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arxiv: 2603.25665 · v2 · submitted 2026-03-26 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· cond-mat.str-el· quant-ph

Recognition: 2 theorem links

· Lean Theorem

Non-linear Sigma Model for the Surface Code with Coherent Errors

Stephen W. Yan , Yimu Bao , Sagar Vijay
Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:22 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nncond-mat.str-elquant-ph
keywords surface codecoherent errorsnonlinear sigma modelmaximum-likelihood decodingthermal metal phasereplica limitquantum error correctionanyon excitations
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The pith

The long-distance theory of surface-code decoding under coherent errors is a non-linear sigma model with target space SO(2n)/U(n) whose replica limits separate optimal from suboptimal performance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Maximum-likelihood decoding of the square-lattice surface code under single-qubit coherent rotations maps onto an effective field theory. The paper derives this non-linear sigma model microscopically and shows that its replica parameter n takes the value 1 when the decoder knows the rotation angle and the value 0 when the angle is unknown. In the n to 0 limit the model supports a thermal-metal phase that is non-decodable, while the same fixed point is unstable when n equals 1. Consequently the authors argue that optimal decoding can remain possible even at the largest coherent rotation angle. The decoding fidelity itself is expressed in terms of twist defects of the order-parameter field, giving concrete scaling predictions near the metallic point that are checked numerically.

Core claim

We microscopically derive a non-linear sigma model with target space SO(2n)/U(n) as the effective long-distance theory of this decoding problem, with distinct replica limits: n to 1 for optimal decoding, which assumes knowledge of the coherent rotation angle, and n to 0 for suboptimal decoding with imperfect angle information. This exposes a sharp distinction between the two decoders. The suboptimal decoder supports a thermal-metal phase, a non-decodable regime that is qualitatively distinct from the conventional non-decodable phase of the surface code under incoherent Pauli errors. By contrast, the metal phase cannot arise in optimal decoding, since the metallic fixed-point becomes unstable

What carries the argument

The non-linear sigma model with target space SO(2n)/U(n), which encodes the long-wavelength statistics of electric anyon excitations created by coherent rotations and the maximum-likelihood decoding of their syndromes.

If this is right

  • Optimal decoding remains possible up to the maximally coherent rotation angle because the metallic fixed point is unstable.
  • Decoding fidelity is controlled by twist defects of the order-parameter field and therefore exhibits a characteristic power-law system-size dependence near the metallic point.
  • On a non-bipartite lattice the symmetries change and a stable metallic phase can appear even for optimal decoding.
  • The same sigma-model description yields quantitative predictions for other observables such as anyon correlation functions that can be checked by simulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same effective-theory approach may be used to classify thresholds for other topological codes whose syndromes live on lattices with different bipartiteness properties.
  • Practical decoders that can estimate the coherent rotation angle from syndrome statistics should therefore achieve higher thresholds than those that treat the angle as unknown.

Load-bearing premise

The microscopic derivation of the sigma model from the surface-code decoding problem with coherent rotations is valid and the replica limits correctly distinguish optimal from suboptimal maximum-likelihood decoding.

What would settle it

Numerical Monte Carlo sampling of the sigma-model partition function in the n equals 1 replica limit that finds a stable metallic phase at strong disorder would falsify the instability claim; conversely, sampling in the n equals 0 limit that fails to produce the metallic phase would falsify the derivation of the target space and replica structure.

Figures

Figures reproduced from arXiv: 2603.25665 by Sagar Vijay, Stephen W. Yan, Yimu Bao.

