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Quantum Neural Ordinary and Partial Differential Equations

Yu Cao, Shi Jin, Nana Liu

TL;DR

The paper introduces Quantum Neural ODEs and PDEs (QNODEs/QNPDEs) as a continuous-time framework in which quantum states $\rho(t,\theta)$ evolve under $\partial_t \rho = -i [H(t,\theta), \rho]$ with a Hamiltonian decomposed as $H(t,\theta)=\sum_{k=1}^K f_k(t,\theta) H_k$, paired with a task-specific loss $\mathcal{L}_{\theta}$. It develops two adjoint-based gradient estimation algorithms—one with partial time discretisation and one fully continuous in time—allowing efficient gradient computation for learning quantum dynamics and mapped classical dynamics, with resource estimates and landscape analysis. The framework supports learning in closed and open quantum systems, Hamiltonian learning from states or observables, and learning of ODEs/PDEs, offering a pathway to continuous-depth quantum neural networks and novel quantum control architectures. Numerical demonstrations on quantum state preparation and Hamiltonian-learning tasks illustrate practical viability and highlight the potential impact on scalable quantum ML and control applications.

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 Schrödinger-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.

Quantum Neural Ordinary and Partial Differential Equations

TL;DR

The paper introduces Quantum Neural ODEs and PDEs (QNODEs/QNPDEs) as a continuous-time framework in which quantum states evolve under with a Hamiltonian decomposed as , paired with a task-specific loss . It develops two adjoint-based gradient estimation algorithms—one with partial time discretisation and one fully continuous in time—allowing efficient gradient computation for learning quantum dynamics and mapped classical dynamics, with resource estimates and landscape analysis. The framework supports learning in closed and open quantum systems, Hamiltonian learning from states or observables, and learning of ODEs/PDEs, offering a pathway to continuous-depth quantum neural networks and novel quantum control architectures. Numerical demonstrations on quantum state preparation and Hamiltonian-learning tasks illustrate practical viability and highlight the potential impact on scalable quantum ML and control applications.

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 Schrödinger-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.

Paper Structure

This paper contains 43 sections, 15 theorems, 171 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Basic notation in quantum information processing
  4. Classical neural ODE and PDE
  5. Quantum neural ODE and PDE: scope and applications
  6. General concepts
  7. Quantum neural ODEs and PDEs
  8. Learning in closed quantum systems
  9. Learning in open quantum systems
  10. Learning ODEs and PDEs
  11. Learning non-autonomous systems
  12. Framework for new quantum neural network architectures
  13. Gradient estimation algorithm for QNODE and QNPDE
  14. Revisiting the adjoint equation: application to QNODEs and QNPDEs
  15. Quantum gradient algorithm: partial time discretisation
  16. ...and 28 more sections

Key Result

Lemma 2

The gradient of the loss function $\mathcal{L}_{\theta} = \mathcal{L}(\rho(T,\theta))$ with respect to the parameter $\theta_m$ where $m = 1, \cdots, M$ can be written as when the state $\rho(s, \theta)$ evolves according to Eq. eq:schrdensity, and $a(s, \theta)$ evolves from a terminal condition with the same unitary dynamics as Eq. eq:schrdensity

Figures (5)

  • Figure 1: The schematic quantum circuit for the gradient computation scheme with respect to parameter $\theta_m$, $1\le m\le M$. We remark that if we parameterise $H(t,\theta)$ as in Eq. \ref{['eqn::H_ansatz']}, one does not need to perform quantum measurement for each $\frac{\partial H(s,\theta)}{\partial \theta_m}$ where $1\le m\le M$; see Section \ref{['sec::cost']}.
  • Figure 2: The error $1-\lvert \langle + | U(T,\theta) |0\rangle\rvert^2$ versus training iterations for different numbers of shots. Shots refer to the number of repeated measurements to estimate the observables for each $s$ in Theorem \ref{['thm::main']}. Each curve represents the median error from five independent experiments; the shaded region (with the same color) indicates the corresponding minimum and maximum error during training. For the case of exact quantum measurement (black curve), there is no fluctuation in optimisation.
  • Figure 3: Training results for Section \ref{['eg::hl']} for different time-independent Hamiltonians using the loss function in Eq. \ref{['eqn::loss_eg2']}. Each curve represents the median error from five independent training runs, and the shaded region indicates the maximum and minimum error. The y-axis range is truncated for better visualization.
  • Figure 4: Training results for Section \ref{['eg::hl']} for two-site time-dependent quantum Ising chain (td-Ising) using the loss function in Eq. \ref{['eqn::loss_eg2']}. (a) Each curve represents the median error from five independent training runs, and the shaded region indicates the maximum and minimum error. (b) We show a typical training result of time-schedule using $10^3$ shots per quantum estimate.
  • Figure 5: Training results for Section \ref{['section::hl_observable']} using the loss function in Eq. \ref{['eqn::loss_observable']}: (a) for different time-independent Hamiltonians; (b) for two-site time-dependent quantum Ising chain (td-Ising). Each curve represents the median error from five independent training runs, and the shaded region indicates the maximum and minimum error.

Theorems & Definitions (33)

  • Definition 1
  • Lemma 2
  • proof
  • Theorem 3: Quantum gradient algorithm: partial time discretisation
  • proof
  • Corollary 4
  • Theorem 5: Quantum gradient algorithm: fully continuous time
  • proof
  • Lemma 7: Hessian for QNODEs for state learning
  • Definition 8
  • ...and 23 more