Networked Realization of Quantum LDPC Codes

Narayanan Rengaswamy; Swayangprabha Shaw

arxiv: 2604.25026 · v1 · submitted 2026-04-27 · 🪐 quant-ph · cs.IT· math.IT

Networked Realization of Quantum LDPC Codes

Swayangprabha Shaw , Narayanan Rengaswamy This is my paper

Pith reviewed 2026-05-08 03:59 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT

keywords quantum LDPC codesbivariate bicycle codesnetworked quantum computingteleported CNOTBP-OSD decodingTanner graph partitioningcircuit-level noisefault-tolerant quantum computing

0 comments

The pith

Bivariate bicycle QLDPC codes can be split across networked nodes while retaining error suppression close to monolithic versions.

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

The paper sets out to show that bivariate bicycle codes, which improve on surface codes by balancing locality with better parameters, remain practical when their qubits are distributed one-per-node across a quantum network. It does so by first rebuilding networked surface-code simulations in Stim, then applying balanced min-cut partitioning to the combined X-Z Tanner graph of the bicycle codes to find good bipartitions. Cross-node stabilizers are realized with teleported CNOT gates whose noise is controlled by the fidelity of supplied Bell pairs. Circuit-level noise simulations with BP-OSD decoding then compare the resulting logical error rates directly to the corresponding monolithic implementations, supplying the first concrete performance data for these networked realizations.

Core claim

By partitioning bivariate bicycle codes into two nodes via balanced min-cut on the combined X-Z Tanner graph and executing stabilizers that span nodes through teleported CNOTs with tunable Bell-pair fidelity, the authors obtain circuit-level error rates under BP-OSD decoding that are competitive with the same codes run monolithically, while also recreating and extending networked surface-code results in Stim.

What carries the argument

Balanced min-cut partitioning on the combined X-Z Tanner graph to obtain near-optimal bipartitions, together with teleported CNOT gates whose noise scales with Bell-pair fidelity.

If this is right

Networked surface codes with one qubit per node can be simulated in Stim and yield additional performance benchmarks beyond prior studies.
Bipartite splits of bivariate bicycle codes become implementable once stabilizers crossing nodes are replaced by teleported CNOTs.
Varying Bell-pair fidelity in the model directly tunes the effective noise of networked gates and reveals sensitivity to entanglement quality.
BP-OSD decoding applied to the full circuit-level noise model supplies the first quantitative comparison between networked and monolithic QLDPC performance.
The outlined advantages and limitations of the approach point to concrete hardware constraints that must be met for the networked realization to remain advantageous.

Where Pith is reading between the lines

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

Modular quantum processor architectures could adopt these bipartitioning techniques to relax on-chip connectivity requirements while preserving fault-tolerance thresholds.
Extending the same min-cut method to multi-way partitions or to other families of QLDPC codes would test whether the competitive performance generalizes beyond two nodes.
Hardware teams could use the reported sensitivity to Bell-pair fidelity to set entanglement-generation targets before attempting physical networked implementations.
If better graph-partitioning algorithms reduce the cut size further, the gap between networked and monolithic logical error rates would shrink even more.

Load-bearing premise

The balanced min-cut method yields splits close enough to optimal that the resulting performance reflects what networked hardware could actually achieve, and the teleported-CNOT noise model with variable Bell-pair fidelity matches real device behavior.

What would settle it

A circuit-level simulation that replaces min-cut partitioning with an exhaustive or heuristic search for a lower-cut bipartition, or that substitutes measured hardware noise parameters for the Bell-pair fidelity model, would show whether the reported error rates rise or fall substantially.

Figures

Figures reproduced from arXiv: 2604.25026 by Narayanan Rengaswamy, Swayangprabha Shaw.

