Quantum Neural Ordinary and Partial Differential Equations

TL;DR

This paper introduces a unified quantum framework for neural ordinary and partial differential equations, enabling efficient gradient computation and broad applications in quantum and classical dynamics learning.

Contribution

It extends classical differential equations into quantum machine learning, providing algorithms for gradient estimation and resource analysis for quantum neural differential equations.

Findings

Quantum algorithms for gradient computation are more efficient than classical methods.

The formalism applies to diverse problems like quantum state preparation and Hamiltonian learning.

Quantum approaches are advantageous when classical simulation is inefficient.

Abstract

We introduce a unified framework -- Quantum Neural Ordinary and Partial Differential Equations (QNODEs and QNPDEs) -- which extends the continuous-time formalism of classical neural ordinary and partial differential equations into quantum machine learning and quantum control. QNODEs denote the evolution of finite-dimensional quantum systems, whereas QNPDEs denote their infinite-dimensional (continuous-variable) counterparts; both are governed by generalised Schr\"odinger-type Hamiltonian dynamics, coupled with a corresponding loss function. This formalism permits gradient estimation via an adjoint-state method, facilitating efficient learning of quantum dynamics, and other dynamics that can be mapped (relatively easily) to quantum dynamics. Using this method, we present quantum algorithms for computing gradients with and without time discretisation, achieving efficient gradient computation that would otherwise be intractable on classical devices. We provide detailed resource estimates for these algorithms and investigate the local energy landscape for training. The formalism subsumes a wide array of applications, including quantum state preparation, Hamiltonian learning, learning dynamics in open systems, and the learning of both autonomous and non-autonomous classical ODEs and PDEs. In many cases of interest, the Hamiltonian is composed of a relatively small number of local operators, yet the corresponding classical simulation remains inefficient, making quantum approaches advantageous for gradient estimation. This continuous-time perspective can also serve as a blueprint for designing novel quantum neural network architectures, generalising discrete-layered models into continuous-depth models.