Phase diagram and optimal threshold of the random alternating eight-vertex model for the toric code under depolarizing noise

Bombin, Andrist, Ohzeki, Katzgraber and Martin-Delgado (Phys. Rev. X 2, 021004 (2012)) showed that Kitaev's toric code under i.i.d. depolarizing noise maps onto a random alternating eight-vertex model: two stacked square-lattice Ising copies (direct and dual lattice) with quenched random-sign pair couplings and a four-spin interaction, an Ashkin-Teller-type coupled-Ising class that, unlike Baxter's exactly solved standard eight-vertex model, has no known exact solution. Combining parallel-tempering Monte Carlo (with finite-size crossings that become marginal near the transition) and an approximate duality method, they estimated the optimal, maximum-likelihood error threshold as $p_c = 0.189(3)$ and $0.1888$ respectively, and noted its unexplained proximity to the depolarizing-channel hashing bound $p_h \approx 0.1893$. No exact solution of the disordered model class is known; no refined thermodynamic-limit determination of $p_c$ has been reported (published estimates still spread over roughly $0.186$-$0.189(3)$ depending on method, with the 2026 topical review of maximum-likelihood decoding still citing $18.9\%$ solely from the 2012 paper); the universality class of the Nishimori multicritical point for this coupled model is undetermined; and whether the hashing-bound near-coincidence is structural or accidental remains unresolved. The problem of a controlled characterization of the phase boundary $T_c(p)$ and the precise threshold $p_c$ therefore remains open.

Background

The toric code (Kitaev, 2003) is the canonical two-dimensional topological stabilizer code: qubits sit on the edges of a square lattice, and commuting Pauli check operators ('star' and 'plaquette' operators) are measured to detect errors. A stabilizer code is a quantum error-correcting code defined by such a commuting set of check operators that pins a protected code space; its code capacity is the error model in which qubits corrupt independently and syndrome measurements are perfect. Under i.i.d. depolarizing noise each qubit independently suffers a Pauli $X$, $Y$, or $Z$ error, each with probability $p/3$. Since $Y = iXZ$, depolarizing noise correlates bit-flip ($X$) and phase-flip ($Z$) errors on the same qubit, so the two error species cannot be corrected independently without loss.

For error rates below a critical value $p_c$, maximum-likelihood (ML) decoding - choosing the most probable logical equivalence class of errors consistent with the measured syndrome - drives the logical failure probability to zero as the lattice grows; $p_c$ is the code-capacity threshold, and it upper-bounds every practical decoder. Dennis, Kitaev, Landahl and Preskill (2002) established that this threshold is a phase transition of a disordered classical spin model: decoding failure is governed by the free energy of a random-bond Ising model (RBIM, an Ising model whose bond signs are quenched random variables fixed by the error pattern), and $p_c$ is the multicritical point where the ferromagnet-paramagnet phase boundary meets the Nishimori line - the line in the disorder-temperature plane on which the disorder distribution and the Boltzmann weight are related by an exact gauge symmetry. Along the Nishimori line certain thermal averages are known exactly, but the location of the critical point on it is not.

For depolarizing noise the $Y$ errors couple the bit-flip and phase-flip sectors, and Bombin, Andrist, Ohzeki, Katzgraber and Martin-Delgado (2012) showed that Kitaev's toric code maps to a random alternating eight-vertex model. The eight-vertex model places arrows on the edges of a square lattice with an even number of arrows pointing into each vertex, giving eight allowed vertex types with weights $a, b, c, d$. Baxter (1972) solved the model with uniform weights exactly; its critical line is $e^{-2\beta J_4} = \sinh(2\beta J)$, where $J$ is a pair coupling, $J_4$ a four-spin coupling, and $\beta$ the inverse temperature, and its critical exponents vary continuously with $J_4/J$ - the classic violation of universality. In the magnetic (Wu-Kadanoff-Wegner) representation the model is two Ising copies, one on the direct and one on the dual lattice, coupled by a four-spin term at each bond crossing. If the pair couplings are assigned by bond orientation one obtains the standard, Baxter-solvable model; if they are assigned by lattice membership (direct versus dual lattice), the vertex weights alternate in a checkerboard pattern - the alternating eight-vertex model, equivalent to an Ashkin-Teller-type pair of coupled Ising models (Ashkin and Teller, 1943), which is in general not exactly solvable. Concretely, the disordered model realized by the toric code under depolarizing noise is

$H_E = -\sum_{+}\left(J_x\,\tau^{x}_{+}\,s_i s_j + J_z\,\tau^{z}_{+}\,s'_k s'_\ell + J_y\,\tau^{y}_{+}\,s_i s_j s'_k s'_\ell\right),$

with spins $s$ on the direct and $s'$ on the dual square lattice (stacked and shifted by half a lattice spacing so the crossings $+$ are in one-to-one correspondence with physical qubits), equal couplings $J_x = J_z = J_y$ for the symmetric depolarizing channel, and quenched signs $\tau^{x,z,y}_{+} = \pm 1$ drawn independently at each crossing according to the depolarizing error distribution. Wen's variant of the toric code instead yields the standard (solvable) model, and topological color codes yield 'interacting' eight-vertex models on triangular-based lattices with three- and six-spin terms.

