Quantum physics informed neural networks for multi-variable partial differential equations

TL;DR

This work presents Quantum Physics-Informed Neural Networks (QPINNs) implemented on continuous-variable quantum computing to solve multi-variable PDEs. It introduces a novel CV neural network architecture designed to compute higher-order derivatives without nested automatic differentiation, enabling solving PDEs such as the Poisson and heat equations. The paper demonstrates Poisson and heat equation solutions in both ideal and noisy photonic simulations, and provides a realistic noise model based on the X8 hardware, along with preliminary error mitigation strategies. The results show competitive accuracy against classical PINNs under comparable parameter counts, highlighting the potential of CV-based QPINNs for scalable quantum-enhanced physics-informed learning and future applications to many-body and sensing-enabled scenarios.

Abstract

Quantum Physics-Informed Neural Networks (QPINNs) integrate quantum computing and machine learning to impose physical biases on the output of a quantum neural network, aiming to either solve or discover differential equations. The approach has recently been implemented on both the gate model and continuous variable quantum computing architecture, where it has been demonstrated capable of solving ordinary differential equations. Here, we aim to extend the method to effectively address a wider range of equations, such as the Poisson equation and the heat equation. To achieve this goal, we introduce an architecture specifically designed to compute second-order (and higher-order) derivatives without relying on nested automatic differentiation methods. This approach mitigates the unwanted side effects associated with nested gradients in simulations, paving the way for more efficient and accurate implementations. By leveraging such an approach, the quantum circuit addresses partial differential equations, a challenge not yet tackled using this approach on continuous-variable quantum computers. As a proof-of-concept, we solve a one-dimensional instance of the heat equation, demonstrating its effectiveness in handling PDEs, both in an ideal and a noisy regime. We report our experiment on a photonic hardware to address a realistic noise scenario for our simulations. Such a framework paves the way for further developments in continuous-variable quantum computing and underscores its potential contributions to advancing quantum machine learning.

Figures (13)

• Figure 1: (a) Quantum circuit building block for CVQC-based quantum neural network strawberryfields2024quantum. Here $\hat{U}$ represents general n-mode interferometers, $\hat{S}$ represents single-mode squeezing gates, $\hat{D}$ represents single-mode displacement gate and $\hat{\Phi}$ represents any non-gaussian gate (b) & (c) Quantum Neural Network Layer of for a 1 and 2 qumode set-up respectively. Specifically the 2 modes interferometer is implemented through a Beam Splitter ($BS$) and a phase shifter ($R$), the squeezing ($S$) and displacement ($D$) are single mode and the non-gaussian gate is implement through a Kerr gate ($K$). The 1 qumode configuration correspond to a 2 node fully connected configuration in a classical network but with a higher number of trainable parameters, 5 instead of 6, while the 2 qumode configuration corresponds to a 4 nodes fully connected structure with 12 trainable parameters instead of 20.
• Figure 2: QPINN overall structure. The encoding is implemented by a set of displacement gates $\hat{D(\boldsymbol{x})}$ while output is obtain through the homodyne detection of the position quadrature of the final state $\hat{X}$. A classical computer then calculates the loss function a carries optimization by updating the weights of the parametrized QNN layers.
• Figure 3: (a) Schematic network setup used to solve the 1D Poisson equation. The input $x$ is encoded as qumode displacement. The network applies linear and non-linear transformation in the form of gaussian ($BS$, $R$, $S$, and $D$) and non-gaussian ($K$) gates to achieve the desired output. The first output of the QNN is measured and used to estimate the implicit function $u(x)$. The second one is used to estimate the first derivative of the implicit function $u'(x)$. (b) Classical equivalent of the actual QNN used to solve the 1D Poisson equation. The layer in (a) corresponds to a 4 nodes layer in a classical network.
• Figure 4: Training progress to solve the case of ODE represented by the 1D Poisson equation without gradient nesting while being able to implement higher order derivation. Top: Evolution of the prediction of the QPINN after 50, 100, 150 and 200 epochs of training. Bottom: the best result obtained by the method corresponding to RMSE $= 1.09\times 10^{-4}$
• Figure 5: Different components of the loss function during the training of the neural network for the solution of the 1D Poisson equation. A black dot is used to indicate the epoch with the lowest loss value. The values are plotted every 20 epochs to make the lines clearer. The optimization is able to achieve at its best a lowest loss given by the RMSE $= 1.09\times 10^{-4}$ as measured in Eq. \ref{['eq:Poisson_NMSE']}. Similarly to what happens with some other machine learning methods, the method can show instabilities, requiring to memorize the best network weights during the training when performances are improved.