78 Commits

Author SHA1 Message Date
4dfb3a7c35 Last changes 2026-05-06 01:58:49 +02:00
10d791fe04 Final readthrough corrections of quantum fundamentals 2026-05-04 23:04:28 +02:00
06852b8e62 Final readthrough corrections of classical fundamentals 2026-05-04 21:07:25 +02:00
400dc47df0 Incorporate Jonathan's corrections to classical fundamentals 2026-05-04 20:56:35 +02:00
ece8fc1715 Center error marker 2026-05-04 20:24:27 +02:00
56e3a0e5ca Consistently capitalize character after semicolon 2026-05-04 20:21:21 +02:00
8d6df8a79d Final readthrough corrections for fault tolerance chapter 2026-05-04 20:06:18 +02:00
c41ac9f61f Incorporate Jonathan's corrections to Fault Tolerance Chapter 2026-05-04 19:45:15 +02:00
a41e0b05fe Add Lia as supervisor 2026-05-04 19:20:08 +02:00
1edc3f301a Final readthrough corrections for decoding chapter 2026-05-04 18:42:39 +02:00
a977860ddb Incorporate Jonathan's correction to sliding-window decoding sections 2026-05-04 17:35:33 +02:00
7bf1b2f8d7 Incorporate Jonathan's corrections to numerical results section 2026-05-04 17:07:41 +02:00
72acea0321 Incorporate Jonathan's corrections to the introduction 2026-05-04 16:31:31 +02:00
f1a5aaf3f8 Make ToC be on one page 2026-05-04 16:20:37 +02:00
23828b671a Minor changes to conclusion 2026-05-04 16:08:56 +02:00
09893d527e Incorporate Jonathan's corrections to Abstract 2026-05-04 15:36:15 +02:00
25789a6bd3 Incorporate Jonathans's corrections to Conclusion 2026-05-04 15:28:11 +02:00
001ca614bb Fix bibliography 2026-05-04 15:17:05 +02:00
9e5eaaf985 Incorporate Lia's corrections to fault tolerance 2026-05-04 14:59:49 +02:00
17191382cf Incorporate Lia's corrections to QM and QEC fundamentals 2026-05-04 13:01:54 +02:00
aa907ef4a3 Incorporate Lia's corrections to classical fundamentals 2026-05-04 12:12:10 +02:00
12036caa91 Fix bibliography titlecase (in clean_bibliography.sh) and a few things in the bibliography itself 2026-05-04 10:53:50 +02:00
4c206ae9c4 Rephrase first sentence of abstract 2026-05-04 10:34:53 +02:00
01a754e5da Reset acronyms after abstract 2026-05-04 10:31:54 +02:00
81292a2644 Add abstract 2026-05-04 10:28:46 +02:00
73958d7850 Check out version of cel thesis template with option for signature 2026-05-04 02:08:32 +02:00
18e3683502 Add signature 2026-05-04 02:06:58 +02:00
1eb4db289e Make main.tex work with signature modification 2026-05-04 02:06:39 +02:00
f56cd05890 Fix bibtex definition for arxiv papers 2026-05-04 01:45:46 +02:00
e9d996155d Write conclusion 2026-05-04 01:21:26 +02:00
5e26179154 Finish intro 2026-05-03 20:51:32 +02:00
9ae98e07d7 Write most of Introduction; Fix citing Intro. 2026-05-03 19:09:29 +02:00
728c8560c7 Fix N_C/N_V notation 2026-05-03 14:07:04 +02:00
dd30b4fc0d Write captions 2026-05-03 04:26:58 +02:00
6e53ed5d1b Complete results chapter text 2026-05-03 04:00:05 +02:00
0016df0004 Add text for second BPGD plot 2026-05-03 03:10:21 +02:00
9ca2698d38 Add text for first BPGD figure 2026-05-03 02:16:22 +02:00
72461fe555 Complete first draft of warm-start sliding-window decoding section 2026-05-03 01:11:22 +02:00
5fabe2e146 Finish first draft of BP warm start subsection 2026-05-02 23:40:29 +02:00
a90458dd8a Write conclusion to BP investigation. BP investigation now done 2026-05-02 19:16:26 +02:00
d2960b8f0e Write text for figure 4.10 2026-05-02 17:59:44 +02:00
6b1821fd6b Add TODOs; Add magnified plot to other figure 2026-05-02 17:28:20 +02:00
5687499b5b Add paragraphs exp. params, description, and interpretation for fig. 4.9 2026-05-02 16:58:02 +02:00
f4718b67e7 Complete text for first two figures 2026-05-02 14:38:50 +02:00
0848e5dea6 Move order of figures 2026-05-02 11:38:04 +02:00
7e18985b86 Add whole decoding line to max_iter plot 2026-05-02 11:22:25 +02:00
152e784546 Refactor the intro to numerical results 2026-05-02 10:38:51 +02:00
15190ccf48 Include claude corrections for first 5 pages of decoding chapter 2026-05-02 09:01:19 +02:00
606d68e2c1 Rephrase to remove 'gate schedule' 2026-05-01 21:56:03 +02:00
47493a6beb Write numerical results intro 2026-05-01 21:51:48 +02:00
76a91c7d32 Add H_overlap to figure 2026-05-01 19:33:37 +02:00
1632f19c47 Add syndrome update equation 2026-05-01 19:31:18 +02:00
3d3556689e Add node set Visualization figure 2026-05-01 19:13:22 +02:00
4555570665 Finish index definitions 2026-05-01 18:08:57 +02:00
3b7618e1d1 Start VN and CN indexing from zero 2026-05-01 17:30:14 +02:00
635c0aab18 Rewrite VN and CN set definition text; Fix earlier TODOs 2026-05-01 16:43:16 +02:00
c555151b9d Wwrite a few paragraphs on the window generation/decoding 2026-05-01 11:47:21 +02:00
05348579f0 Switch figures 2026-05-01 10:50:07 +02:00
1059b4d98f Add QUITS paper to review 2026-05-01 10:41:54 +02:00
682eeb644e Add literature overview figure 2026-04-30 19:59:21 +02:00
27f13c1db0 Write chapter 4 intro 2026-04-30 14:05:32 +02:00
8071c9f485 Fix typos 2026-04-29 21:03:26 +02:00
94e4c9f8c9 Replace autoref by cref 2026-04-29 20:56:41 +02:00
64cf0e2269 Introduce LER 2026-04-29 20:31:41 +02:00
76270695b9 Fix notation 2026-04-29 20:22:26 +02:00
62b4d4838b Write stim subsection 2026-04-29 18:24:21 +02:00
0aa425ae41 Polish per-round logical error rate subsection 2026-04-29 17:40:37 +02:00
b73a66649c Rewrite DEM subsection; Write first draft of practical considerations 2026-04-29 16:12:25 +02:00
d7f05dc5b9 Rework detector matrix and detector error matrix sections 2026-04-29 13:26:35 +02:00
11178436b6 Rewrite parts of measurement syndrome matrix subsection 2026-04-29 09:26:41 +02:00
dc283012ba Move figures in chapter 3 inbetween text 2026-04-28 23:45:44 +02:00
87e48b5ac6 Move 3-qubit repetition code check matrix; Rewrite DEM intro 2026-04-28 18:58:16 +02:00
42a689d811 Polish second paragraphs of noise model subsecions 2026-04-28 18:12:21 +02:00
b46df8120b Rewrite intro to chapter 3; Add subsections for each noise model 2026-04-28 16:22:03 +02:00
5ced7b152e Write first couple of pages of chapter 3 2026-04-28 01:59:52 +02:00
3953320216 Write mathematical fault tolerance definition 2026-04-27 18:03:28 +02:00
4aa4799969 Fix wrong sim results; Add bpgd with decimation info passing over max iter plots 2026-04-27 16:05:54 +02:00
f899942029 Add TODOs 2026-04-27 00:26:08 +02:00
48 changed files with 4411 additions and 1033 deletions

View File

@@ -3,6 +3,16 @@
long=quantum error correction long=quantum error correction
} }
\DeclareAcronym{dem}{
short=DEM,
long=detector error model
}
\DeclareAcronym{ler}{
short=LER,
long=logical error rate
}
\DeclareAcronym{bp}{ \DeclareAcronym{bp}{
short=BP, short=BP,
long=belief propagation long=belief propagation
@@ -13,11 +23,31 @@
long=belief propagation with guided decimation long=belief propagation with guided decimation
} }
\DeclareAcronym{gdg}{
short=GDG,
long=guided decimation guessing
}
\DeclareAcronym{nms}{ \DeclareAcronym{nms}{
short=NMS, short=NMS,
long=normalized min-sum long=normalized min-sum
} }
\DeclareAcronym{osd}{
short=OSD,
long=ordered statistics decoding
}
\DeclareAcronym{aed}{
short=AED,
long=automorphism ensemble decoding
}
\DeclareAcronym{bsc}{
short=BSC,
long=binary symetric channel
}
\DeclareAcronym{spa}{ \DeclareAcronym{spa}{
short=SPA, short=SPA,
long=sum-product algorithm long=sum-product algorithm
@@ -97,3 +127,18 @@
short=BB, short=BB,
long=bivariate bicycle long=bivariate bicycle
} }
\DeclareAcronym{hgp}{
short=HGP,
long=hypergraph product
}
\DeclareAcronym{lp}{
short=LP,
long=lifted-product
}
\DeclareAcronym{bpc}{
short=BPC,
long=balanced product code
}

