  Although near-term quantum devices have no comprehensive solution for
correcting errors, numerous techniques have been proposed for achieving
practical value. Two works have recently introduced the very promising Error
Suppression by Derangements (ESD) and Virtual Distillation (VD) techniques. The
approach exponentially suppresses errors and ultimately allows one to measure
expectation values in the pure state as the dominant eigenvector of the noisy
quantum state. Interestingly this dominant eigenvector is, however, different
than the ideal computational state and it is the aim of the present work to
comprehensively explore the following fundamental question: how significantly
different are these two pure states? The motivation for this work is two-fold.
First, comprehensively understanding the effect of this coherent mismatch is of
fundamental importance for the successful exploitation of noisy quantum
devices. As such, the present work rigorously establishes that in practically
relevant scenarios the coherent mismatch is exponentially less severe than the
incoherent decay of the fidelity -- where the latter can be suppressed
exponentially via the ESD/VD technique. Second, the above question is closely
related to central problems in mathematics, such as bounding eigenvalues of a
sum of two matrices (Weyl inequalities) -- solving of which was a major
breakthrough. The present work can be viewed as a first step towards extending
the Weyl inequalities to eigenvectors of a sum of two matrices -- and
completely resolves this problem for the special case of the considered density
matrices.
