In this paper, we tackle the problem of measuring similarity among graphs that represent real objects with noisy data. To account for noise, we relax the definition of similarity using the maximum weighted co-$k$-plex relaxation method, which allows dissimilarities among graphs up to a predetermined level. We then formulate the problem as a novel quadratic unconstrained binary optimization problem that can be solved by a quantum annealer. The context of our study is molecular similarity where the presence of noise might be due to regular errors in measuring molecular features. We develop a similarity measure and use it to predict the mutagenicity of a molecule. Our results indicate that the relaxed similarity measure, designed to accommodate the regular errors, yields a higher prediction accuracy than the measure that ignores the noise.