Beyond the variational quantum eigensolver: Approximating solutions of 15 PDEs with a variational quantum algorithm in polynomial time

Quantum algorithms have received much attention, not only for being able to potentially replicate computations that classical algorithms have provided, but also for being able to achieve such results with exponential speedup. To further build upon the ensemble of solution approximations to 8 PDEs that were obtained from an adaptation of a recent variational quantum algorithm (VQA) due to Lubasch et al in 2019, solution approximations for PDEs ranging from the Navier-Stokes to Hunter-Saxton equations were provided in the September 2022 posting, arXiv 2209.07714, in which it was demonstrated that the VQA is capable of simulating nonlinearities of several PDEs. Specifically, with modification to the budget and other parameters that are required by either stochastic, deterministic, gradient-based, or constrained, optimizers for approximating the ground state of a cost function, we illustrate that the VQA can produce solution approximations to several variants of PDEs exhibiting oscillatory wave behavior, including generalized Camassa-Holm, KdV-Burgers, non-homogeneous KdV, generalized KdV, KdV, super KdV, and Benney-Luke equations. From such a collection of a PDEs, solutions can either exhibit periodic peakon, cuspon, or combinations of such behaviors. Across the aforementioned list of 7 additional PDEs, in polynomial time as with previous analysis of VQA performance from the 8 PDEs characterized in arXiv 2209.07714, the VQA is able to obtain solution approximations with varying fidelity also in polynomial time, exhibiting similarities with the dynamics of quantum states for approximating Camassa-Holm solutions, with the total number of time steps of evolution varying from 500 to approximately 20,000. Across several initial conditions of the ansatz which correspond to the ZGR-QFT parameters that are reliant upon several iterations of computing Fourier coefficients required for initializing time-evolution, measurements extracted from cost functions for the KdV-Burgers, KdV, and generalized Camassa-Holm equations in particular exhibit good performance with the open-source Nevergrad optimizer provided by Facebook research.