Quantifying Nonstabilizerness of Codeword-Stabilized Codes

Abstract

Fault-tolerant quantum computation requires non-Clifford gates, which stabilizer codes cannot supply transversally. Non-stabilizer codes are the natural place to look for them, yet no quantitative theory of the nonstabilizerness (or magic) carried by such a code has existed. We develop one for codeword-stabilized (CWS) codes and show that the key quantity is classical: a code's nonstabilizerness is fixed by how its codewords collide under translation, a question that belongs to additive combinatorics. We show that the most magical codes are exactly the Sidon sets whenever a Sidon set of the required size exists, whose pairwise differences are all distinct. No code carries more than twice its number of logical qubits of nonstabilizerness however large it is physically. Furthermore, the same reduction gives structural and operational results. Nonstabilizerness is unchanged by coset closure, which yields non-stabilizer codes with arbitrarily many logical qubits and constant nonstabilizerness as the number of logical qubits grows. A diagonal transversal gate with $k$ logic qubits that is non-Clifford on $t$ coordinates forces the code's nonstabilizerness to be at most $2(k-t)$; thus the nonstabilizerness also bounds the non-Clifford gates needed to build the code and the cost of classically simulating it. Finally, entire families become exactly computable, and we obtain closed-form values for the Kerdock codes. Together these results turn the search for magic-rich codes and transversal non-Clifford gates into classical counting problems, which can be approached with standard tools from additive combinatorics.