The surface code represents a promising candidate for fault-tolerant quantum computation due to its high error threshold and experimental accessibility with nearest-neighbor interactions. However, current exact surface code threshold analyses are based on the assumption of independent and identically distributed (i.i.d.) errors. Though there are numerical studies for the threshold with correlated error, they are only the lower bound rather than the exact value, which offers potential for higher error thresholds. Here, we establish an error-edge map, which allows for the mapping of quantum error correction to a square-octagonal random bond Ising model. We then present the maximum-likelihood threshold under a realistic noise model that combines independent single-qubit errors with correlated errors between nearest-neighbor data qubits. Our method is applicable for any ratio of nearest-neighbor correlated errors to i.i.d. errors. We present analytical constraints for the error correction threshold of surface codes, and by applying these constraints, we numerically obtain the error threshold higher than the existing values. Our error threshold is both an upper bound and achievable.The surface code represents a promising candidate for fault-tolerant quantum computation due to its high error threshold and experimental accessibility with nearest-neighbor interactions. However, current exact surface code threshold analyses are based on the assumption of independent and identically distributed (i.i.d.) errors. Though there are numerical studies for the threshold with correlated error, they are only the lower bound rather than the exact value, which offers potential for higher error thresholds. Here, we establish an error-edge map, which allows for the mapping of quantum error correction to a square-octagonal random bond Ising model. We then present the maximum-likelihood threshold under a realistic noise model that combines independent single-qubit errors with correlated errors between nearest-neighbor data qubits. Our method is applicable for any ratio of nearest-neighbor correlated errors to i.i.d. errors. We present analytical constraints for the error correction threshold of surface codes, and by applying these constraints, we numerically obtain the error threshold higher than the existing values. Our error threshold is both an upper bound and achievable.

arXiv:2510.24181npj Quantum InformationMay 23, 20260 citations

Numerical evaluation of the fidelity error threshold for the surface code

Pejman Jouzdani, Eduardo R. Mucciolo

We study how the resilience of the surface code is affected by the coupling to a non-Markovian environment at zero temperature. The qubits in the surface code experience an effective dynamics due to the coupling to the environment that induces correlations among them. The range of the effective induced qubit-qubit interaction depends on parameters related to the environment and the duration of the quantum error correction cycle. We show numerically that different interaction ranges set different intrinsic bounds on the fidelity of the code. These bounds are unrelated to the error thresholds based on stochastic error models. We introduce a definition of stabilizers based on logical operators that allows us to efficiently implement a Metropolis algorithm to determine upper bounds to the fidelity error threshold.We study how the resilience of the surface code is affected by the coupling to a non-Markovian environment at zero temperature. The qubits in the surface code experience an effective dynamics due to the coupling to the environment that induces correlations among them. The range of the effective induced qubit-qubit interaction depends on parameters related to the environment and the duration of the quantum error correction cycle. We show numerically that different interaction ranges set different intrinsic bounds on the fidelity of the code. These bounds are unrelated to the error thresholds based on stochastic error models. We introduce a definition of stabilizers based on logical operators that allows us to efficiently implement a Metropolis algorithm to determine upper bounds to the fidelity error threshold.

arXiv:1403.7245Physical Review AJul 11, 20143 citations

A Minimal Duality Estimate for the Surface-Code Threshold under Nearest-Neighbor Correlated Errors

Masayuki Ohzeki

We apply the single-equation duality criterion to the square-octagonal random-bond Ising model recently obtained from an exact error-edge map for a surface code with nearest-neighbor correlated errors. The calculation is performed for the minimal cell after the error-edge reduction. For the symmetric case $p_1=p_2=p_3=p$ , this gives $p_c=0.0288427147$ , in close agreement with the reported numerical threshold of about $3\%$ .We apply the single-equation duality criterion to the square-octagonal random-bond Ising model recently obtained from an exact error-edge map for a surface code with nearest-neighbor correlated errors. The calculation is performed for the minimal cell after the error-edge reduction. For the symmetric case $p_1=p_2=p_3=p$, this gives $p_c=0.0288427147$, in close agreement with the reported numerical threshold of about $3\%$.

cond-mat.dis-nnCross-listedarXiv:2606.02052Jun 2, 20260 citations

p_1=p_2=p_3=p

, this gives

p_c=0.0288427147

, in close agreement with the reported numerical threshold of about

3\%