Towers of generalized divisible quantum codes

TL;DR

This paper introduces a generalized method for constructing quantum error-correcting codes with transversal gates at higher levels of the Clifford hierarchy, expanding the capabilities of fault-tolerant quantum computation.

Contribution

It generalizes divisibility conditions for CSS codes, enabling the construction of codes with transversal gates at higher Clifford hierarchy levels and improved parameters.

Findings

Constructed CSS codes with divisor $2^{ u+1}$ from codes with divisor $2^ u$.

Achieved codes with parameters $[[O(d^{ u-1}), \, ext{Omega}(d),d]]$ for large $d$ and $ u \, \ge 2$.

Introduced a conversion method from Clifford measurement-based to transversal $T$-gate-based magic state distillation.

Abstract

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 $\nu$-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 $[[O(d^{\nu-1}), \Omega(d),d]]$ admitting a transversal gate at the $\nu$-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.