Finding the disjointness of stabilizer codes is NP-complete

TL;DR

This paper proves that determining the disjointness of stabilizer codes, a key factor in quantum fault-tolerance, is NP-complete, highlighting the computational difficulty in designing certain quantum gates.

Contribution

It establishes the NP-completeness of calculating the disjointness for stabilizer codes and provides bounds and numerical methods for specific code families.

Findings

Disjointness calculation is NP-complete.

Bounds provided for CSS, concatenated, and hypergraph codes.

Numerical methods can rule out certain transversal gates.

Abstract

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.