View File

@@ -7,19 +7,20 @@
language = {en}, language = {en},
number = {3}, number = {3},
journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
author = {Dirac, P. a. M.}, author = {Dirac, P. A. M.},
month = jul, month = jul,
year = {1939}, year = {1939},
pages = {416--418}, pages = {416--418},
} }
@article{huang_improved_2023, @misc{huang_improved_2023,
title = {Improved {Noisy} {Syndrome} {Decoding} of {Quantum} {LDPC} {Codes} with {Sliding} {Window}}, title = {Improved Noisy Syndrome Decoding of Quantum {LDPC} Codes with Sliding Window},
doi = {10.48550/arXiv.2311.03307}, doi = {10.48550/arXiv.2311.03307},
publisher = {arXiv}, publisher = {arXiv},
author = {Huang, Shilin and Puri, Shruti}, author = {Huang, Shilin and Puri, Shruti},
month = nov, month = nov,
year = {2023}, year = {2023},
howpublished = {arXiv:2311.03307},
} }
@article{huang_increasing_2024, @article{huang_increasing_2024,
@@ -35,17 +36,18 @@
pages = {012453}, pages = {012453},
} }
@article{xu_batched_2025, @misc{xu_batched_2025,
title = {Batched high-rate logical operations for quantum {LDPC} codes}, title = {Batched high-rate logical operations for quantum {LDPC} codes},
doi = {10.48550/arXiv.2510.06159}, doi = {10.48550/arXiv.2510.06159},
publisher = {arXiv}, publisher = {arXiv},
author = {Xu, Qian and Zhou, Hengyun and Bluvstein, Dolev and Cain, Madelyn and Kalinowski, Marcin and Preskill, John and Lukin, Mikhail D. and Maskara, Nishad}, author = {Xu, Qian and Zhou, Hengyun and Bluvstein, Dolev and Cain, Madelyn and Kalinowski, Marcin and Preskill, John and Lukin, Mikhail D. and Maskara, Nishad},
month = oct, month = oct,
year = {2025}, year = {2025},
howpublished = {arXiv:2510.06159},
} }
@article{gidney_stability_2022, @article{gidney_stability_2022,
title = {Stability {Experiments}: {The} {Overlooked} {Dual} of {Memory} {Experiments}}, title = {Stability Experiments: The Overlooked Dual of Memory Experiments},
volume = {6}, volume = {6},
issn = {2521-327X}, issn = {2521-327X},
shorttitle = {Stability {Experiments}}, shorttitle = {Stability {Experiments}},
@@ -57,30 +59,33 @@
pages = {786}, pages = {786},
} }
@article{koutsioumpas_colour_2025, @misc{koutsioumpas_colour_2025,
title = {Colour {Codes} {Reach} {Surface} {Code} {Performance} using {Vibe} {Decoding}}, title = {Colour Codes Reach Surface Code Performance using Vibe Decoding},
doi = {10.48550/arXiv.2508.15743}, doi = {10.48550/arXiv.2508.15743},
publisher = {arXiv}, publisher = {arXiv},
author = {Koutsioumpas, Stergios and Noszko, Tamas and Sayginel, Hasan and Webster, Mark and Roffe, Joschka}, author = {Koutsioumpas, Stergios and Noszko, Tamas and Sayginel, Hasan and Webster, Mark and Roffe, Joschka},
month = aug, month = aug,
year = {2025}, year = {2025},
howpublished = {arXiv:2508.15743},
} }
@article{koutsioumpas_automorphism_2025, @misc{koutsioumpas_automorphism_2025,
title = {Automorphism {Ensemble} {Decoding} of {Quantum} {LDPC} {Codes}}, title = {Automorphism Ensemble Decoding of Quantum {LDPC} Codes},
language = {en}, language = {en},
author = {Koutsioumpas, Stergios and Sayginel, Hasan and Webster, Mark and Browne, Dan E}, author = {Koutsioumpas, Stergios and Sayginel, Hasan and Webster, Mark and Browne, Dan E},
month = mar, month = mar,
year = {2025}, year = {2025},
howpublished = {arXiv:2503.01738},
} }
@article{gottesman_heisenberg_1998, @misc{gottesman_heisenberg_1998,
title = {The {Heisenberg} {Representation} of {Quantum} {Computers}}, title = {The Heisenberg Representation of Quantum Computers},
doi = {10.48550/arXiv.quant-ph/9807006}, doi = {10.48550/arXiv.quant-ph/9807006},
publisher = {arXiv}, publisher = {arXiv},
author = {Gottesman, Daniel}, author = {Gottesman, Daniel},
month = jul, month = jul,
year = {1998}, year = {1998},
howpublished = {arXiv:quant-ph/9807006},
} }
@article{gidney_stim_2021, @article{gidney_stim_2021,
@@ -98,8 +103,8 @@
} }
@phdthesis{higgott_practical_2024, @phdthesis{higgott_practical_2024,
type = {Doctoral}, type = {Ph.D. {Thesis}},
title = {Practical and {Efficient} {Quantum} {Error} {Correction}}, title = {Practical and Efficient Quantum Error Correction},
copyright = {open}, copyright = {open},
language = {eng}, language = {eng},
school = {UCL (University College London)}, school = {UCL (University College London)},
@@ -117,16 +122,17 @@
} }
@misc{gong_toward_2024, @misc{gong_toward_2024,
title = {Toward {Low}-latency {Iterative} {Decoding} of {QLDPC} {Codes} {Under} {Circuit}-{Level} {Noise}}, title = {Toward Low-latency Iterative Decoding of {QLDPC} Codes Under Circuit-Level Noise},
language = {en}, language = {en},
journal = {arXiv.org}, journal = {arXiv.org},
author = {Gong, Anqi and Cammerer, Sebastian and Renes, Joseph M.}, author = {Gong, Anqi and Cammerer, Sebastian and Renes, Joseph M.},
month = mar, month = mar,
howpublished = {arXiv:2403.18901},
year = {2024}, year = {2024},
} }
@article{miao_quaternary_2025, @article{miao_quaternary_2025,
title = {Quaternary {Neural} {Belief} {Propagation} {Decoding} of {Quantum} {LDPC} {Codes} with {Overcomplete} {Check} {Matrices}}, title = {Quaternary Neural Belief Propagation Decoding of Quantum {LDPC} Codes with Overcomplete Check Matrices},
volume = {13}, volume = {13},
issn = {2169-3536}, issn = {2169-3536},
doi = {10.1109/ACCESS.2025.3539475}, doi = {10.1109/ACCESS.2025.3539475},
@@ -138,7 +144,7 @@
} }
@misc{tsouchlos_ccam_2024, @misc{tsouchlos_ccam_2024,
title = {{CCAM} {Summary}}, title = {{CCAM} Summary},
author = {Tsouchlos, Andreas}, author = {Tsouchlos, Andreas},
month = oct, month = oct,
year = {2024}, year = {2024},
@@ -158,7 +164,7 @@
} }
@book{griffiths_introduction_1995, @book{griffiths_introduction_1995,
title = {Introduction to {Quantum} {Mechanics}}, title = {Introduction to Quantum Mechanics},
isbn = {0-13-124405-1}, isbn = {0-13-124405-1},
language = {en}, language = {en},
publisher = {Prentice Hall}, publisher = {Prentice Hall},
@@ -167,7 +173,7 @@
} }
@misc{bradley_tensor_2018, @misc{bradley_tensor_2018,
title = {The {Tensor} {Product}, {Demystified}}, title = {The Tensor Product, Demystified},
author = {Bradley, Tai-Danae}, author = {Bradley, Tai-Danae},
month = nov, month = nov,
year = {2018}, year = {2018},
@@ -175,7 +181,7 @@
@book{nielsen_quantum_2010, @book{nielsen_quantum_2010,
address = {Cambridge}, address = {Cambridge},
title = {Quantum {Computation} and {Quantum} {Information}: 10th {Anniversary} {Edition}}, title = {Quantum Computation and Quantum Information: 10th Anniversary Edition},
isbn = {978-0-511-97666-7}, isbn = {978-0-511-97666-7},
shorttitle = {Quantum {Computation} and {Quantum} {Information}}, shorttitle = {Quantum {Computation} and {Quantum} {Information}},
doi = {10.1017/CBO9780511976667}, doi = {10.1017/CBO9780511976667},
@@ -187,7 +193,7 @@
} }
@article{geiselhart_automorphism_2021, @article{geiselhart_automorphism_2021,
title = {Automorphism {Ensemble} {Decoding} of {Reed}{Muller} {Codes}}, title = {Automorphism Ensemble Decoding of ReedMuller Codes},
volume = {69}, volume = {69},
issn = {1558-0857}, issn = {1558-0857},
doi = {10.1109/TCOMM.2021.3098798}, doi = {10.1109/TCOMM.2021.3098798},
@@ -199,19 +205,20 @@
pages = {6424--6438}, pages = {6424--6438},
} }
@article{derks_designing_2025, @misc{derks_designing_2025,
title = {Designing fault-tolerant circuits using detector error models}, title = {Designing fault-tolerant circuits using detector error models},
doi = {10.48550/arXiv.2407.13826}, doi = {10.48550/arXiv.2407.13826},
publisher = {arXiv}, publisher = {arXiv},
author = {Derks, Peter-Jan H. S. and Townsend-Teague, Alex and Burchards, Ansgar G. and Eisert, Jens}, author = {Derks, Peter-Jan H. S. and Townsend-Teague, Alex and Burchards, Ansgar G. and Eisert, Jens},
month = oct, month = oct,
year = {2025}, year = {2025},
howpublished = {arXiv:2407.13826},
} }
@phdthesis{klinke_neural_2025, @phdthesis{klinke_neural_2025,
address = {Karlsruhe}, address = {Karlsruhe},
type = {Bachelor's {Thesis}}, type = {Bachelor's {Thesis}},
title = {Neural {Belief} {Propagation} {Ensemble} {Decoding} of {Quantum} {LDPC} {Codes}}, title = {Neural Belief Propagation Ensemble Decoding of Quantum {LDPC} Codes},
language = {English}, language = {English},
school = {KIT}, school = {KIT},
author = {Klinke, Jeremi}, author = {Klinke, Jeremi},
@@ -219,14 +226,15 @@
year = {2025}, year = {2025},
} }
@article{camps-moreno_toward_2024, @misc{camps-moreno_toward_2024,
title = {Toward {Quantum} {CSS}-{T} {Codes} from {Sparse} {Matrices}}, title = {Toward Quantum {CSS}-{T} Codes from Sparse Matrices},
doi = {10.48550/arXiv.2406.00425}, doi = {10.48550/arXiv.2406.00425},
abstract = {CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair \$(C\_1, C\_2)\$ of binary linear codes \$C\_1\$ and \$C\_2\$ that satisfy certain conditions. We prove that \$C\_1\$ and \$C\_2\$ form a CSS-T pair if and only if \$C\_2 {\textbackslash}subset {\textbackslash}operatorname\{Hull\}(C\_1) {\textbackslash}cap {\textbackslash}operatorname\{Hull\}(C\_1{\textasciicircum}2)\$, where the hull of a code is the intersection of the code with its dual. We show that if \$(C\_1,C\_2)\$ is a CSS-T pair, and the code \$C\_2\$ is degenerated on \${\textbackslash}\{i{\textbackslash}\}\$, meaning that the \$i{\textasciicircum}\{th\}\$-entry is zero for all the elements in \$C\_2\$, then the pair of punctured codes \$(C\_1{\textbar}\_i,C\_2{\textbar}\_i)\$ is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally.}, abstract = {CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair \$(C\_1, C\_2)\$ of binary linear codes \$C\_1\$ and \$C\_2\$ that satisfy certain conditions. We prove that \$C\_1\$ and \$C\_2\$ form a CSS-T pair if and only if \$C\_2 {\textbackslash}subset {\textbackslash}operatorname\{Hull\}(C\_1) {\textbackslash}cap {\textbackslash}operatorname\{Hull\}(C\_1{\textasciicircum}2)\$, where the hull of a code is the intersection of the code with its dual. We show that if \$(C\_1,C\_2)\$ is a CSS-T pair, and the code \$C\_2\$ is degenerated on \${\textbackslash}\{i{\textbackslash}\}\$, meaning that the \$i{\textasciicircum}\{th\}\$-entry is zero for all the elements in \$C\_2\$, then the pair of punctured codes \$(C\_1{\textbar}\_i,C\_2{\textbar}\_i)\$ is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally.},
publisher = {arXiv}, publisher = {arXiv},
author = {Camps-Moreno, Eduardo and López, Hiram H. and Matthews, Gretchen L. and McMillon, Emily}, author = {Camps-Moreno, Eduardo and López, Hiram H. and Matthews, Gretchen L. and McMillon, Emily},
month = jun, month = jun,
year = {2024}, year = {2024},
howpublished = {arXiv:2406.00425},
} }
@article{roffe_quantum_2019, @article{roffe_quantum_2019,
@@ -244,13 +252,14 @@
pages = {226--245}, pages = {226--245},
} }
@article{gottesman_introduction_2009, @misc{gottesman_introduction_2009,
title = {An {Introduction} to {Quantum} {Error} {Correction} and {Fault}-{Tolerant} {Quantum} {Computation}}, title = {An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation},
doi = {10.48550/arXiv.0904.2557}, doi = {10.48550/arXiv.0904.2557},
publisher = {arXiv}, publisher = {arXiv},
author = {Gottesman, Daniel}, author = {Gottesman, Daniel},
month = apr, month = apr,
year = {2009}, year = {2009},
howpublished = {arXiv:0904.2557},
} }
@article{gottesman_theory_1998, @article{gottesman_theory_1998,
@@ -266,35 +275,38 @@
pages = {127--137}, pages = {127--137},
} }
@article{calderbank_quantum_1997, @misc{calderbank_quantum_1997,
title = {Quantum {Error} {Correction} via {Codes} over {GF}(4)}, title = {Quantum Error Correction via Codes over {GF}(4)},
doi = {10.48550/arXiv.quant-ph/9608006}, doi = {10.48550/arXiv.quant-ph/9608006},
publisher = {arXiv}, publisher = {arXiv},
author = {Calderbank, A. R. and Rains, E. M. and Shor, P. W. and Sloane, N. J. A.}, author = {Calderbank, A. R. and Rains, E. M. and Shor, P. W. and Sloane, N. J. A.},
month = sep, month = sep,
year = {1997}, year = {1997},
howpublished = {arXiv:quant-ph/9608006},
} }
@article{gottesman_stabilizer_1997, @misc{gottesman_stabilizer_1997,
title = {Stabilizer {Codes} and {Quantum} {Error} {Correction}}, title = {Stabilizer Codes and Quantum Error Correction},
doi = {10.48550/arXiv.quant-ph/9705052}, doi = {10.48550/arXiv.quant-ph/9705052},
publisher = {arXiv}, publisher = {arXiv},
author = {Gottesman, Daniel}, author = {Gottesman, Daniel},
month = may, month = may,
year = {1997}, year = {1997},
howpublished = {Ph.D. {Thesis}, arXiv:quant-ph/9705052},
} }
@article{shor_fault-tolerant_1997, @misc{shor_fault-tolerant_1997,
title = {Fault-tolerant quantum computation}, title = {Fault-tolerant quantum computation},
doi = {10.48550/arXiv.quant-ph/9605011}, doi = {10.48550/arXiv.quant-ph/9605011},
publisher = {arXiv}, publisher = {arXiv},
author = {Shor, Peter W.}, author = {Shor, Peter W.},
month = mar, month = mar,
year = {1997}, year = {1997},
howpublished = {arXiv:quant-ph/9605011},
} }
@article{divincenzo_fault-tolerant_1996, @article{divincenzo_fault-tolerant_1996,
title = {Fault-{Tolerant} {Error} {Correction} with {Efficient} {Quantum} {Codes}}, title = {Fault-Tolerant Error Correction with Efficient Quantum Codes},
volume = {77}, volume = {77},
issn = {0031-9007, 1079-7114}, issn = {0031-9007, 1079-7114},
doi = {10.1103/PhysRevLett.77.3260}, doi = {10.1103/PhysRevLett.77.3260},
@@ -335,7 +347,7 @@
} }
@article{terhal_quantum_2015, @article{terhal_quantum_2015,
title = {Quantum {Error} {Correction} for {Quantum} {Memories}}, title = {Quantum Error Correction for Quantum Memories},
