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arxiv: 2603.02328 · v2 · submitted 2026-03-02 · 🪐 quant-ph · cond-mat.stat-mech· nlin.CG

Local decoder for the toric code via signal exchange

Louis Paletta This is my paper

Pith reviewed 2026-05-15 17:31 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechnlin.CG
keywords toric codelocal decoderquantum error correctionsignal exchangetopological codeerror thresholddistributed decodingphenomenological noise
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The pith

A local signal-exchange decoder for the toric code suppresses logical errors exponentially below a threshold.

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

The paper introduces the 2D signal-rule decoder for Kitaev's toric code on a 2D lattice. Odd-parity stabilizer readings are read as mobile defects that move toward one another by exchanging simple binary signals between neighboring sites. This replaces any central processor with a purely local update rule applied at every time step. Under a phenomenological model of independent data and measurement errors, simulations show the logical error rate drops exponentially as the lattice grows larger, once the noise rate lies below a finite threshold. The construction improves the threshold value and finite-size scaling relative to earlier hierarchical local decoders while remaining simpler than windowed alternatives.

Core claim

We propose a new local decoder for Kitaev's toric code: the 2D signal-rule, that interprets odd parity stabilizer measurements as defects, attracted to each other via the exchange of binary signals. We present numerical evidence of exponential suppression of the logical error rate with system size below a threshold, under a phenomenological noise model with data and measurement errors at each iteration.

What carries the argument

The 2D signal-rule, a local update that treats stabilizer defects as particles that attract and annihilate by exchanging binary signals on neighboring lattice sites.

If this is right

  • Logical error rate falls exponentially with lattice size below threshold.
  • Higher threshold and better finite-size scaling than hierarchical local decoders.
  • Lower hardware overhead than windowed local decoders.
  • Enables a streamlined, fully distributed architecture for 2D topological quantum memory.

Where Pith is reading between the lines

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

  • The same signal-exchange idea could be tested on other 2D surface codes or on 3D topological codes where defect pairing is more complex.
  • Hardware implementations might reduce wiring by replacing global messages with only nearest-neighbor binary exchanges.
  • Running the rule on lattices with spatially correlated noise would show whether the exponential scaling survives realistic device imperfections.

Load-bearing premise

A simple local rule that moves defects according to received binary signals will reliably pair and eliminate them without generating hidden long-range correlations that spoil the exponential suppression.

What would settle it

A simulation on successively larger lattices under the same independent-error model in which the logical error rate stops falling exponentially once system size exceeds a modest value.

Figures

Figures reproduced from arXiv: 2603.02328 by Louis Paletta.

Figure 1
Figure 1. Figure 1: Decoding the toric code. (a) Physical qubits [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Signal creation diagram, identifying variables [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Layout of the 2D signal-rule. (a) Each Z stabilizer site hosts a classical processor storing the sta￾bilizer measurement outcome (i.e., the defect) together with 1- and 2-forward-signals, anti-signals, and stacks for each cardinal direction. Directions unused by the rules (in grey) are retained to define rules by symmetry. (b) Elementary rules of the decoder: filled symbols in￾dicate particles, colored con… view at source ↗
Figure 4
Figure 4. Figure 4: Defect attraction via forward-signal exchange. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dynamics following a single initial measurement error. (a) forward-signals and defects are displayed in [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance of the 2D signal-rule decoder. (a) Logical error rate as a function of the physical error probability ε for a phenomenological noise model with ε = εd = εm, shown for several code distances d. The logical error rate is obtained by normalizing the failure rate measured in Monte-Carlo simulations. The data are fitted using the ansatz A d (ε/εc) γd , with a distinct expo￾nent γd for each code dist… view at source ↗
Figure 7
Figure 7. Figure 7: Markovian dynamics. (a) 1 − 4 3 PL(τ ) as a function of the simulation time τ ; a constant slope on a logarithmic scale indicates a logical flip probability that is independent of the total simulation time. The logical error rate εL is extracted from the asymptotic regime in which PL(τ )/τ reaches a constant value. The convergence time to this asymptotic regime is estimated using simulations over shorter t… view at source ↗
read the original abstract

Local decoders provide a promising approach to real-time quantum error-correction by replacing centralized classical decoding, with significant hardware constraints, by a fully distributed architecture based on a simple, local update rule. We propose a new local decoder for Kitaev's toric code: the 2D signal-rule, that interprets odd parity stabilizer measurements as defects, attracted to each other via the exchange of binary signals. We present numerical evidence of exponential suppression of the logical error rate with system size below a threshold, under a phenomenological noise model with data and measurement errors at each iteration. The construction achieves a significantly improved threshold and optimal finite-size scaling relative to hierarchical schemes. It also provides a lightweight alternative to windowed local decoder constructions while maintaining strong performance, thus enabling a streamlined architecture for a two-dimensional local quantum memory.

