Revisiting Nishimori multicriticality through the lens of information measures

Abstract

The quantum error correction threshold is closely related to the Nishimori physics of random statistical models. We extend quantum information measures such as coherent information beyond the Nishimori line and establish them as sharp indicators of phase transitions. We derive exact inequalities for several generalized measures, demonstrating that each attains its extremum along the Nishimori line. Using a fermionic transfer matrix method, we compute these quantities in the 2d $\pm J$ random-bond Ising model-corresponding to a surface code under bit-flip noise-on system sizes up to $512$ and over $10^7$ disorder realizations. All critical points extracted from statistical and information-theoretic indicators coincide with high precision at $p_c=0.1092212(4)$, with the coherent information exhibiting the smallest finite-size effects. We further analyze the domain-wall free energy distribution and confirm its scale invariance at the multicritical point.

Figures (7)

• Figure 1: Surface code, RBIM and schematic phase diagram. (a) Surface code with rough (smooth) boundary conditions along the $x$ ($y$) direction, for code distance $d=4$. Blue (yellow) markers indicate plaquette (vertex) stabilizers; The shaded regions indicate qubits defining logical $\bar{X}$ ($\bar{Z}$) operators. The dashed red line shows an error chain $f$, with its endpoint marking the syndrome $\mathbf{s}$. (b) RBIM ($L=4$) mapped from panel (a), with spins on plaquettes operators and qubits are mapped to the bonds between them. The rough (smooth) boundary condition corresponds to free (fixed) boundary conditions in the RBIM, and physical errors appear as $\tau_{ij}$=$-1$ bonds (red dashed lines). Inserting a domain wall (logical error $\bar{X}$) corresponds to imposing a twisted boundary condition. (c) Schematic RBIM phase diagram Ettore2008MC in the $p$-$T$ plane, showing paramagnetic (PM), ferromagnetic (FM) and zero-temperature spin glass (SG) phases. The Nishimori line (red) represents the mapping from panel (a) to (b) in Eq. (\ref{['eq:mapping']}); the thermodynamic-limit behavior of various measures in each phase is indicated. The MNP marks the intersection of the FM–PM phase boundary with the Nishimori line and features two relevant directions (arrows) in the RG sense. Dashed blue curves schematically illustrate the extreme behavior of $\overline{\mathcal{R}^{0.5}(\beta)},\mathcal{P}_\text{succ}^\text{MLD},\mathcal{I}_c(\beta)$ along the Nishimori line.
• Figure 2: Critical behavior of various quantities along the Nishimori line. Finite-size scaling yields $p_c = 0.1092212(4)$ and $1/\nu = 0.652(2)$. (a) Coherent information $\mathcal{I}_c$; the shaded region (red) indicates our critical-point result, compared with previous works Ettore2009strongYoujinPRB2025gy2025nishimoriuniversality. The horizontal region shows the crossing value $\mathcal{I}_c=0.4990(1)$, deviating clearly from 1/2. (b) The success probability of MLD $\mathcal{P}_\text{succ}^{\text{MLD}}$. (c) The success probability of the Bayes decoder $\mathcal{P}_\text{succ}^{\text{Bayes}}$. (d) Disorder-averaged DWFE $d_W$. (e) $q$=$0.5$ moment of partition function ratio $\overline{\mathcal{R}^{0.5}}$. (f),(g) Distribution of $\Delta F$ in the FM ($p$=$0.10$) and PM ($p$=$0.12$) phases. (h) The distribution of $\Delta F$ at MNP, exhibiting a characteristic kink at zero. Quantities shown in (a–e) were computed with an estimator incorporating the Nishimori condition to reduce statistical errors, see Eq. (\ref{['eq:better_estimator']}); Separate fitting results are summarized in Table \ref{['tab:end_matter_res_tab']}, and further details of the finite-size scaling analysis can be found in the SM supplemental.
• Figure 3: Behavior of quantities under perturbations away from the Nishimori line at $p=p_c$. (a) $\mathcal{I}_c$; (b) $1$$-$$\mathcal{S}_\text{DW}$; (c) $\mathcal{P}_\text{succ}^\text{MLD}$; (d) $\overline{R^{0.5}}$. Quantities in (a), (c) and (d) reach an extremum at $T=T_c$, marked by the vertical dashed line. The inset of panel (b) compares $\mathcal{I}_c$ (red) and $1$$-$$\mathcal{S}_\text{DW}$ (black) at $T_c$, with the latter exhibiting substantially smaller error bars. The crossing value, extrapolated to infinite size, deviates from $1/2$. The legends are shared between panels.
• Figure S1: Computational cost of the algorithm. The figure shows the wall-clock time required to compute a single disorder realization—specifically, the outputs $\log Z$ and $\log Z'$—using one core of an AMD EPYC 7742 CPU. The red line corresponds to $y=3\times 10^{-10} x^4$, illustrating the asymptotic scaling behavior of the algorithm.
• Figure S2: Error analysis at Nishimori point for different system sizes $L_x=L_y=L$ with fixed stabilization step $n_\text{stab} =5$. Panels (a),(b) represent the additive error and relative error of $\log Z$ respectively. The reference value $\log Z_\text{stab}$ is calculated by using $n_\text{stab}=1$. The data here is the average of 1000 disorder samples.