Wavefunction Flows: Efficient Quantum Simulation of Continuous Flow Models
David Layden, Ryan Sweke, Vojtěch Havlíček, Anirban Chowdhury, Kirill Neklyudov
TL;DR
This work establishes a direct link between continuous-flow models in ML and quantum dynamics by showing that the continuity equation governing flow models maps to a Schrödinger-type equation with a continuity Hamiltonian, enabling a quantum procedure (the wavefunction flow) to prepare qsamples for distributions learned by flow models. It then develops a digital quantum simulation framework—based on Fourier collocation—for space-time discretization of the continuity Hamiltonian, providing explicit error and resource bounds and a practical product-form time-evolution scheme. The results yield a quantum algorithm to prepare approximate qsamples for a wide class of continuous distributions and enable quantum-accelerated inference tasks, such as mean estimation, under mild regularity conditions. This work thus bridges ML-based distribution learning and quantum computation, with implications for both quantum complexity (which qsamples are efficiently preparable) and ML (leveraging quantum access for continuous distribution problems).
Abstract
Flow models are a cornerstone of modern machine learning. They are generative models that progressively transform probability distributions according to learned dynamics. Specifically, they learn a continuous-time Markov process that efficiently maps samples from a simple source distribution into samples from a complex target distribution. We show that these models are naturally related to the Schrödinger equation, for an unusual Hamiltonian on continuous variables. Moreover, we prove that the dynamics generated by this Hamiltonian can be efficiently simulated on a quantum computer. Together, these results give a quantum algorithm for preparing coherent encodings (a.k.a., qsamples) for a vast family of probability distributions--namely, those expressible by flow models--by reducing the task to an existing classical learning problem, plus Hamiltonian simulation. For statistical problems defined by flow models, such as mean estimation and property testing, this enables the use of quantum algorithms tailored to qsamples, which may offer advantages over classical algorithms based only on samples from a flow model. More broadly, these results reveal a close connection between state-of-the-art machine learning models, such as flow matching and diffusion models, and one of the main expected capabilities of quantum computers: simulating quantum dynamics.