Classical-quantum hybrid algorithms have recently garnered significant attention, which are characterized by combining quantum and classical computing protocols to obtain readout from quantum circuits of interest. Recent progress due to Lubasch et al in a 2019 paper provides readout for solutions to the Schrodinger and Inviscid Burgers equations, by making use of a new variational quantum algorithm (VQA) which determines the ground state of a cost function expressed with a superposition of expectation values and variational parameters. In the following, we analyze additional computational prospects in which the VQA can reliably produce solutions to other PDEs that are comparable to solutions that have been previously realized classically, which are characterized with noiseless quantum simulations. To determine the range of nonlinearities that the algorithm can process for other IVPs, we study several PDEs, first beginning with the Navier-Stokes equations and progressing to other equations underlying physical phenomena ranging from electromagnetism, gravitation, and wave propagation, from simulations of the Einstein, Boussniesq-type, Lin-Tsien, Camassa-Holm, Drinfeld-Sokolov-Wilson (DSW), and Hunter-Saxton equations. To formulate optimization routines that the VQA undergoes for numerical approximations of solutions that are obtained as readout from quantum circuits, cost functions corresponding to each PDE are provided in the supplementary section after which simulations results from hundreds of ZGR-QFT ansatzae are generated.

翻译：近年来，经典-量子混合算法受到广泛关注，其特点在于结合量子与经典计算协议，从目标量子电路中获取读出结果。Lubasch等人于2019年论文中的最新进展，通过利用一种新型变分量子算法（VQA）——该算法通过期望值叠加与变分参数构成的代价函数确定其基态——实现了对薛定谔方程和无粘性Burgers方程解的读出。本文进一步分析了该VQA算法在其他偏微分方程求解中的计算潜力：该算法能够可靠地生成与先前经典方法实现解相当的其他偏微分方程数值解，这些解均通过无噪声量子模拟进行表征。为确定该算法对其他初值问题所能处理的非线性范围，我们研究了若干偏微分方程，首先从Navier-Stokes方程入手，逐步扩展到电磁学、引力理论和波传播等物理现象相关的方程，包括Einstein、Boussinesq型、Lin-Tsien、Camassa-Holm、Drinfeld-Sokolov-Wilson（DSW）及Hunter-Saxton方程的模拟。为构建VQA在数值近似求解过程中所经历的优化流程（这些解作为量子电路的读出结果获得），我们在补充材料中给出了各偏微分方程对应的代价函数，并基于数百个ZGR-QFT拟设生成了模拟结果。