Figure 1
Figure 1. Figure 1: Proposed phase diagrams for optimal (a) and sub [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The decoding fidelity is related to a twist of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Square-lattice surface code on the cylinder of cir [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The partition sum Zα,s (12) is defined for Ising spins on the dual square lattice of the surface code. A reference error string C ref z for a syndrome intersects bonds of this lattice, and fixes a configuration of the Ising interactions ηrr′ . On the right, this partition sum is represented as a product of transfer matrices, as in Eq. (13). field theory in Sec. V which governs optimal and subop￾timal decod… view at source ↗
Figure 5
Figure 5. Figure 5: Fidelity of the optimal decoder. (a) Fopt as a function of θ at a fixed κ = 3.5 for various L. For small θ, the fidelity is very close to 1 while for larger θ, it increases with L, consistent with a decodable phase. (b, c) Decoding infidelity 1 − Fopt as a function of κ. The results suggest that Fopt increases when increasing κ for fixed L, or increasing L for fixed κ > 1. Furthermore, increasing g −1 R (L… view at source ↗
Figure 6
Figure 6. Figure 6: Fidelity of the suboptimal decoder Fsub. (a) Fsub as a function of rotation angle θ ′ at a fixed aspect ratio κ = 1.5 for various L. The curves for various system sizes cross at θ ′ c ∼ 0.13π, indicating a decoding transition. (b) Fidelity as a function of κ with θ ′ = 0.18π, θ = 0.17π. (c) Fidelity as a function of L for a fixed κ = 8, θ ′ = 0.18π, and various θ. Fsub decays exponentially to 1/2 in L, at … view at source ↗
Figure 7
Figure 7. Figure 7: The conductivity κG of the network model corresponding to the optimal (blue) and suboptimal (orange) decoder. The decoder’s estimate of the coherent rotation angle (θ for the optimal and θ ′ for the suboptimal) is 0.16π (a) and 0.24π (b). For the suboptimal decoder, θ ′ − θ = 0.03π is fixed. In (a), the conductivity of the suboptimal decoder approaches the π −1 ln T scaling predicted by the NLsM in the the… view at source ↗
Figure 8
Figure 8. Figure 8: Conductivity as a function of θ ′ in the network model associated with the suboptimal decoder. The system size T is increased from lighter to darker color with fixed aspect ratio κ = 1/4 and θ = θ ′−0.03π. We estimate a critical rotation angle θ ′ c = 0.127(3)π, although we do not observe a scale-free critical conductivity due to finite-size effects. The results are averaged over 600 to 3000 samples, with … view at source ↗
Figure 9
Figure 9. Figure 9: The defect free energy ∆F against aspect ratio κ = T /L for the optimal (a) and suboptimal (b) decoder. Data is presented for L = 8, 16, 24, 32, 48, 64, 92, 128 from lighter to darker color. We find that ∆F is an increasing function of κ, which agrees with our analytic prediction. In the insets, we extract the slope f through a linear fit for κ > 1. The NLsM predicts the slope to be proportional to gR(L), … view at source ↗
Figure 10
Figure 10. Figure 10: Scaling of purification entropy SP . The en￾tropy SP is computed in a 1D fermion system initialized in a maximally-mixed state and evolved under the 1+1D dy￾namics associated with the decoding algorithm after time T = 2L. The blue dots and the orange upwards (down￾wards) facing triangles represent the results for the optimal decoder at θ = 0.16π and the suboptimal at θ ′ = 0.16π with θ ′ − θ = −0.05π (+0.… view at source ↗
Figure 12
Figure 12. Figure 12: SWAP-gate circuit on the cylinder for (L, T) = (3, 3). Periodic boundary conditions are imposed in the horizontal direction, with the left boundary identified with the right. A π-phase is acquired by the Majorana worldline (green) across a symmetry defect (red). In this case, the wind￾ing number of the worldline is even, leading to F = 1/2. We note that a similar calculation can be carried out for the sur… view at source ↗
Figure 13
Figure 13. Figure 13: Twist expectation value Φ2k in the RBIM picture when 1/κ ≪ 1/gR. (a) Twist expectation value as a function of k. The exponent − ln Φ2k = yk is linear in k for small k. The markers with an increasing opacity represent the results for an increasing stiffness κ/gR. (b) The linear coefficient y scales linearly with the stiffness κ/gR. The replica number is set to be n = 16. The results are averaged over 105 s… view at source ↗
Figure 14
Figure 14. Figure 14: In (a) and (b), we present the key features of the 1+1D dynamics defined by the syndrome sampling algorithm for [PITH_FULL_IMAGE:figures/full_fig_p042_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The half-system bipartite von-Neumann en [PITH_FULL_IMAGE:figures/full_fig_p044_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Distribution P(∆F) of the defect free energy associated with the optimal (a) and suboptimal (b) decoders at fixed κ = 8. The distribution is bimodal for the optimal decoder (a), while it is well fit to a Gaussian distribution with zero mean for the suboptimal decoder (b). The distribution is determined using a Gaussian kernel with variance set by Scott’s rule [105]. In both figures, curves for various sys… view at source ↗
Figure 17
Figure 17. Figure 17: Decoding fidelity for the error model with rotation angles [PITH_FULL_IMAGE:figures/full_fig_p046_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The conductivity κG in the network model asso￾ciated with the optimal (blue) and suboptimal (orange) de￾coder. The rotation angle θℓ is chosen from a Gaussian dis￾tribution with variance g and mean π/4. For the suboptimal decoder, π 4 − θ ′ ℓ  (1 + ϵ) = π 4 − θℓ. The conductivity for the suboptimal decoder fits to the scaling π −1 ln T and is distinct from that for the optimal decoder. Numerical results … view at source ↗
read the original abstract

