Efficient ground-state preparation and quantum advantage for the proposed Hamiltonian constructions
Determine whether the homogeneous and projected right-hand-side effective Hamiltonians for differential equations admit efficient quantum ground-state preparation and yield quantum advantage beyond the noise-free statevector demonstrations.
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Several questions remain open. The present noise-free statevector results demonstrate that the Hamiltonians can be constructed and can encode accurate solutions, but they do not establish efficient quantum ground-state preparation or quantum advantage. The overall cost depends on operator decomposition, state preparation, conditioning, the normalised spectral gap, target precision, and solution readout. Variational implementations additionally face limited expressivity, local minima, barren plateaus, sampling noise, and hardware noise. Such an investigation is still required.
The treatment of nonlinearities also requires further study. The repeated-product ansatz replaces the unphysical degeneracy of the unrestricted search with a constrained, non-convex optimisation problem whose convergence is not guaranteed.
While we employ strategies to speed up the variational optimisation by using a multigrid approach, the optimisation process currently remains a practical bottleneck of unclear complexity.
Most results nevertheless concern equation mappings or small-scale proof-of-principle demonstrations. End-to-end quantum speedup for general solid mechanics problems has not been established.