Finding the disjointness of stabilizer codes is NP-complete

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