  The disjointness of a stabilizer code is a quantity used to constrain the
level of the logical Clifford hierarchy attainable by transversal gates and
constant-depth quantum circuits. We show that for any positive integer constant
$c$, the problem of calculating the $c$-disjointness, or even approximating it
to within a constant multiplicative factor, is NP-complete. We provide bounds
on the disjointness for various code families, including the CSS codes,
concatenated codes and hypergraph product codes. We also describe numerical
methods of finding the disjointness, which can be readily used to rule out the
existence of any transversal gate implementing some non-Clifford logical
operation in small stabilizer codes. Our results indicate that finding
fault-tolerant logical gates for generic quantum error-correcting codes is a
computationally challenging task.