volume = {87}, volume = {87},
issn = {0034-6861, 1539-0756}, issn = {0034-6861, 1539-0756},
doi = {10.1103/RevModPhys.87.307}, doi = {10.1103/RevModPhys.87.307},
@@ -353,7 +365,7 @@
title = {Guidelines for snowballing in systematic literature studies and a replication in software engineering}, title = {Guidelines for snowballing in systematic literature studies and a replication in software engineering},
isbn = {978-1-4503-2476-2}, isbn = {978-1-4503-2476-2},
doi = {10.1145/2601248.2601268}, doi = {10.1145/2601248.2601268},
booktitle = {Proceedings of the 18th {International} {Conference} on {Evaluation} and {Assessment} in {Software} {Engineering}}, booktitle = {Proceedings of the 18th International Conference on Evaluation and Assessment in Software Engineering},
publisher = {Association for Computing Machinery}, publisher = {Association for Computing Machinery},
author = {Wohlin, Claes}, author = {Wohlin, Claes},
month = may, month = may,
@@ -374,20 +386,21 @@
pages = {83--84}, pages = {83--84},
} }
@article{blume-kohout_estimating_2025, @misc{blume-kohout_estimating_2025,
title = {Estimating detector error models from syndrome data}, title = {Estimating detector error models from syndrome data},
doi = {10.48550/arXiv.2504.14643}, doi = {10.48550/arXiv.2504.14643},
publisher = {arXiv}, publisher = {arXiv},
author = {Blume-Kohout, Robin and Young, Kevin}, author = {Blume-Kohout, Robin and Young, Kevin},
month = apr, month = apr,
year = {2025}, year = {2025},
howpublished = {arXiv:2504.14643},
} }
@inproceedings{chatterjee_quantum_2023, @inproceedings{chatterjee_quantum_2023,
title = {Quantum {Error} {Correction} {For} {Dummies}}, title = {Quantum Error Correction For Dummies},
volume = {01}, volume = {01},
doi = {10.1109/QCE57702.2023.00017}, doi = {10.1109/QCE57702.2023.00017},
booktitle = {2023 {IEEE} {International} {Conference} on {Quantum} {Computing} and {Engineering} ({QCE})}, booktitle = {2023 {IEEE} International Conference on Quantum Computing and Engineering ({QCE})},
author = {Chatterjee, Avimita and Phalak, Koustubh and Ghosh, Swaroop}, author = {Chatterjee, Avimita and Phalak, Koustubh and Ghosh, Swaroop},
month = sep, month = sep,
year = {2023}, year = {2023},
@@ -395,7 +408,7 @@
} }
@inproceedings{petersen_systematic_2008, @inproceedings{petersen_systematic_2008,
title = {Systematic {Mapping} {Studies} in {Software} {Engineering}}, title = {Systematic Mapping Studies in Software Engineering},
doi = {10.14236/ewic/EASE2008.8}, doi = {10.14236/ewic/EASE2008.8},
language = {en}, language = {en},
publisher = {BCS Learning \& Development}, publisher = {BCS Learning \& Development},
@@ -405,7 +418,7 @@
} }
@article{postler_demonstration_2024, @article{postler_demonstration_2024,
title = {Demonstration of {Fault}-{Tolerant} {Steane} {Quantum} {Error} {Correction}}, title = {Demonstration of Fault-Tolerant Steane Quantum Error Correction},
volume = {5}, volume = {5},
doi = {10.1103/PRXQuantum.5.030326}, doi = {10.1103/PRXQuantum.5.030326},
number = {3}, number = {3},
@@ -418,7 +431,7 @@
} }
@article{cao_exact_2025, @article{cao_exact_2025,
title = {Exact {Decoding} of {Quantum} {Error}-{Correcting} {Codes}}, title = {Exact Decoding of Quantum Error-Correcting Codes},
volume = {134}, volume = {134},
doi = {10.1103/PhysRevLett.134.190603}, doi = {10.1103/PhysRevLett.134.190603},
number = {19}, number = {19},
@@ -431,13 +444,14 @@
} }
@misc{beni_tesseract_2025, @misc{beni_tesseract_2025,
title = {Tesseract: {A} {Search}-{Based} {Decoder} for {Quantum} {Error} {Correction}}, title = {Tesseract: {A} Search-Based Decoder for Quantum Error Correction},
shorttitle = {Tesseract}, shorttitle = {Tesseract},
doi = {10.48550/arXiv.2503.10988}, doi = {10.48550/arXiv.2503.10988},
publisher = {arXiv}, publisher = {arXiv},
author = {Beni, Laleh Aghababaie and Higgott, Oscar and Shutty, Noah}, author = {Beni, Laleh Aghababaie and Higgott, Oscar and Shutty, Noah},
month = aug, month = aug,
year = {2025}, year = {2025},
howpublished = {arXiv:2503.10988},
} }
@article{bausch_learning_2024, @article{bausch_learning_2024,
@@ -457,12 +471,13 @@
} }
@misc{bhardwaj_adaptive_2025, @misc{bhardwaj_adaptive_2025,
title = {Adaptive {Estimation} of {Drifting} {Noise} in {Quantum} {Error} {Correction}}, title = {Adaptive Estimation of Drifting Noise in Quantum Error Correction},
doi = {10.48550/arXiv.2511.09491}, doi = {10.48550/arXiv.2511.09491},
publisher = {arXiv}, publisher = {arXiv},
author = {Bhardwaj, Devansh and Takou, Evangelia and Lin, Yingjia and Brown, Kenneth R.}, author = {Bhardwaj, Devansh and Takou, Evangelia and Lin, Yingjia and Brown, Kenneth R.},
month = nov, month = nov,
year = {2025}, year = {2025},
howpublished = {arXiv:2511.09491},
} }
@article{roffe_decoding_2020, @article{roffe_decoding_2020,
@@ -492,7 +507,7 @@
} }
@article{bausch_learning_2024-1, @article{bausch_learning_2024-1,
title = {Learning to {Decode} the {Surface} {Code} with a {Recurrent}, {Transformer}-{Based} {Neural} {Network}}, title = {Learning to Decode the Surface Code with a Recurrent, Transformer-Based Neural Network},
volume = {635}, volume = {635},
issn = {0028-0836, 1476-4687}, issn = {0028-0836, 1476-4687},
doi = {10.1038/s41586-024-08148-8}, doi = {10.1038/s41586-024-08148-8},
@@ -511,15 +526,17 @@
author = {Lin, Hsiang-Ku and Lim, Pak Kau and Kovalev, Alexey A. and Pryadko, Leonid P.}, author = {Lin, Hsiang-Ku and Lim, Pak Kau and Kovalev, Alexey A. and Pryadko, Leonid P.},
month = aug, month = aug,
year = {2025}, year = {2025},
howpublished = {arXiv:2506.16910},
} }
@misc{fan_accelerating_2025, @misc{fan_accelerating_2025,
title = {Accelerating {BP}-{OSD} {Decoder} for {QLDPC} {Codes} with {Local} {Syndrome}-{Based} {Preprocessing}}, title = {Accelerating {BP}-{OSD} Decoder for {QLDPC} Codes with Local Syndrome-Based Preprocessing},
doi = {10.48550/arXiv.2509.01892}, doi = {10.48550/arXiv.2509.01892},
publisher = {arXiv}, publisher = {arXiv},
author = {Fan, Wenxuan and Suzuki, Yasunari and Ravi, Gokul Subramanian and Ueno, Yosuke and Inoue, Koji and Tanimoto, Teruo}, author = {Fan, Wenxuan and Suzuki, Yasunari and Ravi, Gokul Subramanian and Ueno, Yosuke and Inoue, Koji and Tanimoto, Teruo},
month = sep, month = sep,
year = {2025}, year = {2025},
howpublished = {arXiv:2509.01892},
} }
@misc{senior_scalable_2025, @misc{senior_scalable_2025,
@@ -529,14 +546,16 @@
author = {Senior, Andrew W. and Edlich, Thomas and Heras, Francisco J. H. and Zhang, Lei M. and Higgott, Oscar and Spencer, James S. and Applebaum, Taylor and Blackwell, Sam and Ledford, Justin and Žemgulytė, Akvilė and Žídek, Augustin and Shutty, Noah and Cowie, Andrew and Li, Yin and Holland, George and Brooks, Peter and Beattie, Charlie and Newman, Michael and Davies, Alex and Jones, Cody and Boixo, Sergio and Neven, Hartmut and Kohli, Pushmeet and Bausch, Johannes}, author = {Senior, Andrew W. and Edlich, Thomas and Heras, Francisco J. H. and Zhang, Lei M. and Higgott, Oscar and Spencer, James S. and Applebaum, Taylor and Blackwell, Sam and Ledford, Justin and Žemgulytė, Akvilė and Žídek, Augustin and Shutty, Noah and Cowie, Andrew and Li, Yin and Holland, George and Brooks, Peter and Beattie, Charlie and Newman, Michael and Davies, Alex and Jones, Cody and Boixo, Sergio and Neven, Hartmut and Kohli, Pushmeet and Bausch, Johannes},
month = dec, month = dec,
year = {2025}, year = {2025},
howpublished = {arXiv:2512.07737},
} }
@misc{wang_fully_2025, @misc{wang_fully_2025,
title = {Fully {Parallelized} {BP} {Decoding} for {Quantum} {LDPC} {Codes} {Can} {Outperform} {BP}-{OSD}}, title = {Fully Parallelized {BP} Decoding for Quantum {LDPC} Codes Can Outperform {BP}-{OSD}},
language = {en}, language = {en},
author = {Wang, Ming and Li, Ang and Mueller, Frank}, author = {Wang, Ming and Li, Ang and Mueller, Frank},
month = jun, month = jun,
year = {2025}, year = {2025},
howpublished = {arXiv:2507.00254},
} }
@misc{ye_beam_2025, @misc{ye_beam_2025,
@@ -565,7 +584,7 @@
} }
@article{higgott_improved_2023, @article{higgott_improved_2023,
title = {Improved {Decoding} of {Circuit} {Noise} and {Fragile} {Boundaries} of {Tailored} {Surface} {Codes}}, title = {Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes},
volume = {13}, volume = {13},
doi = {10.1103/PhysRevX.13.031007}, doi = {10.1103/PhysRevX.13.031007},
number = {3}, number = {3},
@@ -578,31 +597,34 @@
} }
@misc{tsubouchi_degeneracy_2025, @misc{tsubouchi_degeneracy_2025,
title = {Degeneracy {Cutting}: {A} {Local} and {Efficient} {Post}-{Processing} for {Belief} {Propagation} {Decoding} of {Quantum} {Low}-{Density} {Parity}-{Check} {Codes}}, title = {Degeneracy Cutting: {A} Local and Efficient Post-Processing for Belief Propagation Decoding of Quantum Low-Density Parity-Check Codes},
shorttitle = {Degeneracy {Cutting}}, shorttitle = {Degeneracy {Cutting}},
doi = {10.48550/arXiv.2510.08695}, doi = {10.48550/arXiv.2510.08695},
publisher = {arXiv}, publisher = {arXiv},
author = {Tsubouchi, Kento and Yamasaki, Hayata and Tamiya, Shiro}, author = {Tsubouchi, Kento and Yamasaki, Hayata and Tamiya, Shiro},
month = oct, month = oct,
year = {2025}, year = {2025},
howpublished = {arXiv:2510.08695},
} }
@misc{lee_scalable_2025, @misc{lee_scalable_2025,
title = {Scalable {Neural} {Decoders} for {Practical} {Real}-{Time} {Quantum} {Error} {Correction}}, title = {Scalable Neural Decoders for Practical Real-Time Quantum Error Correction},
doi = {10.48550/arXiv.2510.22724}, doi = {10.48550/arXiv.2510.22724},
publisher = {arXiv}, publisher = {arXiv},
author = {Lee, Changwon and Hur, Tak and Park, Daniel K.}, author = {Lee, Changwon and Hur, Tak and Park, Daniel K.},
month = oct, month = oct,
year = {2025}, year = {2025},
howpublished = {arXiv:2510.22724},
} }
@misc{maan_decoding_2025, @misc{maan_decoding_2025,
title = {Decoding {Correlated} {Errors} in {Quantum} {LDPC} {Codes}}, title = {Decoding Correlated Errors in Quantum {LDPC} Codes},
doi = {10.48550/arXiv.2510.14060}, doi = {10.48550/arXiv.2510.14060},
publisher = {arXiv}, publisher = {arXiv},
author = {Maan, Arshpreet Singh and Herrero, Francisco-Garcia and Paler, Alexandru and Savin, Valentin}, author = {Maan, Arshpreet Singh and Herrero, Francisco-Garcia and Paler, Alexandru and Savin, Valentin},
month = oct, month = oct,
year = {2025}, year = {2025},
howpublished = {arXiv:2510.14060},
} }
@article{skoric_parallel_2023, @article{skoric_parallel_2023,
@@ -622,7 +644,7 @@
} }
@article{higgott_sparse_2025, @article{higgott_sparse_2025,
title = {Sparse {Blossom}: correcting a million errors per core second with minimum-weight matching}, title = {Sparse Blossom: correcting a million errors per core second with minimum-weight matching},
volume = {9}, volume = {9},
shorttitle = {Sparse {Blossom}}, shorttitle = {Sparse {Blossom}},
doi = {10.22331/q-2025-01-20-1600}, doi = {10.22331/q-2025-01-20-1600},
@@ -636,7 +658,7 @@
} }
@article{breuckmann_quantum_2021, @article{breuckmann_quantum_2021,
title = {Quantum {Low}-{Density} {Parity}-{Check} {Codes}}, title = {Quantum Low-Density Parity-Check Codes},
volume = {2}, volume = {2},
doi = {10.1103/PRXQuantum.2.040101}, doi = {10.1103/PRXQuantum.2.040101},
number = {4}, number = {4},
@@ -649,10 +671,10 @@
} }
@inproceedings{gokduman_erasure_2024, @inproceedings{gokduman_erasure_2024,
title = {Erasure {Decoding} for {Quantum} {LDPC} {Codes} via {Belief} {Propagation} with {Guided} {Decimation}}, title = {Erasure Decoding for Quantum {LDPC} Codes via Belief Propagation with Guided Decimation},
issn = {2836-4503}, issn = {2836-4503},
doi = {10.1109/Allerton63246.2024.10735275}, doi = {10.1109/Allerton63246.2024.10735275},
booktitle = {2024 60th {Annual} {Allerton} {Conference} on {Communication}, {Control}, and {Computing}}, booktitle = {2024 60th Annual Allerton Conference on Communication, Control, and Computing},
author = {Gökduman, Mert and Yao, Hanwen and Pfister, Henry D.}, author = {Gökduman, Mert and Yao, Hanwen and Pfister, Henry D.},
month = sep, month = sep,
year = {2024}, year = {2024},
@@ -660,13 +682,14 @@
} }
@misc{swierkowska_eccentric_2025, @misc{swierkowska_eccentric_2025,
title = {{ECCentric}: {An} {Empirical} {Analysis} of {Quantum} {Error} {Correction} {Codes}}, title = {ECCentric: An Empirical Analysis of Quantum Error Correction Codes},
shorttitle = {{ECCentric}}, shorttitle = {{ECCentric}},
doi = {10.48550/arXiv.2511.01062}, doi = {10.48550/arXiv.2511.01062},
publisher = {arXiv}, publisher = {arXiv},
author = {{\'S}wierkowska, Aleksandra and Pflieger, Jannik and Giortamis, Emmanouil and Bhatotia, Pramod}, author = {{\'S}wierkowska, Aleksandra and Pflieger, Jannik and Giortamis, Emmanouil and Bhatotia, Pramod},
month = nov, month = nov,
year = {2025}, year = {2025},
howpublished = {arXiv:2511.01062},
} }
@phdthesis{guernut_fault-tolerant_2025, @phdthesis{guernut_fault-tolerant_2025,
@@ -693,7 +716,7 @@
} }
@article{tan_scalable_2023, @article{tan_scalable_2023,
title = {Scalable {Surface}-{Code} {Decoders} with {Parallelization} in {Time}}, title = {Scalable Surface-Code Decoders with Parallelization in Time},
volume = {4}, volume = {4},
doi = {10.1103/PRXQuantum.4.040344}, doi = {10.1103/PRXQuantum.4.040344},
number = {4}, number = {4},
@@ -735,21 +758,23 @@
} }
@misc{kuo_fault-tolerant_2024, @misc{kuo_fault-tolerant_2024,
title = {Fault-{Tolerant} {Belief} {Propagation} for {Practical} {Quantum} {Memory}}, title = {Fault-Tolerant Belief Propagation for Practical Quantum Memory},