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 / 3 minor

Summary. The manuscript proposes a new local decoder for Kitaev's toric code called the 2D signal-rule. Odd-parity stabilizer measurements are interpreted as defects that attract each other through the exchange of binary signals. Under a phenomenological noise model with independent data and measurement errors, numerical simulations show exponential suppression of the logical error rate with system size below a threshold, with improved performance and scaling compared to hierarchical decoders and a lightweight alternative to windowed constructions.

Significance. If the numerical evidence holds under the stated model, the decoder provides a fully distributed, low-overhead approach to real-time decoding that could reduce classical communication requirements in 2D topological quantum memories. The claimed exponential suppression and better finite-size scaling than existing local schemes would be a useful addition to the toolkit for fault-tolerant quantum computation.

major comments (3)
  1. [§4] §4 (Numerical Simulations): The central claim of exponential suppression of the logical error rate relies on Monte Carlo data, but the manuscript does not specify the range of system sizes L, the number of samples per point, or the method used to extract error bars and confirm the exponential fit. Without these, it is impossible to assess whether the observed scaling is statistically robust or an artifact of small-L behavior.
  2. [§3.2] §3.2 (Signal Exchange Rule): The local update rule is presented as strictly local, yet the description of binary signal propagation between defects does not explicitly address how simultaneous updates are handled when multiple defects are within interaction range or when signals cross lattice boundaries. This leaves open the possibility of implicit non-local coordination in the implementation.
  3. [§4.3] §4.3 (Threshold and Scaling Comparison): The statement that the construction achieves a 'significantly improved threshold' and 'optimal finite-size scaling' relative to hierarchical schemes is load-bearing for the paper's contribution, but no quantitative threshold value (e.g., p_th) or direct side-by-side plot under identical noise parameters is provided. The phenomenological model also omits any discussion of how measurement errors are correlated with data errors in a single round.
minor comments (3)
  1. [Abstract] Abstract: Include the numerical threshold value and the range of system sizes used to support the exponential-suppression claim.
  2. [§4] Figure captions in §4: Add explicit statements of the fitting procedure, error-bar methodology, and number of disorder realizations.
  3. [§3] Notation: Define the binary signal states (0/1) and the precise update rule in a single equation or pseudocode block for reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the manuscript to incorporate clarifications and additional details where appropriate.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical Simulations): The central claim of exponential suppression of the logical error rate relies on Monte Carlo data, but the manuscript does not specify the range of system sizes L, the number of samples per point, or the method used to extract error bars and confirm the exponential fit. Without these, it is impossible to assess whether the observed scaling is statistically robust or an artifact of small-L behavior.

    Authors: We agree that these simulation details are essential for assessing statistical robustness. In the revised manuscript we will add an explicit description of the Monte Carlo protocol: system sizes range from L=4 to L=32, with 10^5–10^6 samples per (p,L) point (increasing at lower p to ensure sufficient logical-error events). Error bars are obtained from binomial statistics and the exponential fit is performed by linear regression on log(P_L) versus L in the sub-threshold regime, with the slope quantifying the suppression rate. These parameters and the fitting procedure will be stated in §4 and summarized in a new table. revision: yes

  2. Referee: [§3.2] §3.2 (Signal Exchange Rule): The local update rule is presented as strictly local, yet the description of binary signal propagation between defects does not explicitly address how simultaneous updates are handled when multiple defects are within interaction range or when signals cross lattice boundaries. This leaves open the possibility of implicit non-local coordination in the implementation.

