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ADAPT-VQE stands out as a robust algorithm for constructing compact ans\"atzefor molecular simulation. It enables to significantly reduce the circuit depthwith respect to other methods, such as UCCSD, while achieving higher accuracyand not suffering from so-called barren plateaus that hinder the variationaloptimisation of many hardware-efficient ans\"atze. In its standardimplementation, however, it introduces a considerable measurement overhead inthe form of gradient evaluations trough estimations of many commutatoroperators. In this work, we mitigate this measurement overhead by exploiting arecently introduced method for energy evaluation relying on AdaptiveInformationally complete generalised Measurements (AIM). Besides offering anefficient way to measure the energy itself, Informationally Complete (IC)measurement data can be reused to estimate all the commutators of the operatorsin the operator pool of ADAPT-VQE, using only classically efficientpost-processing. We present the AIM-ADAPT-VQE scheme in detail, and investigateits performance with several H4 Hamiltonians and operator pools. Our numericalsimulations indicate that the measurement data obtained to evaluate the energycan be reused to implement ADAPT-VQE with no additional measurement overheadfor the systems considered here. In addition, we show that, if the energy ismeasured within chemical precision, the CNOT count in the resulting circuits isclose to the ideal one. With scarce measurement data, AIM-ADAPT-VQE stillconverges to the ground state with high probability, albeit with an increasedcircuit depth in some cases.

Mitigating the measurement overhead of ADAPT-VQE with optimised informationally complete generalised measurements