Unsupervised Random Quantum Networks for PDEs
This work addresses solving PDEs, which are fundamental in science, by proposing a quantum computing method, but it appears incremental as it adapts existing techniques to a new domain.
The authors tackled solving partial differential equations (PDEs) by using parameterized random quantum circuits as trial solutions, adapting classical physics-informed neural network techniques to a quantum setting, and showed numerically that this approach can outperform traditional quantum and random classical networks.
Classical Physics-informed neural networks (PINNs) approximate solutions to PDEs with the help of deep neural networks trained to satisfy the differential operator and the relevant boundary conditions. We revisit this idea in the quantum computing realm, using parameterised random quantum circuits as trial solutions. We further adapt recent PINN-based techniques to our quantum setting, in particular Gaussian smoothing. Our analysis concentrates on the Poisson, the Heat and the Hamilton-Jacobi-Bellman equations, which are ubiquitous in most areas of science. On the theoretical side, we develop a complexity analysis of this approach, and show numerically that random quantum networks can outperform more traditional quantum networks as well as random classical networks.