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Classical-quantum hybrid algorithms have recently garnered significantattention, which are characterized by combining quantum and classical computingprotocols to obtain readout from quantum circuits of interest. Recent progressdue to Lubasch et al in a 2019 paper provides readout for solutions to theSchrodinger and Inviscid Burgers equations, by making use of a new variationalquantum algorithm (VQA) which determines the ground state of a cost functionexpressed with a superposition of expectation values and variationalparameters. In the following, we analyze additional computational prospects inwhich the VQA can reliably produce solutions to other PDEs that are comparableto solutions that have been previously realized classically, which arecharacterized with noiseless quantum simulations. To determine the range ofnonlinearities that the algorithm can process for other IVPs, we study severalPDEs, first beginning with the Navier-Stokes equations and progressing to otherequations 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), andHunter-Saxton equations. To formulate optimization routines that the VQAundergoes for numerical approximations of solutions that are obtained asreadout from quantum circuits, cost functions corresponding to each PDE areprovided in the supplementary section after which simulations results fromhundreds of ZGR-QFT ansatzae are generated.

Variational quantum algorithm for measurement extraction from the  Navier-Stokes, Einstein, Maxwell, B-type, Lin-Tsien, Camassa-Holm, DSW, H-S,  KdV-B, non-homogeneous KdV, generalized KdV, KdV, translational KdV, sKdV,  B-L and Airy equations