One-Shot any Code

Abstract

We first provide a simple proof showing that any single-shot code is self-correcting. We then show that any $[[n,k]]$ CSS quantum low-density parity-check (QLDPC) code can be transformed into an $[[nm,k]]$ CSS QLDPC code with single-shot (SS) quantum error correction and an efficient local decoder which runs in $O(\log m)$ parallel time. Below a constant threshold $p_{\rm RG}>0$ for joint local stochastic physical and measurement noise, the logical failure probability over $T$ correction rounds is bounded by \begin{equation*} O(T n)\exp[-Ω(m^α)] \end{equation*} for a constant $α>0$; we say that the \textit{threshold} is $p_{\rm RG}$ and \textit{error suppression} is $Ω(m^α)$. Moreover, if the input code is equipped with a single-shot decoder with threshold $p_c>0$ and error suppression $Ω(n^β)$, then the output code \textit{enhances} the error suppression to $Ω(m^α n^β)$, while maintaining the decoder-independent threshold $p_{\rm RG}$, and thus \textit{enhanced} if $p_c < p_{\rm RG}$.