doi = {10.48550/arXiv.2409.18689}, doi = {10.48550/arXiv.2409.18689},
publisher = {arXiv}, publisher = {arXiv},
author = {Kuo, Kao-Yueh and Lai, Ching-Yi}, author = {Kuo, Kao-Yueh and Lai, Ching-Yi},
month = sep, month = sep,
year = {2024}, year = {2024},
howpublished = {arXiv:2409.18689},
} }
@misc{poor_ultra_2025, @misc{poor_ultra_2025,
title = {Ultra {Low} {Overhead} {Syndrome} {Extraction} for the {Steane} code}, title = {Ultra Low Overhead Syndrome Extraction for the Steane code},
doi = {10.48550/arXiv.2511.13700}, doi = {10.48550/arXiv.2511.13700},
publisher = {arXiv}, publisher = {arXiv},
author = {Poór, Boldizsár and Rodatz, Benjamin and Kissinger, Aleks}, author = {Poór, Boldizsár and Rodatz, Benjamin and Kissinger, Aleks},
month = nov, month = nov,
year = {2025}, year = {2025},
howpublished = {arXiv:2511.13700},
} }
@article{feynman_simulating_1982, @article{feynman_simulating_1982,
@@ -770,15 +795,16 @@
title = {Algorithms for quantum computation: discrete logarithms and factoring}, title = {Algorithms for quantum computation: discrete logarithms and factoring},
shorttitle = {Algorithms for quantum computation}, shorttitle = {Algorithms for quantum computation},
doi = {10.1109/SFCS.1994.365700}, doi = {10.1109/SFCS.1994.365700},
booktitle = {Proceedings 35th {Annual} {Symposium} on {Foundations} of {Computer} {Science}}, booktitle = {Proc. Annual Symposium on Foundations of Computer Science},
author = {Shor, P.W.}, author = {Shor, P.W.},
address = {Santa Fe},
month = nov, month = nov,
year = {1994}, year = {1994},
pages = {124--134}, pages = {124--134},
} }
@article{preskill_quantum_2018, @article{preskill_quantum_2018,
title = {Quantum {Computing} in the {NISQ} era and beyond}, title = {Quantum Computing in the {NISQ} era and beyond},
volume = {2}, volume = {2},
doi = {10.22331/q-2018-08-06-79}, doi = {10.22331/q-2018-08-06-79},
language = {en-GB}, language = {en-GB},
@@ -791,7 +817,7 @@
} }
@misc{google_quantum_ai_quantum_nodate, @misc{google_quantum_ai_quantum_nodate,
title = {Quantum {Computing} {Roadmap}}, title = {Quantum Computing Roadmap},
language = {en}, language = {en},
journal = {Google Quantum AI}, journal = {Google Quantum AI},
author = {{Google Quantum AI}}, author = {{Google Quantum AI}},
@@ -811,7 +837,7 @@
} }
@article{zhang_classical_2023, @article{zhang_classical_2023,
title = {A {Classical} {Architecture} for {Digital} {Quantum} {Computers}}, title = {A Classical Architecture for Digital Quantum Computers},
volume = {5}, volume = {5},
doi = {10.1145/3626199}, doi = {10.1145/3626199},
number = {1}, number = {1},
@@ -829,6 +855,7 @@
author = {Caune, Laura and Skoric, Luka and Blunt, Nick S. and Ruban, Archibald and McDaniel, Jimmy and Valery, Joseph A. and Patterson, Andrew D. and Gramolin, Alexander V. and Majaniemi, Joonas and Barnes, Kenton M. and Bialas, Tomasz and Buğdaycı, Okan and Crawford, Ophelia and Gehér, György P. and Krovi, Hari and Matekole, Elisha and Topal, Canberk and Poletto, Stefano and Bryant, Michael and Snyder, Kalan and Gillespie, Neil I. and Jones, Glenn and Johar, Kauser and Campbell, Earl T. and Hill, Alexander D.}, author = {Caune, Laura and Skoric, Luka and Blunt, Nick S. and Ruban, Archibald and McDaniel, Jimmy and Valery, Joseph A. and Patterson, Andrew D. and Gramolin, Alexander V. and Majaniemi, Joonas and Barnes, Kenton M. and Bialas, Tomasz and Buğdaycı, Okan and Crawford, Ophelia and Gehér, György P. and Krovi, Hari and Matekole, Elisha and Topal, Canberk and Poletto, Stefano and Bryant, Michael and Snyder, Kalan and Gillespie, Neil I. and Jones, Glenn and Johar, Kauser and Campbell, Earl T. and Hill, Alexander D.},
month = oct, month = oct,
year = {2024}, year = {2024},
howpublished = {arXiv:2410.05202},
} }
@misc{ye_beam_2025-1, @misc{ye_beam_2025-1,
@@ -838,14 +865,15 @@
author = {Ye, Min and Wecker, Dave and Delfosse, Nicolas}, author = {Ye, Min and Wecker, Dave and Delfosse, Nicolas},
month = dec, month = dec,
year = {2025}, year = {2025},
howpublished = {arXiv:2512.07057},
} }
@misc{noauthor_reproducing_nodate, @misc{noauthor_reproducing_nodate,
title = {Reproducing repetition and {Shor} code simulations using stim}, title = {Reproducing repetition and Shor code simulations using stim},
} }
@misc{noauthor_tutorial_nodate, @misc{noauthor_tutorial_nodate,
title = {Tutorial - {Estimating} the {Surface} {Code} {Threshold}{NordIQuEst} {Application} {Library}}, title = {Tutorial - Estimating the Surface Code Threshold — NordIQuEst Application Library},
} }
@misc{noauthor_simulating_nodate, @misc{noauthor_simulating_nodate,
@@ -853,7 +881,7 @@
} }
@article{ryan-anderson_realization_2021, @article{ryan-anderson_realization_2021,
title = {Realization of {Real}-{Time} {Fault}-{Tolerant} {Quantum} {Error} {Correction}}, title = {Realization of Real-Time Fault-Tolerant Quantum Error Correction},
volume = {11}, volume = {11},
doi = {10.1103/PhysRevX.11.041058}, doi = {10.1103/PhysRevX.11.041058},
number = {4}, number = {4},
@@ -880,11 +908,11 @@
} }
@misc{noauthor_tutorial_nodate-1, @misc{noauthor_tutorial_nodate-1,
title = {Tutorial - {Fault}-{Tolerant} {Quantum} {Computing} with {CSS} codes}, title = {Tutorial - Fault-Tolerant Quantum Computing with {CSS} codes},
} }
@article{panteleev_degenerate_2021, @article{panteleev_degenerate_2021,
title = {Degenerate {Quantum} {LDPC} {Codes} {With} {Good} {Finite} {Length} {Performance}}, title = {Degenerate Quantum {LDPC} Codes With Good Finite Length Performance},
volume = {5}, volume = {5},
doi = {10.22331/q-2021-11-22-585}, doi = {10.22331/q-2021-11-22-585},
language = {en-GB}, language = {en-GB},
@@ -897,27 +925,29 @@
} }
@article{babar_fifteen_2015, @article{babar_fifteen_2015,
title = {Fifteen {Years} of {Quantum} {LDPC} {Coding} and {Improved} {Decoding} {Strategies}}, title = {Fifteen Years of Quantum {LDPC} Coding and Improved Decoding Strategies},
volume = {3}, volume = {3},
issn = {2169-3536}, issn = {2169-3536},
doi = {10.1109/ACCESS.2015.2503267}, doi = {10.1109/ACCESS.2015.2503267},
journal = {IEEE Access}, journal = {IEEE Access},
author = {Babar, Zunaira and Botsinis, Panagiotis and Alanis, Dimitrios and Ng, Soon Xin and Hanzo, Lajos}, author = {Babar, Zunaira and Botsinis, Panagiotis and Alanis, Dimitrios and Ng, Soon Xin and Hanzo, Lajos},
month = nov,
year = {2015}, year = {2015},
pages = {2492--2519}, pages = {2492--2519},
} }
@misc{yao_belief_2024, @misc{yao_belief_2024,
title = {Belief {Propagation} {Decoding} of {Quantum} {LDPC} {Codes} with {Guided} {Decimation}}, title = {Belief Propagation Decoding of Quantum {LDPC} Codes with Guided Decimation},
doi = {10.48550/arXiv.2312.10950}, doi = {10.48550/arXiv.2312.10950},
publisher = {arXiv}, publisher = {arXiv},
author = {Yao, Hanwen and Laban, Waleed Abu and Häger, Christian and Amat, Alexandre Graell i and Pfister, Henry D.}, author = {Yao, Hanwen and Laban, Waleed Abu and Häger, Christian and Amat, Alexandre Graell i and Pfister, Henry D.},
month = jun, month = jun,
year = {2024}, year = {2024},
howpublished = {arXiv:2312.10950},
} }
@article{sharon_efficient_2007, @article{sharon_efficient_2007,
title = {Efficient {Serial} {Message}-{Passing} {Schedules} for {LDPC} {Decoding}}, title = {Efficient Serial Message-Passing Schedules for {LDPC} Decoding},
volume = {53}, volume = {53},
issn = {1557-9654}, issn = {1557-9654},
doi = {10.1109/TIT.2007.907507}, doi = {10.1109/TIT.2007.907507},
@@ -943,7 +973,7 @@
} }
@book{ryan_channel_2009, @book{ryan_channel_2009,
title = {Channel {Codes}: {Classical} and {Modern}}, title = {Channel Codes: Classical and Modern},
isbn = {978-1-139-48301-8}, isbn = {978-1-139-48301-8},
shorttitle = {Channel {Codes}}, shorttitle = {Channel {Codes}},
language = {en}, language = {en},
@@ -954,7 +984,7 @@
} }
@book{macwilliams_theory_1977, @book{macwilliams_theory_1977,
title = {The {Theory} of {Error}-correcting {Codes}}, title = {The Theory of Error-correcting Codes},
isbn = {978-0-444-85010-2}, isbn = {978-0-444-85010-2},
language = {en}, language = {en},
publisher = {Elsevier}, publisher = {Elsevier},
@@ -964,7 +994,7 @@
@book{richardson_modern_2008, @book{richardson_modern_2008,
address = {Cambridge}, address = {Cambridge},
title = {Modern {Coding} {Theory}}, title = {Modern Coding Theory},
isbn = {978-0-521-85229-6}, isbn = {978-0-521-85229-6},
doi = {10.1017/CBO9780511791338}, doi = {10.1017/CBO9780511791338},
publisher = {Cambridge University Press}, publisher = {Cambridge University Press},
@@ -973,7 +1003,7 @@
} }
@phdthesis{gallager_low_1960, @phdthesis{gallager_low_1960,
type = {Thesis}, type = {Ph.D. {Thesis}},
title = {Low density parity check codes}, title = {Low density parity check codes},
copyright = {M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.}, copyright = {M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.},
language = {eng}, language = {eng},
@@ -986,11 +1016,11 @@
title = {Fully parallel window decoder architecture for spatially-coupled {LDPC} codes}, title = {Fully parallel window decoder architecture for spatially-coupled {LDPC} codes},
issn = {1938-1883}, issn = {1938-1883},
doi = {10.1109/ICC.2016.7511553}, doi = {10.1109/ICC.2016.7511553},
booktitle = {2016 {IEEE} {International} {Conference} on {Communications} ({ICC})}, booktitle = {Proc. {IEEE} International Conference on Communications ({ICC})},
author = {Hassan, Najeeb Ul and Schlüter, Martin and Fettweis, Gerhard P.}, author = {Hassan, Najeeb Ul and Schlüter, Martin and Fettweis, Gerhard P.},
address = {Kuala Lumpur},
month = may, month = may,
year = {2016}, year = {2016},
pages = {1--6},
} }
@article{costello_spatially_2014, @article{costello_spatially_2014,
@@ -1019,7 +1049,7 @@
} }
@article{kang_quits_2025, @article{kang_quits_2025,
title = {{QUITS}: {A} modular {Qldpc} code {circUIT} {Simulator}}, title = {{QUITS}: {A} modular Qldpc code circUIT Simulator},
volume = {9}, volume = {9},
issn = {2521-327X}, issn = {2521-327X},
shorttitle = {{QUITS}}, shorttitle = {{QUITS}},
@@ -1033,7 +1063,7 @@
@book{griffiths_consistent_2001, @book{griffiths_consistent_2001,
address = {Cambridge}, address = {Cambridge},
title = {Consistent {Quantum} {Theory}}, title = {Consistent Quantum Theory},
isbn = {978-0-521-53929-6}, isbn = {978-0-521-53929-6},
doi = {10.1017/CBO9780511606052}, doi = {10.1017/CBO9780511606052},
publisher = {Cambridge University Press}, publisher = {Cambridge University Press},
@@ -1042,10 +1072,95 @@
} }
@misc{gottesman_fault-tolerant_2014, @misc{gottesman_fault-tolerant_2014,
title = {Fault-{Tolerant} {Quantum} {Computation} with {Constant} {Overhead}}, title = {Fault-Tolerant Quantum Computation with Constant Overhead},
doi = {10.48550/arXiv.1310.2984}, doi = {10.48550/arXiv.1310.2984},
publisher = {arXiv}, publisher = {arXiv},
author = {Gottesman, Daniel}, author = {Gottesman, Daniel},
month = jul, month = jul,
year = {2014}, year = {2014},
howpublished = {arXiv:1310.2984},
}
@misc{gidney_new_2023,
title = {New circuits and an open source decoder for the color code},
doi = {10.48550/arXiv.2312.08813},
publisher = {arXiv},
author = {Gidney, Craig and Jones, Cody},
month = dec,
year = {2023},
howpublished = {arXiv:2312.08813},
}
@article{gidney_fault-tolerant_2021,
title = {A Fault-Tolerant Honeycomb Memory},
volume = {5},
issn = {2521-327X},
doi = {10.22331/q-2021-12-20-605},
journal = {Quantum},
author = {Gidney, Craig and Newman, Michael and Fowler, Austin and Broughton, Michael},
month = dec,
year = {2021},
pages = {605},
}
@article{chamberland_flag_2018,
title = {Flag fault-tolerant error correction with arbitrary distance codes},
volume = {2},
issn = {2521-327X},
doi = {10.22331/q-2018-02-08-53},
journal = {Quantum},
author = {Chamberland, Christopher and Beverland, Michael E.},
month = feb,
year = {2018},
pages = {53},
}
@article{chen_exponential_2021,
title = {Exponential suppression of bit or phase errors with cyclic error correction},
volume = {595},
copyright = {2021 The Author(s)},
issn = {1476-4687},
doi = {10.1038/s41586-021-03588-y},
language = {en},
number = {7867},
journal = {Nature},
publisher = {Nature Publishing Group},
author = {{Google Quantum AI}},
month = jul,
year = {2021},
pages = {383--387},
}
@article{kelly_state_2015,
title = {State preservation by repetitive error detection in a superconducting quantum circuit},
volume = {519},
issn = {0028-0836, 1476-4687},
doi = {10.1038/nature14270},
number = {7541},
journal = {Nature},
author = {Kelly, J. and Barends, R. and Fowler, A. G. and Megrant, A. and Jeffrey, E. and White, T. C. and Sank, D. and Mutus, J. Y. and Campbell, B. and Chen, Yu and Chen, Z. and Chiaro, B. and Dunsworth, A. and Hoi, I.-C. and Neill, C. and O'Malley, P. J. J. and Quintana, C. and Roushan, P. and Vainsencher, A. and Wenner, J. and Cleland, A. N. and Martinis, John M.},
month = mar,
year = {2015},
pages = {66--69},
}
@misc{bombin_modular_2023,
title = {Modular decoding: parallelizable real-time decoding for quantum computers},
shorttitle = {Modular decoding},
doi = {10.48550/arXiv.2303.04846},
publisher = {arXiv},
author = {Bomb{\'i}n, H{\'e}ctor and Dawson, Chris and Liu, Ye-Hua and Nickerson, Naomi and Pastawski, Fernando and Roberts, Sam},
month = mar,
year = {2023},
howpublished = {arXiv:2303.04846},
}
@misc{leverrier_decoding_2022,
title = {Decoding quantum Tanner codes},
doi = {10.48550/arXiv.2208.05537},
publisher = {arXiv},
author = {Leverrier, Anthony and Z{\'e}mor, Gilles},
month = dec,
year = {2022},
howpublished = {arXiv:2208.05537},
} }