**Figure 1.** Figure 1: Bivariate bicycle (BB) code [15]: (a) toric representa view at source ↗

**Figure 2.** Figure 2: Schematic of networked surface code architecture. The view at source ↗

**Figure 5.** Figure 5: Surface code performance with noisy GHZ (p view at source ↗

**Figure 6.** Figure 6: Logical error-rate comparison between non-partitioned and partitioned BB code implementations for the view at source ↗

read the original abstract

Quantum low-density parity-check (QLDPC) codes with good parameters are promising candidates for low-overhead fault-tolerant quantum computing, but their non-local stabilizers require long-range connectivity and frequent qubit movement, introducing practical challenges. Prior work has studied the networked implementation of topological codes, where each node only holds one or a few qubits of the entire code, and demonstrated competitive performance under practical constraints such as the quality of network-provided entanglement. However, since these codes are already geometrically local, such a networked setting might not be essential. In this work, we propose and study the networked implementation of better QLDPC codes, specifically bivariate bicycle codes due to their similarity to surface codes and the controlled amount of long-range connections in their stabilizers. We begin by recreating networked surface codes in Stim, with one code qubit per node, and provide additional insights into their circuit-level noise performance. We then extend this approach to bipartitions of bivariate bicycle codes, using balanced min-cut partitioning on their combined X-Z Tanner graph to identify optimal qubit splits. For stabilizers spanning nodes, we implement teleported CNOTs and vary the Bell pair fidelity enabling these gates. Through circuit-level noise simulations with BP-OSD decoding, we provide the first insights into networked realizations of these codes and compare their performance with monolithic implementations. We conclude by outlining advantages, limitations, and future directions.

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.

Desk Editor's Note private letter to a colleague

This paper runs the first circuit simulations of networked bivariate bicycle codes via min-cut Tanner partitioning and teleported CNOTs, but the performance edge over monolithic versions rests on two untested modeling choices.

read the letter

The main takeaway is that the authors take the networked topological code setup from prior work and apply it to bivariate bicycle codes, which already mix local and long-range stabilizers. They partition the combined X-Z Tanner graph with a balanced min-cut, assign one qubit per node, and replace cross-node stabilizers with teleported CNOTs whose fidelity they vary. They then run Stim simulations with BP-OSD decoding and compare logical error rates to the single-chip case. That extension is the concrete new piece; the surface-code recreation mostly serves as a sanity check and baseline. The simulations themselves look like standard circuit-level noise workflows, which is fine as far as it goes. The soft spots are exactly where the stress-test note flags them. The min-cut is called optimal without any comparison to other splits or to hardware metrics such as entanglement latency or stabilizer weight distribution. The noise model for the teleported gates only tunes Bell-pair fidelity; everything else is assumed ideal. Without error bars, exact noise parameters, or a sensitivity check on those choices, it is hard to know how much the reported curves would move if the assumptions were relaxed. The abstract claims “first insights,” which is fair on the narrow combination of code family plus partitioning method, but the evidence is still simulation-only and model-dependent. This is the kind of paper that belongs in a reading group for people working on modular fault-tolerant architectures. It is not yet strong enough to cite for quantitative claims, but it is coherent enough and addresses a real engineering gap, so a serious editor should send it out for review rather than desk-reject. The referees can push on the modeling validation and ask for more ablation data.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes and studies networked realizations of bivariate bicycle QLDPC codes. It recreates networked surface codes with one qubit per node in Stim, then extends to bipartitions of bivariate bicycle codes via balanced min-cut partitioning on the combined X-Z Tanner graph to minimize inter-node connections. Stabilizers spanning nodes are implemented with teleported CNOTs whose noise is modeled by varying Bell-pair fidelity. Circuit-level simulations with BP-OSD decoding are used to compare error performance against monolithic implementations and to provide initial insights into networked QLDPC behavior under practical network constraints.

Significance. If the modeling choices prove accurate, the work supplies the first simulation-based comparison of networked versus monolithic bivariate bicycle codes, extending prior networked-surface-code studies to higher-rate QLDPC families. This could help assess whether the controlled long-range connectivity in these codes remains advantageous once network entanglement overhead is included. The use of standard Stim + BP-OSD tooling is a positive, reproducible element, but the absence of raw data, error bars, and validation of the two core modeling assumptions limits immediate impact.

major comments (3)

