PiQuantO · Luca Nenna

$\displaystyle c(x_1,\dots,x_N)=\sum_{i<j}\frac{1}{|x_i-x_j|}$ $\displaystyle F_{\mathrm{SCE}}[\rho]=\min_{\gamma\in\Pi(\rho,\dots,\rho)}\int c\,d\gamma$ $\displaystyle F_L[\rho]=\min_{\Gamma\mapsto\rho}\mathrm{Tr}\!\big[(T+V_{ee})\,\Gamma\big]$ $F_L^\hbar[\rho]\xrightarrow{\hbar\to 0} F_{\mathrm{SCE}}[\rho]$ $E_0=\min_\rho\Big\{F[\rho]+\!\int V\rho\Big\}$ $\rho(x)=N\!\int|\Psi|^2\,dx_2\cdots dx_N$

The project

PiQuantO — Problems in Quantum Optimal Transport is funded by the Inria PiQ programme (Programme Inria Quadrant), within the France 2030 framework.

This project deals with the computation of the Lieb functional, a key quantity in Density Functional Theory which describes the electronic structure of molecules. We want to overcome the curse of dimensionality which affects this problem.

The Lieb functional can be reformulated, in the strongly-correlated limit, as a multi-marginal optimal transport problem with Coulomb cost $c(x_1,\dots,x_N)=\sum_{i<j}|x_i-x_j|^{-1}$ — the bridge between optimal transport and electronic-structure theory at the heart of PiQuantO.

Project page on the Inria PiQ site →

Collaborators

Rodrigue Lelotte — Maître de conférences, Université Paris 1 Panthéon-Sorbonne (SAMM lab)

Related publications

V. Ehrlacher, R. Lelotte, L. Nenna. A land of monotone plenty, bis repetita: from classical to weak optimal transport. Preprint, 2026.
V. Ehrlacher, L. Nenna. A sparse approximation of the Lieb functional with moment constraints. SIAM J. Math. Anal., 2026.
S. Di Marino, M. Lewin, L. Nenna. Grand-Canonical Optimal Transport. Arch. Ration. Mech. Anal., 2025.