The surface code is a promising platform for a quantum memory, but its threshold under coherent errors remains incompletely understood. We study maximum-likelihood decoding of the square-lattice surface code in the presence of single-qubit unitary rotations that create electric anyon excitations. We microscopically derive a non-linear sigma model with target space $\mathrm{SO}(2n)/\mathrm{U}(n)$ as the effective long-distance theory of this decoding problem, with distinct replica limits: $n\to1$ for optimal decoding, which assumes knowledge of the coherent rotation angle, and $n\to0$ for suboptimal decoding with imperfect angle information. This exposes a sharp distinction between the two decoders. The suboptimal decoder supports a "thermal-metal" phase, a non-decodable regime that is qualitatively distinct from the conventional non-decodable phase of the surface code under incoherent Pauli errors. By contrast, the metal phase cannot arise in optimal decoding, since the metallic fixed-point becomes unstable in the $n\to 1$ replica limit. We argue that optimal decoding may be possible up to the maximally-coherent rotation angle. Within the sigma model description, we show that the decoding fidelity is related to twist defects of the order-parameter field, yielding quantitative predictions for its system-size dependence near the metallic fixed point for both decoders. We examine our analytic predictions for the decoding fidelity as well as other physical observables with extensive numerical simulations. We discuss how the symmetries and the target space for the sigma model rely on the lattice of the surface code, and how a stable thermal metal phase can arise in optimal decoding when the syndromes reside on a non-bipartite lattice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to provide a microscopic derivation of a non-linear sigma model (NLSM) with target space SO(2n)/U(n) as the effective long-distance theory for maximum-likelihood decoding of the square-lattice surface code under coherent single-qubit unitary rotations. It distinguishes optimal decoding (n→1 replica limit, assuming knowledge of the rotation angle) from suboptimal decoding (n→0 limit with imperfect angle information), arguing that a thermal-metal phase arises only for the suboptimal decoder while remaining unstable for optimal decoding; decoding fidelity is related to twist defects of the order-parameter field, with quantitative predictions checked against extensive numerics. The target space and phase stability are shown to depend on lattice bipartiteness.

Significance. If the central mapping holds, the work supplies a controlled field-theoretic description of coherent-error decoding thresholds in topological codes, cleanly separating optimal and suboptimal maximum-likelihood decoders and linking fidelity to twist-defect observables. This could guide decoder design and clarify why coherent errors behave differently from Pauli noise, especially on bipartite versus non-bipartite lattices.

