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arxiv: 2607.01364 · v1 · pith:KRI6YXVYnew · submitted 2026-07-01 · 🪐 quant-ph

Decohered toric code under quantum damping noise and its mapping to a classical spin model

Nihar Ranjan Dash , Sanjoy Dutta , R. Srikanth , Subhashish Banerjee This is my paper

Pith reviewed 2026-07-03 20:08 UTC · model grok-4.3

classification 🪐 quant-ph
keywords toric codegeneralized amplitude dampingsqueezed generalized amplitude dampingPauli twirlingstatistical mechanical modelslogical failure probabilityquantum damping noise
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The pith

Decohered toric codes under generalized amplitude-damping channels map to statistical-mechanical models where channel parameters set the spin couplings and logical failure rates depend on temperature and squeezing.

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

The paper maps the action of generalized amplitude-damping and squeezed generalized amplitude-damping channels on the toric code to equivalent statistical-mechanical models. It does this first through the double Hilbert-space formalism and second by applying Pauli twirling to obtain asymmetric depolarizing channels. Logical failure probabilities are then expressed directly in terms of the channel's temperature and squeezing parameters, with those parameters tied to the coupling constants of the resulting spin model. A sympathetic reader would care because the mapping lets standard tools from classical statistical mechanics be used to analyze error rates under realistic damping noise that includes thermal and non-Markovian effects.

Core claim

The action of the GAD and SGAD channels on the toric code is mapped to statistical-mechanical models using the double Hilbert-space formalism, and via Pauli twirling to stochastic Pauli-type errors yielding asymmetric depolarizing channels, with channel parameters related to spin-coupling constants and logical failure probabilities obtained as a function of temperature and squeezing.

What carries the argument

Double Hilbert-space formalism that converts the quantum damping channels into classical spin models, together with Pauli twirling that produces the asymmetric depolarizing error rates.

If this is right

  • Logical failure probabilities are obtained as explicit functions of temperature and squeezing.
  • The parameters of the GAD and SGAD channels become the spin-coupling constants of the equivalent statistical model.
  • The toric code under these channels can be analyzed with classical statistical-mechanics techniques for both the full mapping and the twirled error model.
  • Both the GAD and SGAD channels admit such mappings, allowing comparison of their effects on logical stability.

Where Pith is reading between the lines

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

  • The mapping could be used to estimate error thresholds for toric codes in environments dominated by squeezed thermal noise.
  • Direct comparison of the analytic failure-rate curves against small-code numerics would test the accuracy of the twirling step.
  • Similar mappings might apply to other topological codes or to different non-Markovian damping models.

Load-bearing premise

The double Hilbert-space formalism and Pauli twirling capture the full effect of the non-Markovian and squeezed damping channels without introducing uncontrolled approximations that change the logical failure rates.

What would settle it

A direct simulation of logical error rates on a small toric code under the SGAD channel at fixed temperature but varying squeezing strength that deviates from the temperature-and-squeezing dependence predicted by the mapped spin model.

Figures

Figures reproduced from arXiv: 2607.01364 by Nihar Ranjan Dash, R. Srikanth, Sanjoy Dutta, Subhashish Banerjee.

Figure 1
Figure 1. Figure 1: FIG. 1: (Color online) The 2D toric code with star and pla [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Color online) Coupling constants for the GAD chan [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online) Coupling constants for the PT-GAD [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (Color online) Coupling constants for the PT [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Plot of logical failure probabilities as a function of time [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

We investigate properties of toric codes under realistic damping error channels, which include squeezing, thermal and non-Markovian effects. First, we map the decohered toric code under the generalized amplitude-damping (GAD) and the squeezed generalized amplitude-damping (SGAD) channels to the statistical-mechanical models using the double Hilbert-space formalism. Second, we map the action of the GAD and SGAD channels on the toric code to stochastic Pauli-type errors via Pauli twirling, yielding asymmetric depolarizing channels, and obtain the logical failure probabilities as a function of temperature and squeezing. In both cases, we relate the channel parameters of the GAD and SGAD channels to the spin-coupling constants of the statistical-mechanical model.

