Towers of generalized divisible quantum codes

A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is 2^ν for a positive integer ν, then one can construct a Calderbank-Shor-Steane (CSS) code, where X-stabilizer space is the divisible classical code, that admits a transversal gate in the νth level of Clifford hierarchy. We consider a generalization of the divisibility by allowing a coefficient vector of odd integers with which every code word has zero dot product modulo the divisor. In this generalized sense, we construct a CSS code with divisor 2^ν+1 and code distance d from any CSS code of code distance d and divisor 2^ν where the transversal X is a nontrivial logical operator. The encoding rate of the new code is approximately d times smaller than that of the old code. In particular, for large d and ν≥2, our construction yields a CSS code of parameters [[O(d^ν-1),Ω(d),d]] admitting a transversal gate at the νth level of Clifford hierarchy. For our construction we introduce a conversion from magic state distillation protocols based on Clifford measurements to those based on codes with transversal T gates. Our tower contains, as a subclass, generalized triply even CSS codes that have appeared in so-called gauge fixing or code switching methods.