Abstract
A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is for a positive integer , then one can construct a Calderbank-Shor-Steane (CSS) code, where -stabilizer space is the divisible classical code, that admits a transversal gate in the 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 and code distance from any CSS code of code distance and divisor where the transversal is a nontrivial logical operator. The encoding rate of the new code is approximately times smaller than that of the old code. In particular, for large and , our construction yields a CSS code of parameters admitting a transversal gate at the 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 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
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