One-Shot any Code

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