major comments (3)
  1. [derivation of the effective action] The microscopic derivation of the NLSM action (detailed after the abstract) must explicitly demonstrate that the representation of the likelihood function and the subsequent disorder average introduce no additional relevant operators in the n→1 limit; without these steps it remains possible that the metallic fixed point acquires a stabilizing perturbation even for optimal decoding, contrary to the stated instability.
  2. [replica limits and decoder distinction] The identification of the n→1 replica limit with optimal decoding (knowledge of the coherent angle) and n→0 with suboptimal decoding requires a precise accounting of how the angle information enters the replicated partition function; the current argument leaves open whether the two limits differ only by the target-space symmetry or whether extra averaging terms appear that could alter the RG flow of the metallic fixed point.
  3. [numerical simulations] The numerical verification of fidelity scaling near the metallic fixed point (reported in the simulations section) should include explicit finite-size scaling collapses and data-exclusion criteria; without them it is difficult to confirm that the observed system-size dependence matches the twist-defect prediction rather than crossover effects from the lattice cutoff.
minor comments (2)
  1. [fidelity predictions] The notation for the order-parameter field and its twist defects should be introduced with a short table or diagram to clarify how the decoding fidelity is extracted from the NLSM correlators.
  2. [discussion of target space] A brief comparison paragraph placing the SO(2n)/U(n) target space against the more familiar O(n) or CP^{n-1} models used in prior disordered-system literature would help readers assess the novelty of the symmetry structure.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and valuable comments on our manuscript. We address each of the major comments point by point below, indicating the revisions we plan to make.

read point-by-point responses

  1. Referee: The microscopic derivation of the NLSM action (detailed after the abstract) must explicitly demonstrate that the representation of the likelihood function and the subsequent disorder average introduce no additional relevant operators in the n→1 limit; without these steps it remains possible that the metallic fixed point acquires a stabilizing perturbation even for optimal decoding, contrary to the stated instability.

    Authors: We agree that the derivation can be made more explicit to address this concern. In the revised version, we will include a detailed breakdown of the likelihood function representation and the disorder averaging process. We will demonstrate that in the n→1 limit, the symmetry of the target space SO(2n)/U(n) and the structure of the averaging ensure no additional relevant operators are generated that could stabilize the metallic fixed point. This will be shown through an explicit calculation of the effective action terms and a renormalization group analysis confirming the instability. revision: yes

  2. Referee: The identification of the n→1 replica limit with optimal decoding (knowledge of the coherent angle) and n→0 with suboptimal decoding requires a precise accounting of how the angle information enters the replicated partition function; the current argument leaves open whether the two limits differ only by the target-space symmetry or whether extra averaging terms appear that could alter the RG flow of the metallic fixed point.

    Authors: We will provide a more precise accounting in the revision. Specifically, we will write out the replicated partition function explicitly for both cases. For the n→1 limit corresponding to optimal decoding, the angle is known and fixed, resulting in the standard replica limit without extra disorder terms. For n→0, the imperfect information leads to additional averaging over the angle, which modifies the effective theory. We will show that this distinction is solely in the target space symmetry and does not introduce terms that alter the RG flow in a way that stabilizes the metal for optimal decoding. revision: yes

  3. Referee: The numerical verification of fidelity scaling near the metallic fixed point (reported in the simulations section) should include explicit finite-size scaling collapses and data-exclusion criteria; without them it is difficult to confirm that the observed system-size dependence matches the twist-defect prediction rather than crossover effects from the lattice cutoff.

    Authors: We acknowledge the need for more rigorous presentation of the numerics. In the updated manuscript, we will add explicit finite-size scaling collapses for the fidelity data near the metallic fixed point. Additionally, we will specify the data-exclusion criteria used, such as discarding data from system sizes below a certain threshold where finite-size effects from the lattice cutoff are significant. This will help confirm the agreement with the twist-defect scaling predictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity: microscopic derivation of SO(2n)/U(n) NLSM stands on standard replica techniques and lattice symmetries

full rationale

The paper derives the non-linear sigma model with target space SO(2n)/U(n) directly from the maximum-likelihood decoding problem for coherent rotations on the surface code, using the symmetries of the anyon excitations and the replica trick. The distinct limits n→1 (optimal, angle known) and n→0 (suboptimal, angle unknown) follow from standard replica averaging in disordered systems rather than from any parameter fitted inside the present work or from a self-citation chain that would render the target space or metal-phase stability tautological. No equation reduces a claimed prediction to a quantity defined by the authors' own inputs, and the dependence on lattice bipartiteness is an external geometric fact, not an internal redefinition. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation relies on standard replica-trick techniques and the assumption that the long-distance physics is captured by a sigma model on the given target space. No explicit free parameters are introduced in the abstract; the rotation angle is treated as an input known or unknown to the decoder. The thermal-metal phase is an emergent feature rather than an invented entity with independent evidence.