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

2 major / 1 minor

Summary. The paper maps the toric code under generalized amplitude-damping (GAD) and squeezed generalized amplitude-damping (SGAD) channels to classical statistical-mechanical models via the double Hilbert-space formalism. It also applies Pauli twirling to obtain asymmetric depolarizing channels, relates the damping-channel parameters to spin-coupling constants, and computes logical failure probabilities as functions of temperature and squeezing.

Significance. If the mappings hold without uncontrolled approximations, the work would connect realistic damping noise (including squeezing and non-Markovian effects) on a topological code to classical spin models, enabling computation of logical error rates via established statistical-mechanics techniques. The explicit parameter mapping and functional dependence on squeezing would be a concrete contribution.

major comments (2)
  1. [Abstract (mapping via Pauli twirling)] The Pauli-twirling step that replaces the SGAD channel by an asymmetric depolarizing channel (described in the abstract) averages over phase-sensitive operators. For a squeezed bath these coherences can survive stabilizer measurements and couple to the logical subspace differently than the twirled Pauli errors; no bound is given showing that the resulting logical failure probability remains accurate to the order of the reported squeezing dependence. This is load-bearing for the central claim that logical failure probabilities are obtained as a function of squeezing.
  2. [Abstract (double Hilbert-space formalism)] The double-Hilbert-space construction for the non-Markovian GAD/SGAD case likewise assumes a specific bath-correlation structure. The manuscript does not provide an error bound demonstrating that this structure preserves the logical-subspace dynamics to the precision needed for the claimed temperature and squeezing dependence.
minor comments (1)
  1. [Abstract] Notation for the spin-coupling constants obtained from the channel parameters should be introduced with an explicit equation relating each damping parameter to each coupling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising these points concerning the approximations underlying our mappings. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract (mapping via Pauli twirling)] The Pauli-twirling step that replaces the SGAD channel by an asymmetric depolarizing channel (described in the abstract) averages over phase-sensitive operators. For a squeezed bath these coherences can survive stabilizer measurements and couple to the logical subspace differently than the twirled Pauli errors; no bound is given showing that the resulting logical failure probability remains accurate to the order of the reported squeezing dependence. This is load-bearing for the central claim that logical failure probabilities are obtained as a function of squeezing.

    Authors: The Pauli twirling is used as a distinct second approach to obtain an effective asymmetric depolarizing channel whose parameters are related to temperature and squeezing; the logical failure probabilities are then computed for this effective channel via the spin-model mapping. The abstract presents this as a separate procedure from the double-Hilbert-space construction. While we acknowledge that the twirling neglects coherences that could in principle survive, the reported squeezing dependence is that of the twirled model. We will revise the manuscript to state this scope explicitly and to add a qualitative discussion of why the leading squeezing dependence is expected to be representative for the toric-code logical subspace. revision: partial

  2. Referee: [Abstract (double Hilbert-space formalism)] The double-Hilbert-space construction for the non-Markovian GAD/SGAD case likewise assumes a specific bath-correlation structure. The manuscript does not provide an error bound demonstrating that this structure preserves the logical-subspace dynamics to the precision needed for the claimed temperature and squeezing dependence.

    Authors: The double-Hilbert-space formalism supplies an exact mapping of the decoherence process onto a classical spin model once the bath-correlation structure of the GAD/SGAD channels is fixed. Temperature and squeezing enter directly as parameters of the resulting spin couplings, and the logical failure probabilities follow from this mapping without further approximation. The manuscript states the bath-correlation assumptions; within those assumptions the claimed dependences are exact. We will revise the text to reiterate that the results hold exactly for the assumed structure. revision: partial

Circularity Check

0 steps flagged

No circularity detected; mappings rely on external formalisms without self-referential reductions

full rationale

The abstract and description present two mappings (double Hilbert-space to statistical-mechanical models; Pauli twirling to asymmetric depolarizing channels) that relate channel parameters to spin couplings and yield logical failure probabilities as functions of temperature and squeezing. No equations, self-citations, or fitted-parameter steps are quoted that reduce any claimed prediction to its own inputs by construction. The derivation chain is therefore self-contained against external benchmarks (standard channel mappings and twirling) and receives the default non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted. The double Hilbert-space formalism and Pauli twirling are invoked but their status as standard versus ad-hoc is unknown.

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