This document summarizes the activities and outcomes of the Long Program “Non-commutative Optimal Transport,” which was held at the Institute of Pure and Applied Mathematics (IPAM) from March 10 to June 13, 2025. It also briefly explores some of the current open questions and future directions in the field of electronic structure theory, as well as related fields that were discussed during the program.

This document surveys key topics identified by the program participants:

Bridging Existing NOT Concepts: Generalizing classical OT quantities to the quantum setting is a challenging task due to the barriers posed by entanglement and non-commutativity. This has led to diverging definitions of NOT concepts, which have little in common with each other. A key area of ongoing research, explored by two working groups, lies in the reunification of these concepts and the building of bridges between existing NOT formulations.

NOT in Free Probability: Analogizing from the classical setting, free optimal transport, free entropy, and free optimal control provide a framework for studying random matrix models, their large-N limits, and structural properties of vNAs.

Gradient Structures from Dilations: Recent work has shown the existence of a gradient structure for the Lindblad equation under certain symmetry conditions, which has been effectively used to prove entropy inequalities. We explore a way to derive gradient structures from dilations and microscopic dynamics by using large deviations.

Quantum Thermodynamics: The notion of work, recently adapted to the quantum setting, provides a control-theoretic framework for studying quantum state transportation along a path. Thanks to the entropic nature of the dissipative components of the work along such a path, new estimates can be derived. 

Quantum Optimal Control and Algorithms: Both QOC and variational quantum circuit optimization are known to suffer from barren plateaus and many bad local minima as the number of qubits in a quantum computer increases. Using optimal transport techniques has proven effective in mitigating these issues for certain problems and circuit parameterizations. We explore how optimal transport techniques can be used to enhance the performance of QOC protocols and analyze the computational complexity of quantum algorithms.

Quantum Many-Body Problem: NOT provides a mathematical framework for reconstructing quantum states and their properties directly from their reduced density matrices. This leads to new mathematical and computational techniques that are especially effective for applications where the number of quantum particles becomes (infinitely) large.   

Computational Aspects for Quantum: This section presents recent advances in computational strategies for QOT, highlighting the development of entropy-based regularization methods, SDP approaches, and ML techniques to overcome the challenges posed by QOT’s non-commutative structure and high dimensionality.

Read the full report.