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Quantum chemistry and materials is one of the most promising applications ofquantum computing. Yet much work is still to be done in matchingindustry-relevant problems in these areas with quantum algorithms that cansolve them. Most previous efforts have carried out resource estimations forquantum algorithms run on large-scale fault-tolerant architectures, whichinclude the quantum phase estimation algorithm. In contrast, few have assessedthe performance of near-term quantum algorithms, which include the variationalquantum eigensolver (VQE) algorithm. Recently, a large-scale benchmark study[Gonthier et al. 2020] found evidence that the performance of the variationalquantum eigensolver for a set of industry-relevant molecules may be tooinefficient to be of practical use. This motivates the need for developing andassessing methods that improve the efficiency of VQE. In this work, we predictthe runtime of the energy estimation subroutine of VQE when using robustamplitude estimation (RAE) to estimate Pauli expectation values. Underconservative assumptions, our resource estimation predicts that RAE can reducethe runtime over the standard estimation method in VQE by one to two orders ofmagnitude. Despite this improvement, we find that the runtimes are still toolarge to be practical. These findings motivate two complementary effortstowards quantum advantage: 1) the investigation of more efficient near-termmethods for ground state energy estimation and 2) the development of probleminstances that are of industrial value and classically challenging, but bettersuited to quantum computation.

Reducing the cost of energy estimation in the variational quantum eigensolver algorithm with robust amplitude estimation