Wavefunction Flows: Efficient Quantum Simulation of Continuous Flow Models
This work connects state-of-the-art machine learning models with quantum computing capabilities, potentially offering advantages for statistical problems, though it is incremental in leveraging existing classical learning and quantum simulation methods.
The paper tackles the problem of efficiently simulating continuous flow models on quantum computers by relating them to the Schrödinger equation, proving that the dynamics can be simulated quantumly, enabling quantum algorithms for tasks like mean estimation and property testing using qsamples.
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.