Transport
This work addresses a challenging conditional distribution learning problem in machine learning, representing a first step toward quantum-enhanced optimal transport, though it is incremental as it builds on existing neural OT methods.
The paper tackles the problem of learning global transport maps for conditional distribution learning with covariates by proposing a quantum computing formulation for amortized optimization of contextualized transportation plans, reporting performance on a 24-qubit hardware experiment that cannot be matched by classical neural OT approaches.
Optimal Transport (OT) has fueled machine learning (ML) across many domains. When paired data measurements $(\boldsymbolμ, \boldsymbolν)$ 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.