[Abstract] Abstract and simulation workflow: the central performance claims rest on circuit-level Stim simulations, yet no error bars, exact noise parameters, or complete partitioning results are supplied. Without these, the reported comparisons between networked and monolithic implementations cannot be verified or reproduced.
[Bipartitioning approach] Bipartitioning section: the claim that balanced min-cut partitioning on the combined X-Z Tanner graph yields near-optimal qubit-to-node assignments (minimizing teleported CNOTs) is presented without any comparison to alternative partitioners or hardware-aware metrics such as stabilizer-weight distribution across the cut.
[Teleported CNOT implementation] Teleported-CNOT noise model: performance curves are generated by varying only Bell-pair fidelity while holding other network effects fixed. No sensitivity analysis or hardware reference is given to justify that this single-parameter model captures the dominant error sources (e.g., latency or multi-qubit entanglement distribution) that would appear in real networked hardware.

minor comments (1)

[Abstract] The abstract would benefit from explicitly naming the specific bivariate bicycle codes examined and the numerical range of Bell-pair fidelities simulated.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and constructive feedback. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract and simulation workflow: the central performance claims rest on circuit-level Stim simulations, yet no error bars, exact noise parameters, or complete partitioning results are supplied. Without these, the reported comparisons between networked and monolithic implementations cannot be verified or reproduced.

Authors: We agree that error bars, explicit noise parameters, and partitioning details are necessary for reproducibility. In the revised manuscript we will add statistical error bars to all Stim simulation curves, include a table specifying the exact physical error rates, depolarizing noise strengths, and Bell-pair fidelities used, and append the complete bipartitioning results (including cut sizes and node assignments) for the bivariate bicycle codes examined. revision: yes
Referee: [Bipartitioning approach] Bipartitioning section: the claim that balanced min-cut partitioning on the combined X-Z Tanner graph yields near-optimal qubit-to-node assignments (minimizing teleported CNOTs) is presented without any comparison to alternative partitioners or hardware-aware metrics such as stabilizer-weight distribution across the cut.

Authors: The balanced min-cut was selected because it explicitly minimizes the number of crossing edges in the combined X-Z Tanner graph, which directly reduces the count of teleported CNOTs. While the original submission did not contain side-by-side comparisons with other partitioners, we will add a short justification paragraph citing the suitability of min-cut for this objective and note that metrics such as stabilizer-weight distribution across the cut represent valuable extensions for future study. revision: partial
Referee: [Teleported CNOT implementation] Teleported-CNOT noise model: performance curves are generated by varying only Bell-pair fidelity while holding other network effects fixed. No sensitivity analysis or hardware reference is given to justify that this single-parameter model captures the dominant error sources (e.g., latency or multi-qubit entanglement distribution) that would appear in real networked hardware.

Authors: Our model isolates Bell-pair fidelity as the primary network parameter because it governs the quality of the entanglement resource enabling teleported gates. We acknowledge that latency and multi-qubit distribution effects are not varied. The revision will add a dedicated paragraph discussing these modeling choices, reference recent experimental literature on quantum-network noise, and include a limited sensitivity study by re-running a subset of simulations with small additional latency-like delays. revision: yes

standing simulated objections not resolved

Supplying the complete raw simulation datasets, which are voluminous and not stored in a readily shareable format at present.

Circularity Check

0 steps flagged

No circularity; results are direct outputs of independent circuit simulations

full rationale

The paper's central claims rest on explicit circuit-level noise simulations performed in Stim using BP-OSD decoding, with modeling choices (balanced min-cut partitioning on the X-Z Tanner graph and teleported CNOTs parameterized by Bell-pair fidelity) stated as inputs rather than derived quantities. No equations, predictions, or first-principles results are presented that reduce by construction to fitted parameters, self-citations, or ansatzes. The performance comparisons between networked and monolithic implementations are computed outputs under the stated noise model, not tautological re-expressions of the inputs. This is self-contained empirical work with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Central claims rest on standard quantum circuit noise assumptions and the variable Bell pair fidelity parameter; no new entities are postulated.

free parameters (1)

Bell pair fidelity
Varied across simulations to map performance dependence on network entanglement quality.

axioms (1)

domain assumption Circuit-level depolarizing noise model for gates and measurements
Standard assumption invoked for all Stim simulations.

pith-pipeline@v0.9.0 · 5544 in / 1095 out tokens · 51719 ms · 2026-05-08T03:59:50.196346+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 4 canonical work pages

[1]

Topological quantum computing with a very noisy network and local error rates approaching one percent,

