An exact Error Threshold of Surface Code under Correlated Nearest-Neighbor Errors: A Statistical Mechanical Analysis

TL;DR

This work addresses the quest for an exact error threshold of the surface code under realistic correlated noise, combining i.i.d. errors with nearest-neighbor correlations. It develops an error-edge map (EEM) to transform decoding success probabilities into the partition function of a square-octagonal random-bond Ising model, enabling exact threshold determination via statistical mechanics. For the case $p_1=p_2$, parallel-tempering Monte Carlo simulations yield an exact threshold near $p_c \approx 0.03$, and comparisons show current decoders lag behind this bound under correlation, suggesting room for decoder improvement. By bridging quantum error correction with classical statistical mechanics, this approach provides a principled, algorithm-independent benchmark for fault-tolerance under realistic correlated noise.

Abstract

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 studieds for threshold with correlated error, they are only the lower bond ranther than exact value, this 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 exact 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 investigate the error correction threshold of surface codes and we present analytical constraints giving exact value of error threshold. This means that our error threshold is both upper bound and achievable and hence on the one hand the existing numerical threshold values can all be improved to our threshold value, on the other hand, our threshold value is highest achievable value in principle.

Figures (6)

• Figure 1: Error model. (a) $p_1$ represents the probability of a single qubit $Z$ error due to the channel in Eq.(\ref{['eq:iid_error_model']}) , while $p_2$ represents the probability of $ZZ$ error (enclosed by the pink ellipse) due to the channel in Eq.(\ref{['eq:correlated_error_model']}). In this diagram, neighboring ancilla qubits are connected by edges named in $l^k_2$ and $l^k_3$, defined in the main text letter around Eq.(\ref{['eq:nE2']}). For example, there are two edges of $l^k_2$ (in red) and $l^k_3$ (in green). (b)Four qubits are are denoted as $q_{i,j},q_{i-1,j+1},q_{i+1,j+1},q_{i+2,j}$ where the subscripts represent the spatial coordinates of each qubit's position, and they have be used in the main text around Eq.(\ref{['eq:correlated_error_model']}).
• Figure 2: An example of error-edge map. Four pink diamonds indicate Z errors on the corresponding qubits. The figure shows 3 error sets containing different qubits, corresponding to 5 error chains. The error chains (red polylines) Fig.\ref{['fig:error_case']}(a) and (e) are the same, and we use $E'_1$ to indicate them. Consequently, the errors (pink diamonds) depicted in Fig.\ref{['fig:error_case']}(a) and Fig.\ref{['fig:error_case']}(e), they belong to the set $A_{1}$ according to our definitions for $E'_l,A_l$. Although the errors in Fig. 2(a), (b), and (c) occur on identical qubits, they belong to distinct equivalence classes $A_l$.
• Figure 3: Example of error correction on a surface code. Pink diamonds indicate Z errors occurring on the corresponding qubits, while green diamonds represent Pauli Z operators for recovery of the respective qubits. Red lines represent the error chain $E$, and green lines represent the recovery chain $E'$. The dashed circles represent virtual ancilla qubits. After performing our error-edge map, connected error chains corresponding to connections between two virtual nodes (dashed dots) will produce line $L$ that span the surface code. At the centers of triangulars, Ising spins take values in $\{\pm 1\}$. (a) $E + E' = C$ represents a perfect error correction example, where $C$ encloses a domain with spin value $-1$. (b) An example of ising spins corresponding to unwanted terms obtaining in partition function mapped by $\mathcal{M}$. (c) A failed error correction example where $E+ E' = L$ and L span the surface code.
• Figure 4: Mapping a surface code to a square-octagonal lattice.
• Figure 5: Finite-size correlation length $\xi_m/L$ as a function of temperature $T$ for different error probabilities. The temperature range is $T \in [0.5, 1]$. (a): $p = 0.025$. (b): $p = 0.030$. (c): $p = 0.031$. (d): $p=0.032$. For $p \lesssim p_c$ there is clear evidence of a phase transition (curves for different system sizes $L$ cross) whereas for $p > p_c$ the transition disappears.