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@@ -1 +1,197 @@
\chapter{Introduction} \chapter{Introduction}
\label{ch:Introduction}
\acresetall
% Intro to quantum computing
In 1982, Richard Feynman, motivated by the difficulty of simulating
quantum-mechanical systems on classical hardware, put forward the
idea of building computers that are themselves quantum mechanical
\cite{feynman_simulating_1982}.
The use of such quantum computers has since been shown to offer promising
prospects not only with regard to simulating quantum systems but also
for solving certain kinds of problems that are classically intractable.
The most prominent example is Shor's algorithm for integer
factorization \cite{shor_algorithms_1994}.
Similar to the way classical computers are built from bits and gates,
quantum computers are built from \emph{qubits} and \emph{quantum gates}.
Because of quantum entanglement, it does not suffice to consider the
qubits individually, we also have to consider correlations between them.
For a system of $n$ qubits, this makes the state space grow with
$2^n$ instead of linearly with $n$, as would be the case for a classical system
\cite[Sec.~1]{gottesman_stabilizer_1997}.
This is both the reason quantum systems are difficult to simulate and
what provides them with their power \cite[Sec.~2.1]{roffe_decoding_2020}.
% The need for QEC
Realizing algorithms that leverage these quantum-mechanical effects
requires hardware that can execute long quantum computations reliably.
This poses a problem, because the qubits making up current devices
consistently interact with their environment \cite[Sec.~1]{roffe_quantum_2019}.
This interaction acts as a continuous small-scale measurement, an
effect we call \emph{decoherence} of the stored quantum state, which
results in errors on the qubits.
Decoherence is the reason large systems do not exhibit visible quantum
properties at human scales \cite[Sec.~1]{gottesman_stabilizer_1997}.
% Intro to QEC
\Ac{qec} has emerged as a leading candidate in solving this problem.
It addresses the issue by encoding the information of $k$
\emph{logical qubits} into a larger number $n>k$ of \emph{physical
qubits}, in close analogy to classical channel coding
\cite[Sec.~1]{roffe_quantum_2019}.
The redundancy introduced this way can then be used to detect and
correct a corrupted the quantum state.
The quantum setting imposes some important constraints that do not exist in the
classical case, however \cite[Sec.~2.4]{roffe_quantum_2019}:
\begin{itemize}
\item The no-cloning theorem prohibits the duplication of quantum states.
\item In addition to the bit-flip errors we know from the
classical setting, qubits are subject to \emph{phase-flips}.
\item We are not allowed to directly measure the encoded qubits,
as that would collapse their quantum states.
\end{itemize}
We can deal with the first constraint by not duplicating information, instead
spreading the quantum state across the physical qubits
\cite[Sec.~I]{calderbank_good_1996}.
To deal with phase-flip errors, we must take special care when
constructing \ac{qec} codes.
Using \ac{css} codes, for example, we can use two separate classical
binary linear codes to protect against the two kinds of errors
\cite[Sec. 10.5.6]{nielsen_quantum_2010}.
Finally, we can get around the last issue by using \emph{stabilizer
measurements}.
These are parity measurements that give us information about
potential errors without revealing the underlying qubit states
\cite[Sec.~II.C.]{babar_fifteen_2015}.
This way, we perform a \emph{syndrome extraction} and base the
subsequent decoding process on the measured syndrome.
Another difference between \ac{qec} and classical channel coding is
the resource constraints.
For \ac{qec}, achieving low latency matters more than having a low
overall computational complexity, due to the backlog problem
\cite[Sec.~II.G.3.]{terhal_quantum_2015}: Certain gates turn
single-qubit errors into multi-qubit ones, so errors must be
corrected beforehand.
A \ac{qec} system that is too slow accumulates a backlog at these points,
causing exponential slowdown.
Several code constructions have been proposed for \ac{qec} codes over the years.
Topological codes, such as surface codes, have been the industry
standard for experimental applications for a long time
\cite[Sec.~I]{koutsioumpas_colour_2025}, due to their
reliance on only local connections between qubits
\cite[Sec.~5]{roffe_decoding_2020}.
Recently, \ac{qldpc} codes have been getting increasing
attention as they have been shown to offer comparable thresholds with
substantially improved encoding rates \cite[Sec.~1]{bravyi_high-threshold_2024}.
\ac{qldpc} codes are generally decoded using a syndrome-based variant
of the \ac{bp} algorithm \cite[Sec.~1]{roffe_decoding_2020}.
We focus on \ac{qldpc} codes in our work and specifically \ac{bb} codes,
as they are promising candidates for practical QEC due to their high
encoding rates, large minimum distances, and short-depth syndrome
extraction circuits \cite[Sec.~1]{bravyi_high-threshold_2024}.
% DEMs and fault tolerance
The syndrome extraction itself is implemented on quantum hardware and
is therefore subject to the same noise as the data qubits.
As a consequence, the \ac{qec} procedure, meant to protect the quantum
state, itself introduces new \emph{internal errors}.
A procedure is called \emph{fault-tolerant} if it remains effective
even in the presence of these internal errors
\cite[Sec.~4]{gottesman_introduction_2009}.
To deal with internal errors that flip syndrome bits, multiple rounds
of syndrome measurements are performed.
One approach of implementing fault tolerance is using \acp{dem}.
A \ac{dem} abstracts away the underlying circuit,
focusing only on the relationship between possible errors
and their effects on the syndrome \cite[Sec.~1.4.3]{higgott_practical_2024}.
A \emph{detector error matrix} is generated from the circuit, which is
used for decoding instead of the original check matrix.
The detector error matrix is much larger than the
check matrix of the underlying code, since it needs to represent many
more error locations.
For example, in our experiments using the $\llbracket 144,12,12
\rrbracket$ \ac{bb} code with $12$ syndrome measurement rounds, the
number of \acp{vn} grew from $144$ to $9504$ and the number of
\acp{cn} grew from $72$ to $1008$.
Therefore, decoding under a \ac{dem} poses a challenge with respect to the
latency constraint.
To keep the latency of \ac{dem} decoding manageable, one approach is
\emph{sliding-window decoding}.
Instead of decoding on the entire detector error matrix at once,
it is partitioned into several overlapping windows.
Once decoding of one window is complete, error estimates on the initial part
that is no longer needed are committed, and the next window is processed.
This way, decoding can start as soon as the syndrome bits required
for the first window have been extracted.
The idea originates with the \emph{overlapping recovery} scheme
proposed for the surface code in
\cite[Sec.~IV.B]{dennis_topological_2002} and has since been studied
for surface and toric codes \cite{kuo_fault-tolerant_2024} as well as
for \ac{qldpc} codes under both phenomenological and circuit-level
noise \cite{huang_increasing_2024,gong_toward_2024,kang_quits_2025}.
% Reseach gap + our work
We observe a structural similarity between sliding-window decoding for
\acp{dem} and window decoding for \ac{sc}-\acs{ldpc} codes.
In contrast to the latter, however, where \ac{bp} messages are
carried between windows \cite[Sec.~III.~C.]{hassan_fully_2016},
the existing realizations of sliding-window decoding for \ac{qec}
discard the soft information produced inside one window before moving
to the next.
We propose \emph{warm-start sliding-window decoding}, in which the
\ac{bp} messages from the overlap region of the previous window are
reused to initialize \ac{bp} in the current window in place of the
standard cold-start initialization.
We formulate the warm start for standard \ac{bp} and for
\ac{bpgd}, a variant of \ac{bp} with better convergence properties
for \ac{qec} codes.
The decoders are evaluated by Monte Carlo simulation on the
$\llbracket 144,12,12 \rrbracket$ \ac{bb} code under standard
circuit-based depolarizing noise over $12$ syndrome extraction rounds.
The main finding is that warm-starting yields a consistent
improvement at low iteration budgets, which is the regime relevant for
low-latency operation.
% Outline of the Thesis
This thesis is structured as follows:
\Cref{ch:Fundamentals} reviews the fundamentals of classical and
quantum error correction.
On the classical side, it covers binary linear block codes,
\ac{ldpc} and \ac{sc}-\ac{ldpc} codes, and the \ac{bp} decoding
algorithm.
On the quantum side, it introduces the relevant quantum mechanical
notation, stabilizer measurements, stabilizer codes, \acf{css} codes,
\ac{qldpc} codes, and the \ac{bpgd} algorithm.
\Cref{ch:Fault tolerance} introduces fault-tolerant \ac{qec}.
It formalizes the notion of fault tolerance, presents the noise
models considered in this work, and develops the \ac{dem} formalism
through the measurement syndrome matrix, the detector matrix, and the
detector error matrix.
The chapter closes with a discussion of practical considerations
including the choice of noise model, the per-round \acf{ler}, and the
Stim toolchain.
\Cref{ch:Decoding} considers practical aspects of decoding under \acp{dem}.
It reviews the existing literature on sliding-window decoding for
\ac{qec}, develops the formal windowing construction we build upon,
introduces the proposed warm-start sliding-window decoder for
plain \ac{bp} and for \ac{bpgd}, and reports numerical results on the
$\llbracket 144,12,12 \rrbracket$ \ac{bb} code.
% TODO: Possibly extend to mention specific proposed research directions
\Cref{ch:Conclusion} concludes the thesis and outlines directions for
further research.

