Avoiding symmetry roadblocks and minimizing the measure-ment overhead of adaptive variational quantum eigensolvers

Quantum simulation of strongly corre-lated systems is potentially the most feasi-ble useful application of near-term quan-tum computers [1]. Minimizing quan-tum computational resources is crucial to achieving this goal. A promising class of algorithms for this purpose consists of vari-ational quantum eigensolvers (VQEs) [2- 7]. Among these, problem-tailored ver-sions such as ADAPT-VQE [8, 9] that build variational ansatze step by step from a predefined operator pool perform par-ticularly well in terms of circuit depths and variational parameter counts. How-ever, this improved performance comes at the expense of an additional measurement overhead compared to standard VQEs. Here, we show that this overhead can be reduced to an amount that grows only lin-early with the number n of qubits, instead of quartically as in the original ADAPT-VQE. We do this by proving that operator pools of size 2n - 2 can represent any state in Hilbert space if chosen appropriately. We prove that this is the minimal size of such "complete" pools, discuss their al-gebraic properties, and present necessary and sufficient conditions for their com-pleteness that allow us to find such pools efficiently. We further show that, if the simulated problem possesses symmetries, then complete pools can fail to yield con-vergent results, unless the pool is chosen to obey certain symmetry rules. We demon-strate the performance of such symmetry -adapted complete pools by using them in classical simulations of ADAPT-VQE for several strongly correlated molecules. Our findings are relevant for any VQE that uses an ansatz based on Pauli strings.