How to really measure operator gradients in ADAPT-VQE

ADAPT-VQE is one of the leading VQE algorithms which circumvents the choice-of-ansatz conundrum by iteratively growing compact and arbitrarily accurate problem-tailored ans\"atze. However, for hardware-efficient operator pools, the gradient-measurement step of the algorithm requires the estimation of $O(N^8)$ observables, which may represent a bottleneck for relevant system sizes on real devices. We present an efficient strategy for measuring the pool gradients based on simultaneously measuring commuting observables. We argue that our approach is relatively robust to shot-noise effects, and show that measuring the pool gradients is in fact only $O(N)$ times as expensive as a naive VQE iteration. Our proposed measurement strategy significantly ameliorates the measurement overhead of ADAPT-VQE and brings us one step closer to practical implementations on real devices.