  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
semi-definite 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.
