Scalable accuracy gains from postselection in quantum error correcting codes

TL;DR

This work shows that logical failures in topological stabilizer codes are dominated by exponentially unlikely syndrome patterns, enabling scalable accuracy gains through postselection. By mapping decoding to a disordered statistical mechanics model and invoking a large-deviation principle for the domain-wall free-energy $\Delta F$, the authors show that aborting on dangerous syndromes (small $|\Delta F|$) can exponentially boost the effective code distance, with a universal gain factor $b\ge 2$ and numerically $b\approx 3.1$ for the toric code with perfect measurements. The framework extends to general stabilizer codes and rebuts naive scalability concerns by analyzing both optimal and suboptimal decoders (including MWPM under circuit-level noise), finding substantial but bounded gains (e.g., $b$ in the range $2.9$–$3.4$ depending on the setting). Verification across diverse decoders and code families supports the broad applicability of large-deviation-based postselection as a scalable error-mitigation strategy for quantum error correction. The results also connect to code-splitting ideas and highlight how rare but detectable error syndromes can be systematically exploited to reduce logical failure rates without prohibitive resource overhead.

Abstract

Decoding stabilizer codes such as the surface and toric codes involves evaluating free-energy differences in a disordered statistical mechanics model, in which the randomness comes from the observed pattern of error syndromes. We study the statistical distribution of logical failure rates across observed syndromes in the toric code, and show that, within the coding phase, logical failures are predominantly caused by exponentially unlikely syndromes. Therefore, postselecting on not seeing these exponentially unlikely syndrome patterns offers a scalable accuracy gain. In general, the logical error rate can be suppressed from $p_f$ to $p_f^b$, where $b \geq 2$ in general; in the specific case of the toric code with perfect syndrome measurements, we find numerically that $b = 3.1(1)$. Our arguments apply to general topological stabilizer codes, and can be extended to more general settings as long as the decoding failure probability obeys a large deviation principle.

Figures (11)

• Figure 1: Numerical results for $P(\Delta F)$ in the $\pm J$ RBIM (toric code) with $p = 0.06 < p_c \approx 0.109$. (a) Distributions of $P(\Delta F)$ for different system sizes $d$. The peaks of $P(\Delta F)$ move to larger $\Delta F$ as the system size $d$ increases, while error chains with $\Delta F$ near zero constitute only an exponentially small fraction of all error chains. (b) Verification of large deviation scaling. After rescaling by $d$ and normalizing with $N(d)$, defined as the fitted value of $P(\Delta F=0)$, the curves for different system sizes collapse, consistent with the expected large deviation form. The scaling holds for $|s|<\overline{s}$, while for $|s|>\overline{s}$ the distribution exhibits a different asymptotic behavior. (c) Scaling of $\log p_f$ and $-\langle \Delta F\rangle$ with code distance $d$; $p_f$ is the logical failure probability. Both quantities decrease linearly with $d$, but with different slopes: $-\langle \Delta F\rangle$ decreases faster than $\log p_f$. Ordinary least-squares fits over $d=\{12,16,20,24\}$ to $\log{p_f}=L_0-I(0)d$ gives the estimate $I(0)=0.26(1)$, while fitting to $\Delta F=F_0+\overline{s}d$ gives the estimate $\overline{s}=0.80(1)$. This separation confirms that typical error configurations contribute negligibly to logical failure for large $d$, thereby justifying efficient postselection based on $|\Delta F|$.
• Figure 2: (a) Illustration of postselection with fixed qubit budget ($r=4$). An original code patch with $d = L$ is divided into four $d = L/2$ subpatches. During decoding, the patch with the largest $|\Delta F_i|$ (e.g., $|\Delta F_2|$) is selected. (b) Comparison of logical failure probabilities for a single distance-$2n$ code, for two distance-$n$ subcodes with postselection, and for four distance-$n$ subcodes with postselection, simulated in the $\pm J$ RBIM with disorder parameter $p=0.06$. The two $d=n$ codes exhibit the same scaling of logical failure probability as the single $d=2n$ code, while four $d=n$ codes decay exponentially faster with $n$. These numerical results are fully consistent with our theoretical predictions. (c) Accuracy gain from code splitting as a function of the number of subcodes $r$. Numerical results for $p=0.06$$\pm J$ RBIM show that the $r=4$ strategy enhances the effective distance by a factor around $1.45$, in agreement with order-statistics and rate-function predictions (around $1.44$).
• Figure 3: Verification of large deviation scaling for minimum-weight perfect matching (MWPM) surface code decoding under circuit-level noise, including depolarizing noise, measurement errors, and CNOT gate faults at rate $p=0.005$. The rescaled distribution of the complementary gap $G$, defined as the difference between the minimum weights in the two homology classes (see Supplemental Material suppmat), exhibits clear data collapse across code distances, consistent with large deviation scaling within the range $|s|<\overline{s}$. The behavior is analogous to that of the two-dimensional RBIM, demonstrating that the large deviation form persists under realistic circuit-level noise.
• Figure S1: Graphic representations of matrices $\mathcal{B}_{\sigma_{i}'\sigma_{j}'}^{\sigma_i \sigma_j}$ and $\mathcal{S}_{\sigma_i}^{\sigma_j}$. They play as basic ingredients in constructing tensor network representation of partition function $\mathcal{Z}$. (a) Tensor $\mathcal{B}_{\sigma_{i}'\sigma_{j}'}^{\sigma_i \sigma_j}$ as a 4-leg tensor acting like a two-qubit gate. (b) Tensor $\mathcal{S}_{\sigma_i}^{\sigma_j}$ as a 2-leg tensor acting on a single qubit.
• Figure S2: TN representation of $\mathcal{T}_t$ using local tensors $\mathcal{B}_{\sigma_{i}'\sigma_{j}'}^{\sigma_i \sigma_j}$ and $\mathcal{S}_{\sigma_i}^{\sigma_j}$. The overall action of $\mathcal{T}_t$ works like Trotterized time-evolution driven by two-qubit gates and single-qubit gates.