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arxiv: 2505.10693 · v1 · submitted 2025-05-15 · 🪐 quant-ph

Distributed Realization of Color Codes for Quantum Error Correction

Nitish Kumar Chandra , David Tipper , Reza Nejabati , Eneet Kaur , Kaushik P. Seshadreesan This is my paper

Pith reviewed 2026-05-22 14:13 UTC · model grok-4.3

classification 🪐 quant-ph
keywords color codesquantum error correctiondistributed quantum computingerror thresholdasymmetric noiseMWPM decodertensor network decoder
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The pith

Color codes distributed across quantum processors retain error thresholds under uneven noise with suitable decoding.

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

This work explores realizing color codes for quantum error correction by linking separate patches across different processors with entangled connections. The connections create boundary seams where qubits face higher noise levels than in the interior. Using computer simulations, the authors test how this uneven noise affects the code's ability to correct errors. They find that a decoder based on concatenated matching shows almost no drop in the tolerable noise level, while a tensor network decoder sees a small decrease. The result points to color codes being practical for building fault-tolerant systems from multiple smaller quantum devices.

Core claim

By interconnecting patches of the (6.6.6) color code via entangled pairs and modeling higher bit-flip noise on seam qubits, the paper establishes that the concatenated minimum weight perfect matching decoder maintains the error threshold without notable change under asymmetric noise conditions.

What carries the argument

The seam qubits at patch boundaries subject to elevated bit-flip noise in a distributed color code, decoded using tensor-network methods or concatenated MWPM.

If this is right

  • The concatenated MWPM decoder remains effective even with asymmetric noise from interconnects.
  • Color codes support distributed fault-tolerant quantum computing with noisy links between units.
  • The tensor-network decoder experiences only minor threshold reduction from seam noise.
  • Distributed architectures can realize large color codes without major threshold penalties.

Where Pith is reading between the lines

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

  • This modeling of interconnect noise could be tested in other code families like surface codes for comparison.
  • Reducing entanglement noise might allow even better performance in multi-processor setups.
  • The robustness suggests scaling to more patches could be feasible for larger logical qubits.

Load-bearing premise

The effects of noisy interconnects between patches are accurately represented by increasing only the bit-flip error rate on boundary seam qubits.

What would settle it

Running simulations with noise applied uniformly versus only on seams and observing whether the threshold difference matches the reported values for each decoder.

Figures

Figures reproduced from arXiv: 2505.10693 by David Tipper, Eneet Kaur, Kaushik P. Seshadreesan, Nitish Kumar Chandra, Reza Nejabati.

Figure 1
Figure 1. Figure 1: (a) A triangular color code constructed on a hexagonal lattice with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The spatial separation between QPUs necessitates the use of shared [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Syndrome extraction circuits for Z-type (left) and X-type (right) stabilizers in the color code. Each circuit uses an ancilla qubit to interact with six surrounding data qubits via CNOT gates. Two of the CNOTs are nonlocal, indicated by blue dotted lines, while the remaining are local gates confined within a single QPU. the left shows the process for measuring a Z-type stabilizer. The central ancilla qubit… view at source ↗
Figure 5
Figure 5. Figure 5: Threshold plot for the 6.6.6 color code under a uniform bit-flip noise [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Threshold plot for the 6.6.6 color code under an asymmetric [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Logical failure rate vs physical error rate under symmetric noise [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Threshold fitting region for the symmetric noise case (p [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Logical failure rate vs physical error rate under asymmetric noise [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Threshold fitting region for the asymmetric noise case (p [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
read the original abstract

Color codes are a leading class of topological quantum error-correcting codes with modest error thresholds and structural compatibility with two-dimensional architectures, which make them well-suited for fault-tolerant quantum computing (FTQC). Here, we propose and analyze a distributed architecture for realizing the (6.6.6) color code. The architecture involves interconnecting patches of the color code housed in different quantum processing units (QPUs) via entangled pairs. To account for noisy interconnects, we model the qubits in the color code as being subject to a bit-flip noise channel, where the qubits on the boundary (seam) between patches experience elevated noise compared to those in the bulk. We investigate the error threshold of the distributed color code under such asymmetric noise conditions by employing two decoders: a tensor-network-based decoder and a recently introduced concatenated Minimum Weight Perfect Matching (MWPM) algorithm. Our simulations demonstrate that elevated noise on seam qubits leads to a slight reduction in threshold for the tensor-network decoder, whereas the concatenated MWPM decoder shows no significant change in the error threshold, underscoring its effectiveness under asymmetric noise conditions. Our findings thus highlight the robustness of color codes in distributed architectures and provide valuable insights into the practical realization of FTQC involving noisy interconnects between QPUs.

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 / 2 minor

Summary. The manuscript proposes a distributed architecture for realizing the (6.6.6) color code by interconnecting patches housed in separate QPUs via entangled pairs. Noisy interconnects are modeled by applying an elevated bit-flip noise channel exclusively to boundary seam qubits while bulk qubits experience standard noise. Error thresholds are computed via Monte Carlo simulations for two decoders—a tensor-network decoder and a concatenated MWPM decoder—under this asymmetric noise. The central result is that increased seam noise produces a slight threshold reduction for the tensor-network decoder but no significant change for the concatenated MWPM decoder.

