Variational quantum chemistry requires gate-error probabilities below the fault-tolerance threshold

Abstract

The variational quantum eigensolver (VQE) is a leading contender for useful quantum advantage in the NISQ era. The interplay between quantum processors and classical optimisers is believed to make the VQE noise resilient. Here, we probe this hypothesis. We use full density-matrix simulations to rank the noise resilience of leading gate-based VQE algorithms in ground-state computations on a range of molecules. We find that, in the presence of noise: (i) ADAPT-VQEs that construct ansatz circuits iteratively outperform VQEs that use "fixed" ansatz circuits; and (ii) ADAPT-VQEs perform better when circuits are constructed from gate-efficient elements rather than physically-motivated ones. Our results show that, for a wide range of molecules, even the best-performing VQE algorithms require gate-error probabilities on the order of $10^{-6}$ to $10^{-4}$ to reach chemical accuracy. This is significantly below the fault-tolerance thresholds of most error-correction protocols. Further, we estimate that the maximum allowed gate-error probability scales inversely with the number of noisy (two-qubit) gates. Our results indicate that useful chemistry calculations with current gate-based VQEs are unlikely to be successful on near-term hardware without error correction.Comment: 17 pages, 8 figure