Identifying challenges towards practical quantum advantage through resource estimation: the measurement roadblock in the variational quantum eigensolver

Recent advances in Noisy Intermediate-Scale Quantum (NISQ) devices have brought much attention to the potential of the Variational Quantum Eigensolver (VQE) and related techniques to provide practical quantum advantage in computational chemistry. However, it is not yet clear whether such algorithms, even in the absence of device error, could achieve quantum advantage for systems of practical interest and how large such an advantage might be. To address these questions, we have performed an exhaustive set of benchmarks to estimate number of qubits and number of measurements required to compute the combustion energies of small organic molecules to within chemical accuracy using VQE as well as state-of-the-art classical algorithms. We consider several key modifications to VQE, including the use of Frozen Natural Orbitals, various Hamiltonian decomposition techniques, and the application of fermionic marginal constraints. Our results indicate that although Frozen Natural Orbitals and low-rank factorizations of the Hamiltonian significantly reduce the qubit and measurement requirements, these techniques are not sufficient to achieve practical quantum computational advantage in the calculation of organic molecule combustion energies. This suggests that new approaches to estimation leveraging quantum coherence, such as Bayesian amplitude estimation [arxiv:2006.09350, arxiv:2006.09349], may be required in order to achieve practical quantum advantage with near-term devices. Our work also highlights the crucial role that resource and performance assessments of quantum algorithms play in identifying quantum advantage and guiding quantum algorithm design.Comment: 27 pages, 18 figure