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：经典-量子混合算法近年来引起了广泛关注，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数。Lubasch等人在2019年的论文中通过使用一种新型变分量子算法（VQA）取得了进展，该算法通过求解由期望值与变分参数叠加表示的代价函数的基态，为Schrödinger方程和无黏Burgers方程的解提供了读数。本文进一步分析了VQA能够可靠产生其他偏微分方程解的计算前景，这些解与先前通过经典方式实现的解具有可比性，且特征为无噪声量子模拟。为确定该算法在其它初值问题中可处理的非线性范围，我们研究了多个偏微分方程，首先从Navier-Stokes方程入手，进而涵盖电磁学、引力及波传播等物理现象中涉及的其他方程，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对量子电路读数所得数值近似解进行优化所需的程序，我们在补充材料中提供了各偏微分方程对应的代价函数，并在其后给出了基于数百个ZGR-QFT变分拟设生成的模拟结果。