Exact Solutions to the Quantum Schrödinger Bridge Problem

TL;DR

This work introduces the Quantum Schrödinger Bridge Problem (QSBP), a quantum-inspired extension of entropy-regularized transport that links forward–backward stochastic dynamics to the Schrödinger equation via the Guerra–Morato Lagrangian and the Bohm quantum potential. It derives exact closed-form Gaussian solutions, providing time-evolving means and covariances, and extends to Gaussian mixtures with a Gaussian Wave Packet approach. The authors implement a Gaussian Mixture Model evolution algorithm and demonstrate the method across single-cell trajectory inference, unpaired image translation, latent-space molecular toxicity translation, and mean-field game scenarios, achieving competitive performance and practical transport capabilities. This framework offers a versatile, analytically tractable bridge for generative modeling between arbitrary distributions, with broad applicability in physics-informed diffusion processes and data-driven sequence transport.

Abstract

The Quantum Schrödinger Bridge Problem (QSBP) describes the evolution of a stochastic process between two arbitrary probability distributions, where the dynamics are governed by the Schrödinger equation rather than by the traditional real-valued wave equation. Although the QSBP is known in the mathematical literature, we formulate it here from a Lagrangian perspective and derive its main features in a way that is particularly suited to generative modeling. We show that the resulting evolution equations involve the so-called Bohm (quantum) potential, representing a notion of non-locality in the stochastic process. This distinguishes the QSBP from classical stochastic dynamics and reflects a key characteristic typical of quantum mechanical systems. In this work, we derive exact closed-form solutions for the QSBP between Gaussian distributions. Our derivation is based on solving the Fokker-Planck Equation (FPE) and the Hamilton-Jacobi Equation (HJE) arising from the Lagrangian formulation of dynamical Optimal Transport. We find that, similar to the classical Schrödinger Bridge Problem, the solution to the QSBP between Gaussians is again a Gaussian process; however, the evolution of the covariance differs due to quantum effects. Leveraging these explicit solutions, we present a modified algorithm based on a Gaussian Mixture Model framework, and demonstrate its effectiveness across several experimental settings, including single-cell evolution data, image generation, molecular translation and applications in Mean-Field Games.

Figures (5)

• Figure 1: Example of 1d (a) and 2d distributions (b) with the corresponding Bohm potentials. (c) Learned scores of the data distribution $\nabla \log p(x)$ (top) and learned Bohm potential (bottom) \ref{['eq:bohm']} for the Swiss roll dataset. The Bohm potential peaks at the data points and drops for points out of distribution ($Q(x)<-5 = -5$ is applied for visualization purposes).
• Figure 2: Visualization of Gaussian propagation: (a) 1d examples for different values of $\beta$ with relative total internal potential energy (b); (c) 2d examples for different values of $\beta$.
• Figure 3: Evolution of a GMM for the moon-to-swiss role dataset with 500 Gaussians.
• Figure 4: Pairs of original images (top row) and corresponding de-aged pairs (bottom row).
• Figure 5: Learned Gaussian evolution dynamics in the S-tunnel (top) and the U-tunnel environment (bottom). The populations at $t = 0,1$ are Gaussian distributions. The RRT* algorithm is used to construct a tree (shown in gray), from which an initial path (dashed blue line) is generated. The solid black line represents the optimal trajectory of the Gaussian mean $\boldsymbol{\mu}(t)$.

Theorems & Definitions (10)

• Definition 1
• Proposition 2
• Proposition 3
• Theorem 4
• Proposition 5
• proof
• Proposition 6
• proof
• Theorem 7
• proof