For the pure ($p = 0$) model, self-duality fixes the critical point at $\beta J_c = (1/4)\log 3 \approx 0.2746$, i.e. a critical temperature about 60% above that of the square-lattice Ising model - the origin of the enhancement of the depolarizing threshold over the bit-flip threshold $p_c \approx 0.109$. Indeed, correcting the $X$ and $Z$ sectors separately ignores their correlation and yields only $(3/2)\times 0.109 \approx 16.4\%$. For $p > 0$, Bombin et al. computed the phase boundary $T_c(p)$ by parallel-tempering Monte Carlo of the sublattice magnetization and finite-size correlation-length crossings, finding crossings that become marginal near $p = 0.189$ and estimating the threshold as $p_c = 0.189(3)$; a cluster duality approximation gave $p_c = 0.1888$ for the toric code (and $0.1878$-$0.1914$ for two color codes). Both methods carry uncontrolled systematic errors. Notably, $p_c$ lies close to the hashing bound of the depolarizing channel, $p_h \approx 0.1893$, where the channel's coherent information $1 + (1-p)\log_2(1-p) + p\log_2(p/3)$ vanishes (Bennett et al., 1996); in the bit-flip case the RBIM threshold $\approx 0.109$ is likewise just below the binary hashing bound $\approx 0.110$. Whether this near-coincidence for the toric code is exact, structural, or accidental is not known.

Problem Statement

Determine the phase diagram and optimal threshold of the random alternating eight-vertex (equivalently, disordered Ashkin-Teller / random-bond coupled-Ising) model defined by

$H_E = -\sum_{+}\left(J\,\tau^{x}_{+}\,s_i s_j + J\,\tau^{z}_{+}\,s'_k s'_\ell + J\,\tau^{y}_{+}\,s_i s_j s'_k s'_\ell\right),$

on two stacked, half-shifted square lattices (spins $s$ on the direct and $s'$ on the dual lattice; crossings $+$ in one-to-one correspondence with physical qubits), with symmetric couplings $J_x = J_z = J_y = J$, quenched independent signs $\tau^{x,z,y}_{+} = \pm 1$ drawn at each crossing according to the depolarizing distribution - the pattern $(+,+,+)$ with probability $1-p$ and each of the three patterns with exactly two minus signs with probability $p/3$ (Eq. (18) of the source; equivalently $\tau_\sigma = -1$ precisely when the single-qubit Pauli $\sigma$ anticommutes with the error at that crossing) - and the Nishimori condition $3\,e^{-4\beta J} = p/(1-p)$ - the model that governs maximum-likelihood error correction of Kitaev's toric code under i.i.d. depolarizing noise. The scientific objective is a controlled characterization of this one model class in the thermodynamic limit:

1. The ferromagnet-paramagnet phase boundary $T_c(p)$ for the full range $0 \le p \le p_c$, including its approach to the multicritical region.
2. The location of the multicritical point where the boundary meets the Nishimori line - the optimal (ML) depolarizing error threshold $p_c$ of the toric code - with controlled, quantified precision, resolving the present method spread among Monte Carlo ($0.189(3)$, with marginal finite-size crossings), duality approximations ($0.1885$-$0.1888$), and small-code coherent-information crossings ($\approx 0.186$). Both enumerated items are required. A determination of $p_c$ whose quoted uncertainty remains consistent with both thermodynamic-limit estimates quoted above - the Monte Carlo result $0.189(3)$ and the duality result $0.1885$-$0.1888$ - does not resolve the present spread; resolving it requires a thermodynamic-limit determination precise enough to exclude at least one of those two estimates, and finite-size or small-code values such as $\approx 0.186$ do not count toward this test.

Finite-size or small-code results are inputs to, not substitutes for, the thermodynamic characterization; a practical decoding algorithm alone does not answer the question unless it pins the thermodynamic-limit transition itself.

