Addressing Stopping Failures for Small Set Flip Decoding of Hypergraph Product Codes

Understand

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, Z\'emor 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.