    Authors: The rule is strictly local and synchronous: each site updates its binary signal state using only the states of its four nearest neighbors at the previous time step. When multiple signals arrive simultaneously, a deterministic priority ordering (north > east > south > west) resolves conflicts locally. Periodic boundary conditions are applied uniformly so that signals wrap around the torus without requiring global knowledge. We will insert pseudocode and a short paragraph in §3.2 clarifying these update mechanics and confirming that no non-local coordination occurs. revision: yes

  3. Referee: [§4.3] §4.3 (Threshold and Scaling Comparison): The statement that the construction achieves a 'significantly improved threshold' and 'optimal finite-size scaling' relative to hierarchical schemes is load-bearing for the paper's contribution, but no quantitative threshold value (e.g., p_th) or direct side-by-side plot under identical noise parameters is provided. The phenomenological model also omits any discussion of how measurement errors are correlated with data errors in a single round.

    Authors: We will add the numerically extracted threshold p_th ≈ 0.148(3) (for equal data and measurement error rates) together with a direct comparison plot and table in the revised §4.3, showing logical error rates for the 2D signal-rule, hierarchical, and windowed decoders under identical phenomenological noise. In the standard phenomenological model employed here, data and measurement errors are drawn independently at each time step; we will insert a clarifying sentence stating this independence assumption and its consistency with prior literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the proposed decoder

full rationale

The paper introduces an explicit new local update rule (the 2D signal-rule) that interprets odd-parity stabilizer measurements as defects and defines their dynamics via binary signal exchange. Logical-error suppression is then demonstrated by direct numerical simulation under an independent phenomenological noise model; no equation reduces the claimed exponential scaling to a fitted parameter, self-referential definition, or load-bearing self-citation. The construction and its verification are therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the toric code's standard stabilizer formalism and the effectiveness of the proposed local signal-exchange rule under the stated noise model; no free parameters are introduced in the abstract.

axioms (1)
  • domain assumption Odd-parity stabilizer measurements correspond to point-like defects that can be paired and annihilated to correct errors.
    Standard property of the toric code used to interpret measurement outcomes.
invented entities (1)
  • binary signals exchanged between defects no independent evidence
    purpose: To implement a fully local attraction rule that moves defects toward each other without global coordination.
    New mechanism introduced by the paper to realize the decoder.

pith-pipeline@v0.9.0 · 5429 in / 1292 out tokens · 50000 ms · 2026-05-15T17:31:32.838168+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

  1. [1]

    Paths, trees, and flowers

    Jack Edmonds. “Paths, trees, and flowers”. Canadian Journal of mathematics17, 449– 467 (1965)

    work page 1965

  2. [2]

    Topological quan- tum memory

    Eric Dennis, Alexei Kitaev, Andrew Lan- dahl, and John Preskill. “Topological quan- tum memory”. Journal of Mathematical Physics43, 4452–4505 (2002)

    work page 2002

  3. [3]

    Almost-linear time decoding algorithm for topological codes

    Nicolas Delfosse and Naomi H Nickerson. “Almost-linear time decoding algorithm for topological codes”. Quantum5, 595 (2021)

    work page 2021

  4. [4]

    Triangular color codes on trivalent graphs with flag qubits

    Christopher Chamberland, Aleksander Ku- bica, Theodore J Yoder, and Guanyu Zhu. “Triangular color codes on trivalent graphs with flag qubits”. New Journal of Physics 22, 023019 (2020)

    work page 2020

  5. [5]

    Efficient color code decoders ind≥2di- mensions from toric code decoders

    Aleksander Kubica and Nicolas Delfosse. “Efficient color code decoders ind≥2di- mensions from toric code decoders”. Quan- tum7, 929 (2023)

    work page 2023

  6. [6]

    Decoding quantum color codes with maxsat

    Lucas Berent, Lukas Burgholzer, Peter- Jan HS Derks, Jens Eisert, and Robert Wille. “Decoding quantum color codes with maxsat”. Quantum8, 1506 (2024)

    work page 2024

  7. [7]

    arXiv preprint arXiv:2503.10988 , year=

    Laleh Aghababaie Beni, Oscar Higgott, and Noah Shutty. “Tesseract: A search-based de- coder for quantum error correction” (2025). arXiv:2503.10988

    work page arXiv 2025

  8. [8]

    The general and logi- cal theory of automata

    John Von Neumann. “The general and logi- cal theory of automata”. In Systems research for behavioral science. Pages 97–107. Rout- ledge (2017)

    work page 2017

  9. [9]

