dynamical-quantum-optimal-transport

name	dynamical-quantum-optimal-transport
description	Dynamical quantum optimal transport (QOT) methodology based on Benamou-Brenier formulation for computing geodesics between positive semidefinite matrices. Use when: computing quantum state transport distances, solving quantum chemistry problems via optimal transport, analyzing numerical convergence of QOT distances, or implementing interior-point methods for quantum density matrix geodesics. Activation: quantum optimal transport, dynamical QOT, Benamou-Brenier, quantum chemistry optimal transport, density matrix geodesic, interior-point quantum transport, positive semidefinite transport
metadata	{"arxiv_id":"2606.10075","published":"2026-06-08","authors":"Genevieve Dusson, Virginie Ehrlacher, Etienne Obermeyer"}

Dynamical Quantum Optimal Transport

Source: arXiv:2606.10075 — "An algorithm for dynamical quantum optimal transport with applications to quantum chemistry" by Genevieve Dusson, Virginie Ehrlacher, Etienne Obermeyer (2026-06-08)

Overview

Numerical study of dynamical quantum optimal transport distances based on the Benamou-Brenier formulation adapted to spaces of density matrices. Introduces an interior-point regularized method to compute geodesics between positive semidefinite matrices with applications to quantum chemistry.

Core Methodology

Benamou-Brenier Formulation for Quantum States

The classical Benamou-Brenier formulation expresses the Wasserstein-2 distance as a dynamical optimization problem over continuous paths of probability measures. The quantum extension replaces probability measures with density matrices (positive semidefinite matrices with unit trace):

W_2(ρ_0, ρ_1)² = inf ∫₀¹ Tr(ρ_t · L_{ρ_t}⁻¹(v_t)²) dt

where ρ_t is a path of density matrices, v_t is the velocity field, and L_{ρ_t} is a quantum transport operator.

Interior-Point Regularized Method

1. Parameterize the path: Represent ρ_t as a smooth curve in the space of PSD matrices
2. Add barrier function: Log-det barrier to enforce positive definiteness: -μ · log det(ρ_t)
3. Discretize time: Split [0,1] into N intervals, optimize over discrete sequence {ρ_k}
4. Solve via interior-point: Use Newton-type method with barrier parameter μ → 0

Numerical Properties

• Convergence: Objects converge as matrix size increases
• Visualization: Results expressed as integral kernels and densities
• Parameter tuning: Appropriate parameters approximate certain quantum chemistry problems

Implementation Steps

Step 1: Define the QOT Problem

import numpy as np
from scipy.linalg import sqrtm

def qot_cost(rho0, rho1, n_steps=10, mu=1e-4):
    """Compute dynamical QOT distance between two density matrices."""
    # Discretize path: rho_k for k = 0, ..., n_steps
    # Interior-point objective: sum of kinetic + barrier terms
    pass

Step 2: Interior-Point Optimization

def interior_point_qot(rho0, rho1, n_steps, mu_init, tol=1e-8):
    """Interior-point method for dynamical QOT."""
    mu = mu_init
    # Initialize path as linear interpolation
    path = [rho0 + (rho1 - rho0) * k / n_steps for k in range(n_steps + 1)]
    
    while mu > tol:
        # Newton step on regularized objective
        # Decrease mu
        # Check convergence
        pass
    return path

Step 3: Quantum Chemistry Application

The dynamical QOT distance can approximate:

• Electronic structure comparisons
• Molecular orbital transport costs
• State preparation costs in quantum algorithms

Key Results

1. Interior-point method successfully computes geodesics between PSD matrices
2. Numerical convergence established as matrix size increases
3. Quantum chemistry approximation possible with appropriate parameter tuning
4. Visualization framework via integral kernels and densities

Pitfalls

1. Parameter sensitivity: QOT distances depend critically on the choice of regularization parameter μ and discretization resolution N
2. Positive definiteness: The barrier method requires strictly positive definite matrices — near-singular density matrices need regularization
3. Computational cost: Interior-point methods scale as O(n³) per Newton step for n×n matrices
4. Not a replacement for all quantum chemistry methods: QOT approximation is complementary to standard quantum chemistry methods (DFT, coupled cluster)

Activation

• quantum optimal transport, dynamical QOT, Benamou-Brenier
• quantum chemistry optimal transport
• density matrix geodesic
• interior-point quantum transport
• positive semidefinite transport
• 量子最优传输, 动力学量子最优传输, 量子化学最优传输