A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is $2^{\nu }$ for a positive integer $\nu$, 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 $\nu \mathrm{th}$ 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 $2^{\nu +1}$ and code distance $d$ from any CSS code of code distance $d$ and divisor $2^{\nu }$ 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 $\nu \ge 2$, our construction yields a CSS code of parameters $\left[[O\left(d^{\nu -1}\right),\mathrm{\Omega }\left(d\right),d]\right]$ admitting a transversal gate at the $\nu \mathrm{th}$ 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.

• Received 19 October 2017

DOI:https://doi.org/10.1103/PhysRevA.97.042327