N. H. Nickerson, Y . Li, and S. C. Benjamin, “Topological quantum computing with a very noisy network and local error rates approaching one percent,”Nature Commun., vol. 4, no. 1, p. 1756, 2013

2013
[2]

Thresholds for the distributed surface code in the presence of memory decoherence,

S. de Bone, P. M ¨oller, C. E. Bradley, T. H. Taminiau, and D. Elkouss, “Thresholds for the distributed surface code in the presence of memory decoherence,”arXiv preprint arXiv:2401.10770, 2024

work page arXiv 2024
[3]

Hierarchical surface code for network quantum computing with modules of arbitrary size,

Y . Li and S. C. Benjamin, “Hierarchical surface code for network quantum computing with modules of arbitrary size,”Physical Review A, vol. 94, p. 042303, 10 2016

2016
[4]

Fault-tolerant connection of error-corrected qubits with noisy links,

J. Ramette, J. Sinclair, N. P. Breuckmann, and V . Vuleti´c, “Fault-tolerant connection of error-corrected qubits with noisy links,”npj Quantum Information, vol. 10, no. 1, p. 58, 2024

2024
[5]

Long-range- enhanced surface codes,

Y . Hong, M. Marinelli, A. M. Kaufman, and A. Lucas, “Long-range- enhanced surface codes,”arXiv preprint arXiv:2309.11719, 2023

work page arXiv 2023
[6]

An architecture for improved surface code connectivity in neutral atoms,

J. Viszlai, S. F. Lin, S. Dangwal, J. M. Baker, and F. T. Chong, “An architecture for improved surface code connectivity in neutral atoms,” arXiv preprint arXiv:2309.13507, 2023

work page arXiv 2023
[7]

Guinn, S

C. Guinn, S. Stein, E. Tureci, G. Avis, C. Liu, S. Krastanov, A. A. Houck, and A. Li, “Co-designed superconducting architecture for lattice surgery of surface codes with quantum interface routing card,”arXiv preprint arXiv:2312.01246, 2023

work page arXiv 2023
[8]

Distributed realization of color codes for quantum error correction,

N. K. Chandra, D. Tipper, R. Nejabati, E. Kaur, and K. P. Seshadreesan, “Distributed realization of color codes for quantum error correction,” 2025

2025
[9]

Surface codes: Towards practical large-scale quantum computation,

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,”Physical Review A, vol. 86, no. 3, 2012

2012
[10]

Optimized entanglement purification,

S. Krastanov, V . V . Albert, and L. Jiang, “Optimized entanglement purification,”Quantum, vol. 3, p. 123, 2019

2019
[11]

Protocols for creating and distilling multipartite ghz states with bell pairs,

S. de Bone, R. Ouyang, K. Goodenough, and D. Elkouss, “Protocols for creating and distilling multipartite ghz states with bell pairs,”IEEE Transactions on Quantum Engineering, vol. 1, p. 1–10, 2020

2020
[12]

Stim: a fast stabilizer circuit simulator,

C. Gidney, “Stim: a fast stabilizer circuit simulator,”Quantum, vol. 5, p. 497, July 2021

2021
[13]

Sparse blossom: correcting a million errors per core second with minimum-weight matching,

O. Higgott and C. Gidney, “Sparse blossom: correcting a million errors per core second with minimum-weight matching,”Quantum, vol. 9, p. 1600, Jan. 2025

2025
[14]

Tour de gross: A modular quantum computer based on bivariate bicycle codes,

T. J. Yoder, E. Schoute, P. Rall, E. Pritchett, J. M. Gambetta, A. W. Cross, M. Carroll, and M. E. Beverland, “Tour de gross: A modular quantum computer based on bivariate bicycle codes,” 2025

2025
[15]

High-threshold and low-overhead fault-tolerant quantum memory,

S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory,”Nature, vol. 627, p. 778–782, Mar. 2024

2024
[16]

Toward a 2D Local Implementation of Quantum Low-Density Parity-Check Codes,

N. Berthusen, D. Devulapalli, E. Schoute, A. M. Childs, M. J. Gullans, A. V . Gorshkov, and D. Gottesman, “Toward a 2D Local Implementation of Quantum Low-Density Parity-Check Codes,”PRX Quantum, vol. 6, p. 010306, 1 2025

2025
[17]