Significance. If the noise model and simulation results hold, the work demonstrates that color codes remain viable in distributed multi-QPU settings and that the concatenated MWPM decoder is particularly robust to boundary-dominated noise. The explicit comparison of two decoder families and the focus on asymmetric noise provide concrete guidance for FTQC architectures. The numerical threshold extraction itself constitutes a falsifiable prediction that can be tested against more detailed physical interconnect models.

major comments (2)
  1. [Modeling section (abstract and §III)] Modeling section (abstract and §III): the claim that interconnect noise is captured by an elevated bit-flip channel applied only to seam qubits is load-bearing for all threshold conclusions, yet the model omits correlated two-qubit errors, phase noise on virtual links, and loss channels that arise in entanglement generation, distribution, and fusion. If these additional channels are present at comparable rates, both decoders’ thresholds could shift in ways not captured by the reported runs.
  2. [Results section (abstract and §V)] Results section (abstract and §V): no simulation parameters, decoder implementation details, Monte Carlo sample counts, or statistical error bars on the reported thresholds are provided. Without these, it is impossible to assess whether the “slight reduction” for the tensor-network decoder and “no significant change” for concatenated MWPM are statistically distinguishable from zero or from each other.
minor comments (2)
  1. [Abstract] Abstract: the phrase “elevated noise on seam qubits” should be accompanied by the explicit ratio or functional form used for the seam-to-bulk noise rate.
  2. [Figure captions (throughout)] Figure captions (throughout): ensure all threshold plots include both the fitted curves and the raw data points with error bars so that the “no significant change” statement can be visually verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and describe the changes we will incorporate in the revised version.

read point-by-point responses

  1. Referee: Modeling section (abstract and §III): the claim that interconnect noise is captured by an elevated bit-flip channel applied only to seam qubits is load-bearing for all threshold conclusions, yet the model omits correlated two-qubit errors, phase noise on virtual links, and loss channels that arise in entanglement generation, distribution, and fusion. If these additional channels are present at comparable rates, both decoders’ thresholds could shift in ways not captured by the reported runs.

    Authors: We agree that the noise model is a deliberate simplification chosen to isolate the impact of elevated seam noise arising from interconnects. The bit-flip channel on boundary qubits is intended to represent the dominant error mechanism in this initial study rather than a complete physical model of entanglement distribution. In the revised manuscript we will expand Section III with an explicit discussion of the model assumptions, including the omission of correlated errors, phase noise, and loss channels, and we will add a forward-looking statement that more detailed interconnect models remain an important topic for subsequent work. This clarification will not alter the numerical results but will better contextualize their scope. revision: yes

  2. Referee: Results section (abstract and §V): no simulation parameters, decoder implementation details, Monte Carlo sample counts, or statistical error bars on the reported thresholds are provided. Without these, it is impossible to assess whether the “slight reduction” for the tensor-network decoder and “no significant change” for concatenated MWPM are statistically distinguishable from zero or from each other.

    Authors: The referee is correct that the current manuscript does not supply the requested numerical details. We will revise Section V to include a new subsection that reports the Monte Carlo sample counts, decoder hyperparameters (including tensor-network bond dimension and MWPM concatenation depth), the precise method used for threshold extraction, and statistical error bars on all threshold values. These additions will allow readers to evaluate the statistical significance of the observed differences between the two decoders under asymmetric seam noise. revision: yes

Circularity Check

0 steps flagged

No significant circularity: thresholds from direct simulations under explicit noise model

full rationale

The paper selects an explicit modeling ansatz (elevated bit-flip noise applied only to seam qubits) to represent interconnect errors and then reports numerical thresholds obtained by running two decoders on instances generated from that model. These thresholds are computed outputs of the simulation procedure rather than quantities fitted to or predicted from the same data set. No self-definitional reduction, fitted-input-called-prediction, load-bearing self-citation, or ansatz smuggled via prior work appears in the derivation chain. The results remain self-contained numerical experiments on the stated noise model and are therefore not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work rests on standard assumptions from quantum error correction literature about noise channels and decoder performance; no new free parameters or invented entities are introduced beyond the seam-noise model.

pith-pipeline@v0.9.0 · 5768 in / 978 out tokens · 28968 ms · 2026-05-22T14:13:32.426210+00:00 · methodology

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Boundary-Aware Stabilizer Scheduling for Distributed Quantum Error Correction

    quant-ph 2026-04 unverdicted novelty 6.0

    SS-τ and AST scheduling policies for seam checks in distributed triangular color codes reduce remote-operation overhead and achieve lower logical error rates with fault-tolerant scaling in specific EGR regimes under c...

  2. Near-Term Reduction in Nonlocal Gate Count from Distributed Logical Qubits

    quant-ph 2026-04 unverdicted novelty 5.0

    Qubit allocation techniques for distributed color-code logical qubits achieve a 10% reduction in nonlocal gates that scales with more qubits, plus evaluations of methods for universal gate sets including a logical-swa...

  3. Architectural Approaches to Fault-Tolerant Distributed Quantum Computing and Their Entanglement Overheads

    quant-ph 2025-11 unverdicted novelty 5.0

    Three architectural types for fault-tolerant distributed quantum computing exhibit distinct scaling of Bell-pair consumption and generation attempts with code distance in planar surface and toric codes.

Reference graph

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