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@@ -1 +1,115 @@
\chapter{Conclusion and Outlook} \chapter{Conclusion and Outlook}
\label{ch:Conclusion}
% Recap of motivation
This thesis investigates decoding under \acp{dem} for fault-tolerant
\ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes.
The repetition of the syndrome measurements, especially under
consideration of circuit-level noise, leads to a significant increase
in decoding complexity: In our experiments on the $\llbracket
144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction
rounds, the check matrix grows from 144 \acp{vn} and 72
\acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}.
% Recap of research gap and own work
Sliding-window decoding addresses the latency constraint by
exploiting the time-like locality of the syndrome extraction circuit.
This manifests as a block-diagonal structure in the detector error
matrix when detectors are defined as the difference of consecutive
syndrome measurement rounds.
We draw a comparison to windowed decoding for \ac{sc}-\ac{ldpc}
codes, but note that the existing realizations of sliding-window
decoding discard the soft information produced inside one window
before moving to the next.
Building on this observation, we proposed warm-start sliding-window
decoding, in which the \ac{bp} messages on the edges crossing into
the overlap region of the previous window are reused to initialise
the corresponding messages of the next window in place of the
standard cold-start initialisation.
We formulate the warm start for standard \ac{bp} and for
\ac{bpgd}.
The latter is particularly attractive as an inner decoder because it
addresses the convergence problems caused by short cycles and
degeneracy in \ac{qldpc} Tanner graphs.
The decoders are evaluated by conducting Monte Carlo simulations on the
$\llbracket 144,12,12 \rrbracket$ \ac{bb} code over $12$ syndrome
extraction rounds under standard circuit-based depolarizing noise.
We focus on a qualitative analysis, refraining from further
optimizations such as introducing a normalization parameter for the
min-sum algorithm.
% Recap of experimental conclusions
For standard min-sum \ac{bp}, the warm start is consistently
beneficial to the cold start, across the considered parameter ranges.
The size of the gain depends on the overlap between consecutive
windows: Enlarging $W$ or shrinking $F$, both of which enlarge the
overlap, result in larger gains of the warm-start.
We observe that the underlying mechanism is an effective increase in
the number of \ac{bp} iterations spent on the \acp{vn} in the overlap
region: Each such \ac{vn} is processed by multiple consecutive window
invocations, and the warm start lets these invocations accumulate
iterations on the same \acp{vn} rather than restarting from scratch.
The gain was most pronounced at low numbers of maximum iterations, where
every additional iteration carries proportionally more information.
For \ac{bpgd}, we note that more information is available in the
overlap region of a window: In addition to the \ac{bp} messages,
there is information about which \acp{vn} were decimated and to what value.
Passing this decimation information to the next window in addition to
the messages turned out to worsen the performance considerably, which
we attributed to a premature hard decision of the \acp{vn} in the
overlap region.
Restricting the warm start to the \ac{bp} messages alone, removed this effect.
The resulting message-only warm start recovered a consistent
improvement over cold-start that followed the same qualitative
behaviour as for standard \ac{bp}: Larger overlap, achieved by larger
$W$ or smaller $F$, yielded a larger gain, and the
performance difference is most pronounced at low numbers of maximum iterations.
% Implications from experimental results
These observations imply that the warm-start modification to
sliding-window decoding can provide a consistent improvement, as long as
some care is taken with specifying the information to be passed to
the subsequent window.
Note that this comes at no additional cost to the decoding complexity,
since the only difference between warm- and cold-start sliding-window
decoding is the initialization of the \ac{bp} messages.
We expect similar behavior with other inner decoders that support
soft information initialization in the overlap region.
% Further research
Several directions for further research emerge from this work.
The most immediate is an extension of the evaluation to other
\ac{qldpc} code families, to other circuit-level noise models such as
SI1000 or EM3, and to a range of code sizes.
This would clarify the generality of the gain due to the warm-start
initialization.
We expect the qualitative findings to carry over, since the
underlying mechanism is structural rather than code-specific, but
quantifying the gain across code families and noise models is left to
future work.
A second direction is a systematic study of other inner decoders under the
warm-start framework, such as automorphism ensemble decoding
\cite{koutsioumpas_automorphism_2025} or neural \ac{bp}
\cite{miao_quaternary_2025}.
A final direction is suggested by the structural similarity between
sliding-window decoding for \acp{dem} and windowed decoding for
\ac{sc}-\ac{ldpc} codes.
The current approach to generating the syndrome extraction circuitry
necessarily leads to a coupling width of one between adjacent
syndrome measurement rounds.
A natural question is whether the coupling width could be
increased, e.g., by interleaving two separate realizations of the
syndrome measurement circuitry instead of always repeating the same one.
Work in this direction would also be a step toward bringing
sliding-window decoding under DEMs within the scope of the analytical
machinery developed for SC-LDPC codes.