Good quantum error-correcting codes exist,

A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,”Phys. Rev. A, vol. 54, pp. 1098–1105, Aug 1996

1996
[18]

Multiple-particle interference and quantum error correc- tion,

A. M. Steane, “Multiple-particle interference and quantum error correc- tion,”Proceedings of the Royal Society of London. Series A: Mathe- matical, Physical and Engineering Sciences, vol. 452, pp. 2551 – 2577, 1996

1996
[19]

Fault-tolerant quantum computation by anyons,

A. Kitaev, “Fault-tolerant quantum computation by anyons,”Annals of Physics, vol. 303, p. 2–30, Jan. 2003

2003
[20]

How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits,

C. Gidney and M. Eker ˚a, “How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits,”Quantum, vol. 5, p. 433, Apr. 2021

2048
[21]

Quantum kronecker sum-product low-density parity-check codes with finite rate,

A. A. Kovalev and L. P. Pryadko, “Quantum kronecker sum-product low-density parity-check codes with finite rate,”Phys. Rev. A, vol. 88, p. 012311, Jul 2013

2013
[22]

Degenerate quantum ldpc codes with good finite length performance,

P. Panteleev and G. Kalachev, “Degenerate quantum ldpc codes with good finite length performance,”Quantum, vol. 5, p. 585, Nov. 2021

2021
[23]

Karypis and V

G. Karypis and V . Kumar,METIS: A Software Package for Partition- ing Unstructured Graphs, Partitioning Meshes, and Computing Fill- Reducing Orderings of Sparse Matrices, September 1998

1998
[24]

Logical operators of quantum codes,

M. M. Wilde, “Logical operators of quantum codes,”Physical Review A, vol. 79, no. 6, 2009

2009
[25]

Network requirements for distributed quantum computation,

H. Jacinto, E. Gouzien, and N. Sangouard, “Network requirements for distributed quantum computation,”Physical Review Research, vol. 8, Feb. 2026

2026

[1] [1]

Topological quantum computing with a very noisy network and local error rates approaching one percent,

N. H. Nickerson, Y . Li, and S. C. Benjamin, “Topological quantum computing with a very noisy network and local error rates approaching one percent,”Nature Commun., vol. 4, no. 1, p. 1756, 2013

2013

[2] [2]

Thresholds for the distributed surface code in the presence of memory decoherence,

S. de Bone, P. M ¨oller, C. E. Bradley, T. H. Taminiau, and D. Elkouss, “Thresholds for the distributed surface code in the presence of memory decoherence,”arXiv preprint arXiv:2401.10770, 2024

work page arXiv 2024

[3] [3]

Hierarchical surface code for network quantum computing with modules of arbitrary size,

Y . Li and S. C. Benjamin, “Hierarchical surface code for network quantum computing with modules of arbitrary size,”Physical Review A, vol. 94, p. 042303, 10 2016

2016

[4] [4]

Fault-tolerant connection of error-corrected qubits with noisy links,

J. Ramette, J. Sinclair, N. P. Breuckmann, and V . Vuleti´c, “Fault-tolerant connection of error-corrected qubits with noisy links,”npj Quantum Information, vol. 10, no. 1, p. 58, 2024

2024

[5] [5]

Long-range- enhanced surface codes,

Y . Hong, M. Marinelli, A. M. Kaufman, and A. Lucas, “Long-range- enhanced surface codes,”arXiv preprint arXiv:2309.11719, 2023

work page arXiv 2023

[6] [6]

An architecture for improved surface code connectivity in neutral atoms,

J. Viszlai, S. F. Lin, S. Dangwal, J. M. Baker, and F. T. Chong, “An architecture for improved surface code connectivity in neutral atoms,” arXiv preprint arXiv:2309.13507, 2023

work page arXiv 2023

[7] [7]

Guinn, S

C. Guinn, S. Stein, E. Tureci, G. Avis, C. Liu, S. Krastanov, A. A. Houck, and A. Li, “Co-designed superconducting architecture for lattice surgery of surface codes with quantum interface routing card,”arXiv preprint arXiv:2312.01246, 2023

work page arXiv 2023

[8] [8]