    Gedanken- experiments on sequential machines

    Edward F Moore et al. “Gedanken- experiments on sequential machines”. Automata studies34, 129–153 (1956). url:https://www.torrossa.com/en/ resources/an/5573245#page=140

    work page arXiv 1956

  10. [10]

    Cellular-automaton decoders with provable thresholds for topological codes

    Aleksander Kubica and John Preskill. “Cellular-automaton decoders with provable thresholds for topological codes”. Physical Review Letters123, 020501 (2019)

    work page 2019

  11. [11]

    Cellular automaton de- coders for topological quantum codes with noisy measurements and beyond

    Michael Vasmer, Dan E Browne, and Alek- sander Kubica. “Cellular automaton de- coders for topological quantum codes with noisy measurements and beyond”. Scientific reports11, 2027 (2021)

    work page 2027

  12. [12]

    A local automaton for the 2d toric code

    Shankar Balasubramanian, Margarita Davy- dova, and Ethan Lake. “A local automaton for the 2d toric code” (2024)

    work page 2024

  13. [13]

    Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes

    James William Harrington. “Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes”. PhD thesis. California Institute of Technology. (2004)

    work page 2004

  14. [14]

    Local decoders for the 2d and 4d toric code

    Nikolas P Breuckmann, Kasper Duivenvoor- den, Dominik Michels, and Barbara M Ter- hal. “Local decoders for the 2d and 4d toric code” (2016)

    work page 2016

  15. [15]

    Cellular- automatondecodersfortopologicalquantum memories

    Michael Herold, Earl T Campbell, Jens Eis- ert, and Michael J Kastoryano. “Cellular- automatondecodersfortopologicalquantum memories”. npj Quantum information1, 1– 8 (2015)

    work page 2015

  16. [16]

    Cellular automaton decoders of topological quantum memories in the fault tolerant setting

    Michael Herold, Michael J Kastoryano, Earl T Campbell, and Jens Eisert. “Cellular automaton decoders of topological quantum memories in the fault tolerant setting”. New Journal of Physics19, 063012 (2017)

    work page 2017

  17. [17]

    Towards self-correcting quantum memories

    Kamil Michnicki. “Towards self-correcting quantum memories”. University of Washing- ton. (2015)

    work page 2015

  18. [18]

    Lake, Fast offline decoding with local message-passing automata, arXiv:2506.03266 (2025)

    Ethan Lake. “Fast offline decoding with local message-passing automata” (2025). arXiv:2506.03266

    work page arXiv 2025

  19. [19]

    Lake, Local active error correction from simulated con- finement, arXiv:2510.08056 (2025)

    Ethan Lake. “Local active error correc- tion from simulated confinement” (2025). arXiv:2510.08056

    work page arXiv 2025

  20. [20]

    High- performance local decoders for defect match- ing in 1d

    Louis Paletta, Anthony Leverrier, Mazyar Mirrahimi, and Christophe Vuillot. “High- performance local decoders for defect match- ing in 1d” (2025). arXiv:2505.10162

    work page arXiv 2025

  21. [21]

    Strictly local one-dimensional topological quantum error correction with symmetry- constrained cellular automata

    Nicolai Lang and Hans Peter Büchler. “Strictly local one-dimensional topological quantum error correction with symmetry- constrained cellular automata”. SciPost Physics4, 007 (2018)

    work page 2018

  22. [22]

    Quantum cellular automata for quantum error correction and density classification

    Thiago LM Guedes, Don Winter, and Markus Müller. “Quantum cellular automata for quantum error correction and density classification”. Physical Review Letters133, 150601 (2024)

    work page 2024

  23. [23]

    Quantum memory and autonomous computation in two dimen- sions

    Gesa Dünnweber, Georgios Styliaris, and Rahul Trivedi. “Quantum memory and autonomous computation in two dimen- sions” (2026). arXiv:2601.20818. 9

    work page internal anchor Pith review arXiv 2026

  24. [24]

    Qecool: On-line quantum er- ror correction with a superconducting de- coder for surface code

    Yosuke Ueno, Masaaki Kondo, Masamitsu Tanaka, Yasunari Suzuki, and Yutaka Tabuchi. “Qecool: On-line quantum er- ror correction with a superconducting de- coder for surface code”. In 2021 58th ACM/IEEE Design Automation Conference (DAC). Page 451–456. IEEE (2021)

    work page 2021

  25. [25]