References

1. H. Bombin, R. S. Andrist, M. Ohzeki, H. G. Katzgraber, M. A. Martin-Delgado, 'Strong Resilience of Topological Codes to Depolarization', Phys. Rev. X 2, 021004 (2012). DOI: 10.1103/PhysRevX.2.021004, arXiv:1202.1852. https://doi.org/10.1103/PhysRevX.2.021004
2. E. Dennis, A. Kitaev, A. Landahl, J. Preskill, 'Topological quantum memory', J. Math. Phys. 43, 4452 (2002). DOI: 10.1063/1.1499754, arXiv:quant-ph/0110143. https://doi.org/10.1063/1.1499754
3. A. Yu. Kitaev, 'Fault-tolerant quantum computation by anyons', Ann. Phys. 303, 2 (2003). DOI: 10.1016/S0003-4916(02)00018-0, arXiv:quant-ph/9707021. https://doi.org/10.1016/S0003-4916(02)00018-0
4. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, 'Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels', Phys. Rev. Lett. 76, 722 (1996). DOI: 10.1103/PhysRevLett.76.722, arXiv:quant-ph/9511027. https://doi.org/10.1103/PhysRevLett.76.722
5. J. Ashkin, E. Teller, 'Statistics of Two-Dimensional Lattices with Four Components', Phys. Rev. 64, 178 (1943). DOI: 10.1103/PhysRev.64.178. https://doi.org/10.1103/PhysRev.64.178
6. R. J. Baxter, 'Partition function of the Eight-Vertex lattice model', Ann. Phys. 70, 193 (1972). DOI: 10.1016/0003-4916(72)90335-1. https://doi.org/10.1016/0003-4916(72)90335-1
7. G. Duclos-Cianci, D. Poulin, 'Fast Decoders for Topological Quantum Codes', Phys. Rev. Lett. 104, 050504 (2010). DOI: 10.1103/PhysRevLett.104.050504, arXiv:0911.0581. https://doi.org/10.1103/PhysRevLett.104.050504
8. M. Ohzeki, 'Error threshold estimates for surface code with loss of qubits', Phys. Rev. A 85, 060301 (2012). DOI: 10.1103/PhysRevA.85.060301, arXiv:1202.2593. https://doi.org/10.1103/PhysRevA.85.060301
9. S. Bravyi, M. Suchara, A. Vargo, 'Efficient Algorithms for Maximum Likelihood Decoding in the Surface Code', Phys. Rev. A 90, 032326 (2014). DOI: 10.1103/PhysRevA.90.032326, arXiv:1405.4883. https://doi.org/10.1103/PhysRevA.90.032326
10. C. T. Chubb, S. T. Flammia, 'Statistical mechanical models for quantum codes with correlated noise', Ann. Inst. Henri Poincare D 8, 269 (2021). DOI: 10.4171/AIHPD/105, arXiv:1809.10704. https://doi.org/10.4171/AIHPD/105
11. C. T. Chubb, 'General tensor network decoding of 2D Pauli codes', arXiv:2101.04125 (2021). https://arxiv.org/abs/2101.04125
12. L. Colmenarez, S. Kim, M. Müller, 'Fundamental thresholds for computational and erasure errors via the coherent information', PRX Quantum 6, 040327 (2025). DOI: 10.1103/d8rx-srpn, arXiv:2412.16727. https://doi.org/10.1103/d8rx-srpn
13. F. Eckstein, B. Han, S. Trebst, G.-Y. Zhu, 'Robust teleportation of a surface code and cascade of topological quantum phase transitions', PRX Quantum 5, 040313 (2024). DOI: 10.1103/PRXQuantum.5.040313, arXiv:2403.04767. https://doi.org/10.1103/PRXQuantum.5.040313
14. Y. Xu, Y. Zhou, J. P. Sethna, E.-A. Kim, 'Rigorous estimation of error thresholds of transversal Clifford logical circuits', arXiv:2510.10835 (2025). https://arxiv.org/abs/2510.10835
15. Z.-Q. Wan, X.-D. Dai, G.-Y. Zhu, 'Revisiting Nishimori multicriticality through the lens of information measures', Phys. Rev. Research 8, 023059 (2026). DOI: 10.1103/b8y5-k3y6, arXiv:2511.02907. https://doi.org/10.1103/b8y5-k3y6
16. A. Honecker, M. Picco, P. Pujol, 'Universality Class of the Nishimori Point in the 2D ±J Random-Bond Ising Model', Phys. Rev. Lett. 87, 047201 (2001). DOI: 10.1103/PhysRevLett.87.047201. https://doi.org/10.1103/PhysRevLett.87.047201
17. D. Gottesman, 'Keeping One Step Ahead of Errors', Physics 5, 50 (2012). DOI: 10.1103/Physics.5.50. https://doi.org/10.1103/Physics.5.50
18. H. Cao, G. Yan, Y. Du, F. Pan, 'Maximum Likelihood Decoding of Quantum Error Correction Codes' (topical review), arXiv:2605.17230 (2026). https://arxiv.org/abs/2605.17230
19. Z. Tobias, N. P. Breuckmann, B. Placke, 'Optimal Decoding with the Worm', arXiv:2603.05428 (2026). https://arxiv.org/abs/2603.05428
20. J. Y. Lee, 'Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fundamental Error Threshold', Phys. Rev. Lett. 134, 250601 (2025). DOI: 10.1103/hlfh-86yz, arXiv:2402.16937. https://doi.org/10.1103/hlfh-86yz
21. J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, B. J. Brown, 'The XZZX Surface Code', Nat. Commun. 12, 2172 (2021). DOI: 10.1038/s41467-021-22274-1, arXiv:2009.07851. https://doi.org/10.1038/s41467-021-22274-1
22. Y. Xiao, B. Srivastava, M. Granath, 'Exact results on finite size corrections for surface codes tailored to biased noise', Quantum 8, 1468 (2024). DOI: 10.22331/q-2024-09-11-1468, arXiv:2401.04008. https://doi.org/10.22331/q-2024-09-11-1468