Towers of generalized divisible quantum codes
Jeongwan Haah · Physical Review A · 2018
A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is ${2}^{\ensuremath{ u}}$ for a positive integer $\ensuremath{ u}$, 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 $\ensuremath{ u}\mathrm{th}$ 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 ${2}^{\ensuremath{ u}+1}$ and code distance $d$ from any CSS code of code distance $d$ and divisor ${2}^{\ensuremath{ u}}$ 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 $\ensuremath{ u}\ensuremath{\ge}2$, our construction yields a CSS code of parameters $[[O({d}^{\ensuremath{ u}\ensuremath{-}1}),\mathrm{\ensuremath{\Omega}}(d),d]]$ admitting a transversal gate at the $\ensuremath{ u}\mathrm{th}$ 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.