An iterative Ising decoder for quantum error correction codes

2026-06-10

AI Summary

This work addresses the challenges posed by high-order interactions in Ising-based quantum error correction joint decoding, which suffers from poor convergence, substantial computational overhead, and complex hardware embedding. The authors propose an Iterative Low-Order Decoding (ILOD) algorithm that alternately optimizes X-type and Z-type sub-Hamiltonians while incorporating a Bayesian prior to approximate cross-type error correlations, effectively reducing eight- or ten-body interactions to four- or five-body terms. This approach significantly improves convergence, reduces runtime—scaling as (0.81)ᵈ with code distance d—and lowers spin embedding overhead. On the surface code, ILOD achieves a threshold of 4.73%, closely approaching the 4.83% of full joint decoding, and demonstrates stable convergence on large-distance 6.6.6 color codes where joint decoding fails.

Abstract

The Ising framework maps the decoding problem in quantum error correction onto ground-state optimization of a classical Hamiltonian, in which $X$-$Z$ error correlations enter as cross terms. Under phenomenological depolarizing noise, the exact joint formulation contains up to 8-body interactions for the toric code and 10-body for the $6.6.6$ color code. These high-order terms degrade solver convergence, inflate runtime, and raise the auxiliary spin overhead when embedding into native 2-body Ising hardware. In this work, we propose the iterative low-order decoding (ILOD) algorithm, which alternates between $X$- and $Z$-type sub-Hamiltonians, approximating cross-type correlations through Bayesian priors that reweight each type's couplings using the other type's inferred error configuration. This halves the maximum body count of interaction terms in the Hamiltonian, accelerating the solver, restoring convergence at larger code distances, and reducing the total spin count for 2-body embedding by a factor of $2.5$. For the toric code, ILOD attains a threshold of $4.73%$ versus $4.83%$ for the joint formulation, with the empirical runtime ratio scaling as $(0.81)^d$. For the $6.6.6$ color code, their thresholds agree within statistical uncertainty for small code distances, and ILOD remains convergent for larger distances where the joint formulation fails to converge despite a larger annealing budget.

Problem

• quantum error correction
• Ising decoder
• high-order interactions
• X-Z error correlations
• toric code

Innovation

• iterative low-order decoding
• Ising decoder
• quantum error correction
• Bayesian priors
• high-order interactions