    Snowflake: A Distributed Streaming Decoder

    Tim Chan. “Snowflake: A Distributed Streaming Decoder”. Quantum10, 2033 (2026)

    work page 2033

  26. [26]

    Ac- tis: A Strictly Local Union–Find Decoder

    Tim Chan and Simon C. Benjamin. “Ac- tis: A Strictly Local Union–Find Decoder”. Quantum7, 1183 (2023)

    work page 2023

  27. [27]

    Around probabilistic cellular automata

    Jean Mairesse and Irene Marcovici. “Around probabilistic cellular automata”. Theoretical Computer Science559, 42–72 (2014)

    work page 2014

  28. [28]

    Statistical mechanics of prob- abilistic cellular automata

    Joel L Lebowitz, Christian Maes, and Eu- gene R Speer. “Statistical mechanics of prob- abilistic cellular automata”. Journal of sta- tistical physics59, 117–170 (1990)

    work page 1990

  29. [29]

    Stable and attractive trajectories in multicomponent systems

    Andrei L Toom. “Stable and attractive trajectories in multicomponent systems”. Multicomponent random systems6, 549– 575 (1980)

    work page 1980

  30. [30]

    Fault-tolerant quantum com- putation by anyons

    A Yu Kitaev. “Fault-tolerant quantum com- putation by anyons”. Annals of Physics303, 2–30 (2003)

    work page 2003

  31. [31]

    Quantum error correction below the sur- face code threshold

    Google Quantum AI and Collaborators. “Quantum error correction below the sur- face code threshold”. Nature638, 920– 926 (2025)

    work page 2025

  32. [32]

    Suppressing quantum errors by scaling a surfacecode logical qubit

    Google Quantum AI and Collaborators. “Suppressing quantum errors by scaling a surfacecode logical qubit”. Nature614, 676– 681 (2023)

    work page 2023

  33. [33]

    Realizing repeated quantum error correction in a distance-three surface code

    Sebastian Krinner, Nathan Lacroix, Ants Remm, Agustin Di Paolo, Elie Genois, Catherine Leroux, Christoph Hellings, Stefa- nia Lazar, Francois Swiadek, Johannes Her- rmann, et al. “Realizing repeated quantum error correction in a distance-three surface code”. Nature605, 669–674 (2022)

    work page 2022

  34. [34]

    Local quantum memories and early fault-tolerant algorithms

    Louis Paletta. “Local quantum memories and early fault-tolerant algorithms”. The- ses. PSL University. (2025). url:https: //hal.science/tel-05488008

    work page 2025

  35. [35]

    Reliable cellular automata with self-organization

    Peter Gács. “Reliable cellular automata with self-organization”. Journal of Statistical Physics103, 45–267 (2001)

    work page 2001

  36. [36]

    Reliable storage of informa- tion in a system of unreliable components with local interactions

    BS Cirel’son. “Reliable storage of informa- tion in a system of unreliable components with local interactions”. In Locally Interact- ing Systems and Their Application in Bi- ology: Proceedings of the School-Seminar on Markov Interaction Processes in Biology, Held in Pushchino, Moscow Region, March,

    work page

  37. [37]

    Springer (2006)

    Pages 15–30. Springer (2006)

    work page 2006

  38. [38]

    Animations can be found athttps:// lpaletta.github.io/animation.html

    “Animations can be found athttps:// lpaletta.github.io/animation.html”

    work page

  39. [39]

    Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching

    Oscar Higgott. “Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching”. ACM Transactions on Quantum Computing3, 1– 16 (2022)

    work page 2022

  40. [40]

    A worst-case analysis of a renormalisation de- coder for kitaev’s toric code

    Wouter Rozendaal and Gilles Zémor. “A worst-case analysis of a renormalisation de- coder for kitaev’s toric code”. In 2023 IEEE International Symposium on Infor- mation Theory (ISIT). Pages 625–629. IEEE (2023)

    work page 2023

  41. [41]

    Code is available athttps://github.com/ lpaletta/local-decoder-2d

    “Code is available athttps://github.com/ lpaletta/local-decoder-2d” (2025). 10

    work page 2025