View File

@@ -0,0 +1,58 @@
\chapter*{Abstract}
% Current state of the art
\Ac{qec} protects fragile quantum states against decoherence by
encoding logical information into a larger number of physical qubits.
To obtain parity information on an encoded state without disturbing it, a
syndrome extraction is performed.
Because the syndrome extraction circuitry is itself implemented on
noisy quantum hardware, practical \ac{qec} must be fault-tolerant,
accounting for errors introduced by the correction procedure itself.
Fault tolerance considerations and the syndrome extraction circuit
are captured by \acp{dem}, which provide a unified framework for passing
this information to the decoder.
Accounting for fault tolerance substantially inflates the
decoding problem.
At the same time, \ac{qec} imposes strict latency constraints due to
the backlog problem, where syndrome data accumulates faster than it
can be decoded.
Together, these factors pose a serious challenge for practical decoders.
Sliding-window decoding addresses this challenge by exploiting the
repeated structure of the syndrome extraction circuitry, partitioning
the check matrix of the \ac{dem} into overlapping windows that can be
decoded sequentially.
Therefore, decoding can begin as soon as the syndrome components
associated with the first window have been measured.
% Our work: Identify research gap
In this thesis, we perform a review of the existing literature on
sliding-window decoding and draw an analogy to windowed
decoding of classical spatially-coupled low-density parity-check
(\acs{sc}-\acs{ldpc}) codes.
We recognize that in contrast to the latter, existing realizations
of sliding-window decoding for \ac{qec} discard the soft information
produced inside one window before moving to the subsequent window.
% Our work: Warm-start
% TODO: Quantify improvement. Also for conclusion
To take this information into account, we propose warm-start
sliding-window decoding, in which the \ac{bp} messages on the edges
crossing into the overlap region of the previous window are reused to
initialize the corresponding messages of the next window.
The warm start is formulated first for standard \ac{bp} and then extended to
\ac{bp} with guided decimation (\acs{bpgd}).
For both standard \ac{bp} and \ac{bpgd} decoding, the warm-start
initialization provides a consistent improvement across all examined
parameter settings.
We attribute this to an effective increase in \ac{bp} iterations on
variable nodes in the overlap regions: Each such VN is processed by
multiple consecutive windows, and warm-starting lets these
invocations accumulate iterations rather than restart from scratch.
Crucially, the warm-start modification incurs no additional
computational cost relative to cold-start decoding, as it differs
only in the initialization of the \ac{bp} messages.

View File

@@ -1,2 +1,111 @@
sed -i "s/Świerkowska/{\\\\'S}wierkowska/" bibliography.bib sed -i "s/Świerkowska/{\\\\'S}wierkowska/" bibliography.bib
sed -i "s/Héctor/H{\\\\'e}ctor/" bibliography.bib
sed -i "s/Bombín/Bomb{\\\\'i}n/" bibliography.bib
sed -i "s/Zémor/Z{\\\\'e}mor/" bibliography.bib
sed -Ezi "s/\s(abstract|note|urldate|url|keywords|file) = \{[^}]*(\{[^}]*\}[^}]*)*\},?\n//g" bibliography.bib sed -Ezi "s/\s(abstract|note|urldate|url|keywords|file) = \{[^}]*(\{[^}]*\}[^}]*)*\},?\n//g" bibliography.bib
# Normalize arXiv-only entries to @misc with howpublished = {arXiv:<id>}.
# Detection: doi matches 10.48550/arXiv.<id>. The IEEEtranSA .bst's @article
# handler needs a journal field (which preprints lack) and ignores publisher,
# so for arXiv preprints we coerce the type to @misc and add howpublished
# (the field the .bst actually prints for @misc).
python3 - <<'PY'
import re
path = "bibliography.bib"
with open(path) as f:
text = f.read()
doi_re = re.compile(r"doi\s*=\s*\{10\.48550/arXiv\.([^}]+)\}")
type_re = re.compile(r"^@([A-Za-z]+)\{", re.MULTILINE)
howpublished_re = re.compile(r"^\s*howpublished\s*=\s*\{", re.MULTILINE)
title_field_re = re.compile(r"\b(title|booktitle)\s*=\s*\{", re.IGNORECASE)
inner_brace_re = re.compile(r"\{([A-Za-z0-9]+)\}")
# Split into entries by scanning for top-level "@type{...}" blocks. We walk
# brace depth so that the closing "}" of the entry is matched correctly even
# if internal fields contain braces.
def split_entries(s):
out, i, n = [], 0, len(s)
while i < n:
m = type_re.search(s, i)
if not m:
out.append(("text", s[i:]))
break
if m.start() > i:
out.append(("text", s[i:m.start()]))
depth, j = 0, m.start()
while j < n:
c = s[j]
if c == "{":
depth += 1
elif c == "}":
depth -= 1
if depth == 0:
j += 1
break
j += 1
out.append(("entry", s[m.start():j]))
i = j
return out
def normalize_arxiv(entry):
doi_m = doi_re.search(entry)
if not doi_m:
return entry
arxiv_id = doi_m.group(1)
entry = type_re.sub("@misc{", entry, count=1)
if not howpublished_re.search(entry):
# insert howpublished as the last field, before the entry-closing "}"
entry = re.sub(
r"(,?)(\s*)\}\s*$",
lambda m: ("," if m.group(1) != "," else m.group(1))
+ m.group(2) + "\thowpublished = {arXiv:" + arxiv_id + "},\n}",
entry,
count=1,
)
return entry
# Strip protective braces around words inside title/booktitle values.
# BibTeX uses "{Word}" inside titles to preserve case against the bibliography
# style's title-casing rules. We keep that protection only when every character
# inside the braces is non-lowercase (e.g. acronyms like {NASA}); for ordinary
# words like {Quantum} we drop the braces so the style's casing applies.
def strip_title_braces(entry):
out, i, n = [], 0, len(entry)
while True:
m = title_field_re.search(entry, i)
if not m:
out.append(entry[i:])
break
out.append(entry[i:m.end()])
depth, j = 1, m.end()
while j < n and depth > 0:
c = entry[j]
if c == "{":
depth += 1
elif c == "}":
depth -= 1
if depth == 0:
break
j += 1
value = entry[m.end():j]
cleaned = inner_brace_re.sub(
lambda mm: mm.group(1) if any(c.islower() for c in mm.group(1)) else mm.group(0),
value,
)
out.append(cleaned)
if j < n:
out.append(entry[j])
i = j + 1
return "".join(out)
def transform(entry):
return strip_title_braces(normalize_arxiv(entry))
parts = split_entries(text)
new_text = "".join(transform(p) if kind == "entry" else p for kind, p in parts)
with open(path, "w") as f:
f.write(new_text)
PY

