  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.
