  Answering complex queries on incomplete knowledge graphs is a challenging
task where a model needs to answer complex logical queries in the presence of
missing knowledge. Prior work in the literature has proposed to address this
problem by designing architectures trained end-to-end for the complex query
answering task with a reasoning process that is hard to interpret while
requiring data and resource-intensive training. Other lines of research have
proposed re-using simple neural link predictors to answer complex queries,
reducing the amount of training data by orders of magnitude while providing
interpretable answers. The neural link predictor used in such approaches is not
explicitly optimised for the complex query answering task, implying that its
scores are not calibrated to interact together. We propose to address these
problems via CQD$^{\mathcal{A}}$, a parameter-efficient score \emph{adaptation}
model optimised to re-calibrate neural link prediction scores for the complex
query answering task. While the neural link predictor is frozen, the adaptation
component -- which only increases the number of model parameters by $0.03\%$ --
is trained on the downstream complex query answering task. Furthermore, the
calibration component enables us to support reasoning over queries that include
atomic negations, which was previously impossible with link predictors. In our
experiments, CQD$^{\mathcal{A}}$ produces significantly more accurate results
than current state-of-the-art methods, improving from $34.4$ to $35.1$ Mean
Reciprocal Rank values averaged across all datasets and query types while using
$\leq 30\%$ of the available training query types. We further show that
CQD$^{\mathcal{A}}$ is data-efficient, achieving competitive results with only
$1\%$ of the training complex queries, and robust in out-of-domain evaluations.
