  F. Pastawski and J. Preskill discussed error correction of quantum annealing
(QA) based on a parity-encoded spin system, known as the
Sourlas-Lechner-Hauke-Zoller (SLHZ) system. They pointed out that the SLHZ
system is closely related to a classical low-density parity-check (LDPC) code
and demonstrated its error-correcting capability through a belief propagation
(BP) algorithm assuming independent random spin-flip errors. In contrast,
Ablash et al. suggested that the SLHZ system does not receive the benefits of
post-readout decoding. The reason is that independent random spin-flips are not
the most relevant error arising from sampling excited states during the
annealing process, whether in closed or open system cases. In this work, we
revisit this issue: we propose a very simple decoding algorithm to eliminate
errors in the readout of SLHZ systems and show experimental evidence suggesting
that SLHZ system exhibits error-correcting capability in decoding annealing
readouts. Our new algorithm can be thought of as a bit-flipping algorithm for
LDPC codes. Assuming an independent and identical noise model, we found that
the performance of our algorithm is comparable to that of the BP algorithm. The
error correcting-capability for the sampled readouts was investigated using
Monte Carlo calculations that simulate the final time distribution of QA. The
results show that the algorithm successfully eliminates errors in the sampled
readouts under conditions where error-free state or even code state is not
sampled at all. Our simulation suggests that decoding of annealing readouts
will be successful if the correctable states can be sampled by annealing, and
annealing can be considered to play a role as a pre-process of the classical
decoding process. This knowledge will be useful for designing and developing
practical QA based on the SLHZ system in the near future.
