Toward reliability in the NISQ era: Robust interval guarantee for quantum measurements on approximate states

Near-term quantum computation holds potential across multiple application domains. However, imperfect preparation and evolution of states due to algorithmic and experimental shortcomings, characteristic in the near-term implementation, would typically result in measurement outcomes deviating from the ideal setting. It is thus crucial for any near-term application to quantify and bound these output errors. We address this need by deriving robustness intervals which are guaranteed to contain the output in the ideal setting. The first type of interval is based on formulating robustness bounds as semidefinite programs, and uses only the first moment and the fidelity to the ideal state. Furthermore, we consider higher statistical moments of the observable and generalize bounds for pure states based on the non-negativity of Gram matrices to mixed states, thus enabling their applicability in the NISQ era where noisy scenarios are prevalent. Finally, we demonstrate our results in the context of the variational quantum eigensolver (VQE) on noisy and noiseless simulations.