axioms (2)
  • domain assumption The effective long-distance theory of the decoding problem is a non-linear sigma model with target space SO(2n)/U(n)
    Invoked in the derivation of the effective theory from the microscopic model
  • domain assumption Replica limits n to 1 and n to 0 correctly distinguish optimal and suboptimal maximum-likelihood decoding
    Central to the distinction between the two decoders

pith-pipeline@v0.9.0 · 5611 in / 1698 out tokens · 35683 ms · 2026-05-15T00:22:25.480860+00:00 · methodology

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Forward citations

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Reference graph

Works this paper leans on

119 extracted references · 119 canonical work pages · cited by 1 Pith paper · 4 internal anchors

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    Effective field theory for the suboptimal decoder We consider the suboptimal decoder, in which the estimated rotation angleθ′ is related to the rotation angleθin the surface code by(π/4−θ ′)(1 +ϵ) =π/4−θ. In what follows, we show that the partition functionY0 is described 33 by the field theory of the matrix fieldQ∈SO(2n)/U(n). The fluctuation out of the ...

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    We note that the diagonal matrixDbreaks the symmetry of the path integral fromO(2n+ 2)toO(2)×O(2n). Next, we derive an effective theory by expanding the action around the saddle point. We note that, forϵ > 0, the saddle points are characterized by the anti-symmetric matrix fieldQ, which belongs to the target space Γ1 ×Γ n = Γ n. In the case of the optimal...

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    We analyze the twist expectation values in four different regimes: (1)κ≫1,κ≫1/g R(L); (2)κ≫1,κ≪1/g R(L); (3)κ≪1,1/κ≫1/g R(T); (4)κ≪1,1/κ≪1/g R(T)

    Twist in the RBIM picture In the RBIM picture, the twist is inserted in the tem- poral direction. We analyze the twist expectation values in four different regimes: (1)κ≫1,κ≫1/g R(L); (2)κ≫1,κ≪1/g R(L); (3)κ≪1,1/κ≫1/g R(T); (4)κ≪1,1/κ≪1/g R(T). a.κ≫1 In the case that the aspect ratioκ≫1, we first coarse- grain up to scaleLand obtain an effective sigma mod...

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    Twist in the dual picture The analysis of the twist field correlation in the dual picture is similar to that in the RBIM picture. The only difference is that the twist is inserted in the spatial di- rection. In what follows, we present the results in four regimes. a.κ≫1 We again coarse-grain up to scaleL. In this case, the twist is inserted in the spatial...

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    Scaling of the replicated fidelity We remark on how the twist expectation value de- termines the replicated fidelity of the optimal decoder. First, we consider the surface code with a fixed aspect ratio. The twist expectation values predict how the de- coding fidelity changes while increasing the overall scale. •κ≫1.— The twist expectation value in this r...

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    a.κ≫1,κ≪1/g R In this regime, the replica limit of the decoding fidelity can be obtained in the dual picture

    Fidelity in the replica limit We now analyze the fidelity in the replica limitn→1 in various regimes. a.κ≫1,κ≪1/g R In this regime, the replica limit of the decoding fidelity can be obtained in the dual picture. The twist expectation values in the dual picture (D19) has an upper and a lower bound,8 e−˜ck⩽ ˜Φ2k ⩽e −˜ck+e −˜c(n−k),(E1) where˜c=α n +β n/(κgR...

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    ,(F2) whereG(z)is the “Barnes G-function” defined byG(z) = zΓ(z)forz∈CandG(1) = 1. The calculation ofVolSO(2n)(U(n)), the volume of the embeddingU(n),→SO(2n), is more complicated. We begin by computing the volume ofSU(n)in the met- ric induced by the fundamental representation. Now, SU(n)/SU(n−1) ∼= S2n−1 implies Vol(SU(n)) = r n 2(n−1) ×Vol(S 2n−1)(F3) ×...

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