How to Correct Small Quantum Errors

Lev Stambler, Anirudh Krishna, Michael E. Beverland

For a quantum error correcting code to be used in practice, it needs to be equipped with an efficient decoding algorithm, which identifies corrections given the observed syndrome of errors.Hypergraph product codes are a promising family of constant-rate quantum LDPC codes that have a linear-time decoding algorithm called Small-Set-Flip ($\texttt{SSF}$) (Leverrier, Tillich, Zémor FOCS 2015). The algorithm proceeds by iteratively applying small corrections which reduce the syndrome weight. Together, these small corrections can provably correct large errors for sufficiently large codes with sufficiently large (but constant) stabilizer weight. However, this guarantee does not hold for small codes with low stabilizer weight. In this case, $\texttt{SSF}$ can terminate with stopping failures, meaning it encounters an error for which it is unable to identify a small correction. We find that the structure of errors that cause stopping failures have a simple form for sufficiently small qubit failure rates. We propose a new decoding algorithm called the Projection-Along-a-Line ($\texttt{PAL}$) decoder to supplement $\texttt{SSF}$ after stopping failures. Using $\texttt{SSF}+\texttt{PAL}$ as a combined decoder, we find an order-of-magnitude improvement in the logical error rate.

Andrew M. Steane

Alan Robertson, Christopher Granade, Stephen D. Bartlett +1 more

Error correction is essential to quantum information processing, but the demanding performance requirements for $u\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}f\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}l$ error correction make it a difficult proposition. This study shows that incorporating prior knowledge of physical error models can dramatically improve the efficacy of quantum error correction, in a small code. This progress significantly expands the range in which quantum error correction can be usefully applied, facilitating interesting experiments that use accurate device models to protect quantum memories.

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Abhishek Parakh

This paper discusses correcting small rotation errors in the three-stage quantum key distribution (QKD) protocol using only two logical qubits thereby increasing communication efficiency. Further, we can correct errors before they accumulate at every stage ensuring that the errors remain small. Three-stage QKD is tolerant of multi-photon pulses for key exchange and therefore more applicable in real world scenarios. However, given that it requires back and forth communication, any channel errors that occur accumulate. We discuss the proposed error correction protocol in detail and workout each step. We show how the two transacting parties can correct small rotation errors using just two logical qubits in a majority of cases. We compute the efficiency increase and bound on the magnitude of rotation errors that can be tolerated.

Rajendra Kumar Bera