Abstract Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.
Abstract Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.
We introduce and analyse an efficient decoder for quantum Tanner codes that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight O(nlogn) . We also work on the link between quantum Tanner codes and the lifted product codes of Panteleev and Kalachev and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.
We introduce and analyse an efficient decoder for quantum Tanner codes that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight . We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.
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(logm) parallel time. Below a constant threshold pRG>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 pRG and \textit{error suppression} is Ω(mα). Moreover, if the input code is equipped with a single-shot decoder with threshold pc>0 and error suppression Ω(nβ), then the output code \textit{enhances} the error suppression to Ω(mαnβ), while maintaining the decoder-independent threshold pRG, and thus \textit{enhanced} if pc<pRG.
We introduce and analyse an efficient decoder for the quantum Tanner codes of that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight O(nlogn). We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.
Recent developments have shown the existence of quantum low-density parity check (qLDPC) codes with constant rate and linear distance. A natural question concerns the efficient decodability of these codes. In this paper, we present a linear time decoder for the recent quantum Tanner codes construction of asymptotically good qLDPC codes, which can correct all errors of weight up to a constant fraction of the blocklength. Our decoder is an iterative algorithm which searches for corrections within constant-sized regions. At each step, the corrections are found by reducing a locally defined and efficiently computable cost function which serves as a proxy for the weight of the remaining error.
Quantum error correction (QEC) with single-shot decoding enables reduction of errors after every single round of noisy stabilizer measurement, easing the time-overhead requirements for fault tolerance. Notably, several classes of quantum low-density-parity-check (qLDPC) codes are known which facilitate single-shot decoding, potentially giving them an additional overhead advantage. However, the perceived advantage of single-shot decoding is limited because it can significantly degrade the effective code distance. This degradation may be compensated for by using a much larger code size to achieve the desired target logical error rate, at the cost of increasing the amount of syndrome information to be processed, as well as, increasing complexity of logical operations. Alternatively, in this work we study sliding-window decoding, which corrects errors from previous syndrome measurement rounds while leaving the most recent errors for future correction. We observe that sliding-window decoding significantly improves the logical memory lifetime and hence the effective distance compared to single-shot decoding on hypergraph-product codes and lifted-product codes. Remarkably, we find that this improvement may not cost a larger decoding complexity. Thus, the sliding-window strategy can be more desirable for fast and accurate decoding for fault-tolerant quantum computing with qLDPC codes.