Exact Solutions to the Quantum Schrödinger Bridge Problem

2025-09-30

AI Summary

This paper investigates the Quantum Schrödinger Bridge Problem (QSBP): a quantum stochastic evolution governed by the complex-valued Schrödinger equation, interpolating between two given probability distributions. Unlike its classical counterpart, QSBP incorporates a nonlocal quantum potential, inherently encoding wavefunction phase and quantum interference. Methodologically, we reformulate QSBP from a Lagrangian variational perspective and integrate dynamic optimal transport with Fokker–Planck and Hamilton–Jacobi equations. We derive, for the first time, an analytical closed-form solution for QSBP between Gaussian distributions, explicitly characterizing how quantum effects modulate covariance matrix dynamics. Leveraging Gaussian mixture models, we design a scalable algorithm. Experiments on single-cell trajectory inference, image generation, molecular conformational transitions, and mean-field games demonstrate superior modeling fidelity and generalization. This work establishes the first rigorous quantum dynamical framework for generative modeling.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• Deriving exact solutions for quantum Schrödinger bridge between Gaussian distributions
• Formulating quantum bridge problem from Lagrangian perspective for generative modeling
• Developing modified algorithm for applications in data evolution and image generation

Innovation

• Lagrangian formulation for quantum Schrödinger bridge problem
• Exact closed-form solutions for Gaussian distributions
• Gaussian mixture model algorithm with quantum effects