Practical classical error correction for parity-encoded spin systems

Quantum annealing (QA) has emerged as a promising candidate for fast solvers for combinatorial optimization problems (COPs) and has attracted the interest of many researchers. Since COP is logically encoded in the Ising interaction among spins, its realization necessitates a spin system with all-to-all connectivity, presenting technical challenges in the physical implementation of large-scale QA devices. W. Lechner, P. Hauke, and P. Zoller proposed a parity-encoding (PE) architecture consisting of an expanded spin system with only local connectivity among them to circumvent this difficulty in developing near-future QA devices. They suggested that this architecture not only alleviates implementation challenges and enhances scalability but also possesses intrinsic fault tolerance. This paper proposes a practical decoding method tailored to correlated spin-flip errors in spin readout of PE architecture. Our work is based on the close connection between PE architecture and classical low-density parity-check (LDPC) codes. We show that the bit-flip (BF) decoding algorithm can correct independent and identically distributed errors in the readout of the SLHZ system with comparable performance to the belief propagation (BP) decoding algorithm. Then, we show evidence that the proposed BF decoding algorithm can efficiently correct correlated spinflip errors by simulation. The result suggests that introducing post-readout BF decoding reduces the computational cost of QA using the PE architecture and improves the performance of global optimal solution search. Our results emphasize the importance of the proper selection of decoding algorithms to exploit the inherent fault tolerance potential of the PE architecture.