Distributed realization of color codes for quantum error correction,

N. K. Chandra, D. Tipper, R. Nejabati, E. Kaur, and K. P. Seshadreesan, “Distributed realization of color codes for quantum error correction,” 2025

2025

[9] [9]

Surface codes: Towards practical large-scale quantum computation,

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,”Physical Review A, vol. 86, no. 3, 2012

2012

[10] [10]

Optimized entanglement purification,

S. Krastanov, V . V . Albert, and L. Jiang, “Optimized entanglement purification,”Quantum, vol. 3, p. 123, 2019

2019

[11] [11]

Protocols for creating and distilling multipartite ghz states with bell pairs,

S. de Bone, R. Ouyang, K. Goodenough, and D. Elkouss, “Protocols for creating and distilling multipartite ghz states with bell pairs,”IEEE Transactions on Quantum Engineering, vol. 1, p. 1–10, 2020

2020

[12] [12]

Stim: a fast stabilizer circuit simulator,

C. Gidney, “Stim: a fast stabilizer circuit simulator,”Quantum, vol. 5, p. 497, July 2021

2021

[13] [13]

Sparse blossom: correcting a million errors per core second with minimum-weight matching,

O. Higgott and C. Gidney, “Sparse blossom: correcting a million errors per core second with minimum-weight matching,”Quantum, vol. 9, p. 1600, Jan. 2025

2025

[14] [14]

Tour de gross: A modular quantum computer based on bivariate bicycle codes,

T. J. Yoder, E. Schoute, P. Rall, E. Pritchett, J. M. Gambetta, A. W. Cross, M. Carroll, and M. E. Beverland, “Tour de gross: A modular quantum computer based on bivariate bicycle codes,” 2025

2025

[15] [15]

High-threshold and low-overhead fault-tolerant quantum memory,

S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory,”Nature, vol. 627, p. 778–782, Mar. 2024

2024

[16] [16]

Toward a 2D Local Implementation of Quantum Low-Density Parity-Check Codes,

N. Berthusen, D. Devulapalli, E. Schoute, A. M. Childs, M. J. Gullans, A. V . Gorshkov, and D. Gottesman, “Toward a 2D Local Implementation of Quantum Low-Density Parity-Check Codes,”PRX Quantum, vol. 6, p. 010306, 1 2025

2025

[17] [17]

Good quantum error-correcting codes exist,

A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,”Phys. Rev. A, vol. 54, pp. 1098–1105, Aug 1996

1996

[18] [18]

Multiple-particle interference and quantum error correc- tion,

A. M. Steane, “Multiple-particle interference and quantum error correc- tion,”Proceedings of the Royal Society of London. Series A: Mathe- matical, Physical and Engineering Sciences, vol. 452, pp. 2551 – 2577, 1996

1996

[19] [19]

Fault-tolerant quantum computation by anyons,

A. Kitaev, “Fault-tolerant quantum computation by anyons,”Annals of Physics, vol. 303, p. 2–30, Jan. 2003

2003

[20] [20]

How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits,

C. Gidney and M. Eker ˚a, “How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits,”Quantum, vol. 5, p. 433, Apr. 2021

2048

[21] [21]

Quantum kronecker sum-product low-density parity-check codes with finite rate,

A. A. Kovalev and L. P. Pryadko, “Quantum kronecker sum-product low-density parity-check codes with finite rate,”Phys. Rev. A, vol. 88, p. 012311, Jul 2013

2013

[22] [22]

Degenerate quantum ldpc codes with good finite length performance,

P. Panteleev and G. Kalachev, “Degenerate quantum ldpc codes with good finite length performance,”Quantum, vol. 5, p. 585, Nov. 2021

2021

[23] [23]

Karypis and V

G. Karypis and V . Kumar,METIS: A Software Package for Partition- ing Unstructured Graphs, Partitioning Meshes, and Computing Fill- Reducing Orderings of Sparse Matrices, September 1998

1998

[24] [24]

Logical operators of quantum codes,

M. M. Wilde, “Logical operators of quantum codes,”Physical Review A, vol. 79, no. 6, 2009

2009

[25] [25]

Network requirements for distributed quantum computation,

H. Jacinto, E. Gouzien, and N. Sangouard, “Network requirements for distributed quantum computation,”Physical Review Research, vol. 8, Feb. 2026

2026