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 $p_{\mathrm{RG}}>0$ 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 $\alpha >0$; we say that the \textit{threshold} is $p_{\mathrm{RG}}$ and \textit{error suppression} is $\Omega (m^{\alpha })$. Moreover, if the input code is equipped with a single-shot decoder with threshold $p_c>0$ and error suppression $\Omega (n^{\beta })$, then the output code \textit{enhances} the error suppression to $\Omega (m^{\alpha }{n}^{\beta })$, while maintaining the decoder-independent threshold $p_{\mathrm{RG}}$, and thus \textit{enhanced} if $p_c<p_{\mathrm{RG}}$.

arXiv commentThe simple proof (single-shot implies self-correction) is only 2 pages, and independent of everything else