Abelian multi-cycle codes for single-shot error correction

Abstract

We construct a family of quantum low-density parity-check codes locally equivalent to higher-dimensional quantum hypergraph-product (QHP) codes. Similarly to QHP codes, the proposed codes have highly redundant sets of low-weight stabilizer generators, which improves decoding accuracy in a fault-tolerant regime and gives them single-shot properties. The advantage of the new construction is that it gives shorter codes. We derive simple expressions for the dimension of the proposed codes in two important special cases, give bounds on the distances, and explicitly construct some relatively short codes. Circuit simulations for codes locally equivalent to 4-dimensional toric codes show a (pseudo)threshold close to 1.1%, better than for toric or surface codes with a similar noise model.

Figures (2)

• Figure 1: Logical error rate as a function of the circuit noise parameter $p$, using the syndrome-addressing scheme " 1212" and $Z$-basis final measurements. For faster simulations, only $Z$-measurement detector events have been included. Results for short even-distance codes $[[42,6,4]]$, $[66,6,6]]$, and $[[96,6,8]]$ (see Tab. \ref{['tab:toric-2222']}) are shown. Inset shows the success probability $1-p_L$ as a function of $p$ in the vicinity of the crossing point(s) at $p\approx 1.1\%$
• Figure 2: Logical error rates computed with full detector error models (both $X$ and $Z$ detector events) for the same codes as in Fig. \ref{['fig:threshold']}, plotted as a function of the decoding window size $T$. Shading indicates the statistical error at one standard deviation. Here $T=T_{\rm max}=9$, the total number of measurement rounds in our circuits, corresponds to (most accurate) full-block decoding, while $T<T_{\rm max}$ are the number of rounds used in SW decoding protocols. The data labeled as " 1111", " 1212", and " 1234" correspond to different syndrome measurement cycles as explained in the text. At larger $T$, syndrome measurement errors are less important, and the " 1111" cycle gives lower error rates due to shorter measurement cycle. In contrast, with single- ($T=1$) and two-shot ($T=2$) decoding, the error rates are lower with the other two cycles which better utilize the syndrome redundancy. In comparison, in the "1111" measurement cycle, most of the redundant checks are discarded. The difference is substantial for the code $[[96,6,8]]$ with a larger minimal distance $d=8$, see a discussion in Sec. \ref{['sec:disc-sim']}.

Theorems & Definitions (2)

• Lemma 6
• proof