Finding the disjointness of stabilizer codes is NP-complete

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 approximatin...

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Bibliographic Details
Published in:Physical review research 2021-12, Vol.3 (4), p.043192, Article 043192
Main Authors: Bostanci, John, Kubica, Aleksander
Format: Article
Language:English
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Summary: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 Calderbank-Shor-Steane 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.
ISSN:2643-1564
2643-1564
DOI:10.1103/PhysRevResearch.3.043192