188
src/thesis/copy_sim_results.sh Executable file
View File

@@ -0,0 +1,188 @@
#!/bin/bash
BASE_PATH="/home/andreas/workspace/private/ma-sw-results/outputs/"
# Copy BP param exploration results
function post_process_LERs() {
local filename="$1"
python3 -c "
import pandas as pd
import numpy as np
df = pd.read_csv('${filename}')
df['LER_per_round'] = 1 - (1 - df['LER'])**(1/12)
df['num_errors'] = df['num_trials'] * df['LER']
df.to_csv('${filename}', index=False)
"
}
i=1
sp="/-\|"
# echo "Copying BP param exploration results..."
# echo -n ' '
# for decoder in "WindowingSyndromeMinSumDecoder" "WindowingSyndromeSpaDecoder"; do
# for max_iter in 32 200 5000; do
# for pass_soft_info in "True" "False"; do
# for F in 1 2 3; do
# for W in 3 4 5; do
# SRC_PATH="${BASE_PATH}+rust_exp=soft_v_hard_bp,decoder.class_name=${decoder},decoder.max_iter=${max_iter},decoder.pass_soft_info=${pass_soft_info},system.F=${F},system.W=${W}/"
# LATEST_RESULTS_DIR=$(ls -t ${SRC_PATH} | head -1)
# SRC_FILE="${SRC_PATH}/${LATEST_RESULTS_DIR}/LERs.csv"
# DEST_DIR="res/sim/WF/${decoder}/max_iter_${max_iter}/pass_soft_info_${pass_soft_info}/F_${F}/W_${W}/"
# mkdir -p ${DEST_DIR}
# DEST_FILE="${DEST_DIR}/LERs.csv"
# cp ${SRC_FILE} ${DEST_FILE}
# post_process_LERs ${DEST_FILE}
# printf "\b${sp:i++%${#sp}:1}"
# done
# done
# done
# done
# done
#
# # Copy BPGD param exploration results
#
# echo -e "\rCopying BPGD param exploration results..."
# echo -n ' '
# for max_iter in 32 200 5000; do
# for pass_soft_info in "True" "False"; do
# for F in 1 2 3; do
# for W in 3 4 5; do
# SRC_PATH="${BASE_PATH}/+rust_exp=soft_v_hard_bpgd,decoder.class_name=WindowingSyndromeSpaGdDecoder,decoder.max_iter=${max_iter},decoder.pass_soft_info=${pass_soft_info},system.F=${F},system.W=${W}/"
# LATEST_RESULTS_DIR=$(ls -t ${SRC_PATH} | head -1)
# SRC_FILE="${SRC_PATH}/${LATEST_RESULTS_DIR}/LERs.csv"
# DEST_DIR="res/sim/WF/WindowingSyndromeSpaGdDecoder/max_iter_${max_iter}/pass_soft_info_${pass_soft_info}/F_${F}/W_${W}/"
# mkdir -p ${DEST_DIR}
# DEST_FILE="${DEST_DIR}/LERs.csv"
# cp ${SRC_FILE} ${DEST_FILE}
# post_process_LERs ${DEST_FILE}
# printf "\b${sp:i++%${#sp}:1}"
# done
# done
# done
# done
#
# # Copy BP over max iter. results
#
# echo -e "\rCopying BP over max. iter. results..."
# echo -n ' '
# for decoder in "WindowingSyndromeMinSumDecoder" "WindowingSyndromeSpaDecoder"; do
# for p in 0.001 0.0025 0.004; do
# for pass_soft_info in "True" "False"; do
# for F in 1 2 3; do
# for W in 3 4 5; do
# SRC_PATH="${BASE_PATH}+rust_exp=max_iter_bp,decoder.class_name=${decoder},decoder.pass_soft_info=${pass_soft_info},simulation.phy_err_rate=${p},system.F=${F},system.W=${W}/"
# LATEST_RESULTS_DIR=$(ls -t ${SRC_PATH} | head -1)
# SRC_FILE="${SRC_PATH}/${LATEST_RESULTS_DIR}/LERs.csv"
# DEST_DIR="res/sim/max_iter/${decoder}/p_${p}/pass_soft_info_${pass_soft_info}/F_${F}/W_${W}"
# mkdir -p ${DEST_DIR}
# DEST_FILE="${DEST_DIR}/LERs.csv"
# cp ${SRC_FILE} ${DEST_FILE}
# post_process_LERs ${DEST_FILE}
# printf "\b${sp:i++%${#sp}:1}"
# done
# done
# done
# done
# done
#
# # Copy BPGD over max iter. results
#
# echo -e "\rCopying BPGD over max. iter. results..."
# echo -n ' '
# for p in 0.001 0.0025 0.004; do
# for pass_soft_info in "True" "False"; do
# for F in 1 2 3; do
# for W in 3 4 5; do
# SRC_PATH="${BASE_PATH}+rust_exp=max_iter_bpgd,decoder.class_name=WindowingSyndromeSpaGdDecoder,decoder.pass_soft_info=${pass_soft_info},simulation.phy_err_rate=${p},system.F=${F},system.W=${W}/"
# LATEST_RESULTS_DIR=$(ls -t ${SRC_PATH} | head -1)
# SRC_FILE="${SRC_PATH}/${LATEST_RESULTS_DIR}/LERs.csv"
# DEST_DIR="res/sim/max_iter/WindowingSyndromeSpaGdDecoder/p_${p}/pass_soft_info_${pass_soft_info}/F_${F}/W_${W}"
# mkdir -p ${DEST_DIR}
# DEST_FILE="${DEST_DIR}/LERs.csv"
# cp ${SRC_FILE} ${DEST_FILE}
# post_process_LERs ${DEST_FILE}
# printf "\b${sp:i++%${#sp}:1}"
# done
# done
# done
# done
#
# # Copy BP over max iter. results
#
# echo -e "\rCopying one-shot simulation results..."
# echo -n ' '
# for decoder in "SyndromeMinSumDecoder" "SyndromeSpaDecoder" "SyndromeSpaGdDecoder"; do
# for max_iter in 32 200 5000; do
# SRC_PATH="${BASE_PATH}+rust_exp=whole_bp_bpgd,decoder.class_name=${decoder},decoder.max_iter=${max_iter},system.F=1,system.W=5/"
# LATEST_RESULTS_DIR=$(ls -t ${SRC_PATH} | head -1)
# SRC_FILE="${SRC_PATH}/${LATEST_RESULTS_DIR}/LERs.csv"
# DEST_DIR="res/sim/one-shot/${decoder}/max_iter_${max_iter}/"
# mkdir -p ${DEST_DIR}
# DEST_FILE="${DEST_DIR}/LERs.csv"
# cp ${SRC_FILE} ${DEST_FILE}
# post_process_LERs ${DEST_FILE}
# printf "\b${sp:i++%${#sp}:1}"
# done
# done
# Copy whole BP over max iter. results
echo -e "\rCopying whole over max_iter simulation results..."
echo -n ' '
for decoder in "SyndromeMinSumDecoder"; do
for p in 0.001 0.0025 0.004; do
SRC_PATH="${BASE_PATH}+rust_exp=max_iter_bp,decoder.class_name=${decoder},simulation.phy_err_rate=${p}/"
LATEST_RESULTS_DIR=$(ls -t ${SRC_PATH} | head -1)
SRC_FILE="${SRC_PATH}/${LATEST_RESULTS_DIR}/LERs.csv"
DEST_DIR="res/sim/max_iter/${decoder}/p_${p}/"
mkdir -p ${DEST_DIR}
DEST_FILE="${DEST_DIR}/LERs.csv"
cp ${SRC_FILE} ${DEST_FILE}
post_process_LERs ${DEST_FILE}
printf "\b${sp:i++%${#sp}:1}"
done
done
# # Copy BPGD decimation passing
#
# echo -e "\rCopying BPGD param exploration results..."
# echo -n ' '
# for max_iter in 32 200 5000; do
# for F in 1 2 3; do
# for W in 3 4 5; do
# SRC_PATH="${BASE_PATH}+rust_exp=soft_v_hard_bpgd_pass_channel,decoder.class_name=WindowingSyndromeSpaGdDecoder,decoder.max_iter=${max_iter},decoder.pass_soft_info=True,system.F=${F},system.W=${W}"
# LATEST_RESULTS_DIR=$(ls -t ${SRC_PATH} | head -1)
# SRC_FILE="${SRC_PATH}/${LATEST_RESULTS_DIR}/LERs.csv"
# DEST_DIR="res/sim/WF/WindowingSyndromeSpaGdDecoderPassDecimation/max_iter_${max_iter}/pass_soft_info_True/F_${F}/W_${W}/"
# mkdir -p ${DEST_DIR}
# DEST_FILE="${DEST_DIR}/LERs.csv"
# cp ${SRC_FILE} ${DEST_FILE}
# post_process_LERs ${DEST_FILE}
# printf "\b${sp:i++%${#sp}:1}"
# done
# done
# done
# Copy BPGD with decimation info passing over max iter. results
# echo -e "\rCopying BPGD over max. iter. results..."
# echo -n ' '
# for pass_soft_info in "True" "False"; do
# for F in 1 2 3; do
# for W in 3 4 5; do
# SRC_PATH="${BASE_PATH}+rust_exp=max_iter_bpgd_pass_channel,decoder.class_name=WindowingSyndromeSpaGdDecoder,decoder.pass_soft_info=${pass_soft_info},simulation.phy_err_rate=0.0025,system.F=${F},system.W=${W}/"
# LATEST_RESULTS_DIR=$(ls -t ${SRC_PATH} | head -1)
# SRC_FILE="${SRC_PATH}/${LATEST_RESULTS_DIR}/LERs.csv"
# DEST_DIR="res/sim/max_iter/WindowingSyndromeSpaGdDecoderPassDecimation/p_0.0025/pass_soft_info_${pass_soft_info}/F_${F}/W_${W}"
# mkdir -p ${DEST_DIR}
# DEST_FILE="${DEST_DIR}/LERs.csv"
# cp ${SRC_FILE} ${DEST_FILE}
# post_process_LERs ${DEST_FILE}
# printf "\b${sp:i++%${#sp}:1}"
# done
# done
# done

View File

@@ -27,6 +27,9 @@
\usepackage[noEnd=false]{algpseudocodex} \usepackage[noEnd=false]{algpseudocodex}
\usepackage{nicematrix} \usepackage{nicematrix}
\usepackage{colortbl} \usepackage{colortbl}
\usepackage{cleveref}
\usepackage{lipsum}
\usepackage{booktabs}
\usetikzlibrary{calc, positioning, arrows, fit} \usetikzlibrary{calc, positioning, arrows, fit}
\usetikzlibrary{external} \usetikzlibrary{external}
@@ -38,6 +41,11 @@
\setcounter{MaxMatrixCols}{20} \setcounter{MaxMatrixCols}{20}
\Crefname{equation}{}{}
\Crefname{section}{Section}{Sections}
\Crefname{subsection}{Section}{Sections}
\Crefname{figure}{Figure}{Figures}
% %
% %
% Custom commands % Custom commands
@@ -45,7 +53,7 @@
% %
\newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\content}[1]{\noindent\indent\red{[#1]}\\} \newcommand{\content}[1]{\noindent\indent\red{[#1]\\}}
\newcommand{\figwidth}{10cm} \newcommand{\figwidth}{10cm}
\newcommand{\figheight}{7.5cm} \newcommand{\figheight}{7.5cm}
@@ -82,10 +90,12 @@
% \thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost} % \thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost}
%\thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost\\Prof. Dr.-Ing. %\thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost\\Prof. Dr.-Ing.
% Laurent Schmalen} % Laurent Schmalen}
\thesisSupervisor{Jonathan Mandelbaum} \thesisSupervisor{Dr.-Ing. Hedongliang Liu\\ && M.Sc. Jonathan Mandelbaum}
\thesisStartDate{01.11.2025} \thesisStartDate{Nov. 1st, 2025}
\thesisEndDate{04.05.2026} \thesisEndDate{May 4th, 2026}
\thesisSignatureDate{Signature date} \thesisSignatureDate{May 4th, 2026}
\thesisSignature{res/Unterschrift_AT_blue.png}
\thesisSignatureHeight{2.4cm}
\thesisLanguage{english} \thesisLanguage{english}
\begin{document} \begin{document}
@@ -94,13 +104,16 @@
\maketitle \maketitle
\newpage \newpage
% \include{chapters/abstract} \include{chapters/abstract}
\cleardoublepage \cleardoublepage
\pagenumbering{arabic} \pagenumbering{arabic}
\tableofcontents \newgeometry{a4paper,left=3cm,right=3cm,top=2cm,bottom=2.5cm}
\addtocontents{toc}{\protect\vspace*{-9mm}}
\tableofcontents
\cleardoublepage \cleardoublepage
\restoregeometry
\input{chapters/1_introduction.tex} \input{chapters/1_introduction.tex}
\input{chapters/2_fundamentals.tex} \input{chapters/2_fundamentals.tex}
@@ -113,6 +126,11 @@
% \listoftables % \listoftables
% \include{abbreviations} % \include{abbreviations}
\cleardoublepage
\phantomsection
\addcontentsline{toc}{chapter}{List of Abbreviations}
\printacronyms
\bibliography{lib/cel-thesis/IEEEabrv,src/thesis/bibliography} \bibliography{lib/cel-thesis/IEEEabrv,src/thesis/bibliography}
\end{document} \end{document}

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@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,10000,0.0238,0.0020053029820819024,238.00000000000003 0.001,10000,0.0225,0.0018946185336699006,225.0
0.0015,4000,0.072,0.006207614833215747,288.0 0.0015,4000,0.07025,0.006051577178537326,281.0
0.002,2000,0.1545,0.013888248487303967,309.0 0.002,2000,0.1515,0.01359714508496701,303.0
0.0025,2000,0.316,0.03115416825283479,632.0 0.0025,2000,0.317,0.03127228419350614,634.0
0.003,2000,0.5005,0.05620437958483415,1000.9999999999999 0.003,2000,0.494,0.05518696485931962,988.0
0.0035,2000,0.6885,0.09262215539223495,1377.0 0.0035,2000,0.6805,0.09070269229398742,1361.0
0.004,2000,0.8325,0.13834270815997607,1665.0 0.004,2000,0.8305,0.1374899943124941,1661.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 10000 0.0238 0.0225 0.0020053029820819024 0.0018946185336699006 238.00000000000003 225.0
3 0.0015 4000 0.072 0.07025 0.006207614833215747 0.006051577178537326 288.0 281.0
4 0.002 2000 0.1545 0.1515 0.013888248487303967 0.01359714508496701 309.0 303.0
5 0.0025 2000 0.316 0.317 0.03115416825283479 0.03127228419350614 632.0 634.0
6 0.003 2000 0.5005 0.494 0.05620437958483415 0.05518696485931962 1000.9999999999999 988.0
7 0.0035 2000 0.6885 0.6805 0.09262215539223495 0.09070269229398742 1377.0 1361.0
8 0.004 2000 0.8325 0.8305 0.13834270815997607 0.1374899943124941 1665.0 1661.0

View File

@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,14000,0.0152142857142857,0.0012767850842765327,212.9999999999998 0.001,14000,0.0149285714285714,0.0012526417987225313,208.9999999999996
0.0015,6000,0.0423333333333333,0.003598136858263712,253.9999999999998 0.0015,6000,0.0413333333333333,0.0035114743705089158,247.9999999999998
0.002,4000,0.09125,0.007942064662371462,365.0 0.002,4000,0.09075,0.00789658974957197,363.0
0.0025,2000,0.2125,0.019710810011481006,425.0 0.0025,2000,0.2125,0.019710810011481006,425.0
0.003,2000,0.3495,0.03520004381148534,699.0 0.003,2000,0.3505,0.03532372820929974,701.0
0.0035,2000,0.5385,0.062407102537387016,1077.0 0.0035,2000,0.545,0.06351473218845116,1090.0
0.004,2000,0.73,0.10336921268218224,1460.0 0.004,2000,0.733,0.10420368457269413,1466.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 14000 0.0152142857142857 0.0149285714285714 0.0012767850842765327 0.0012526417987225313 212.9999999999998 208.9999999999996
3 0.0015 6000 0.0423333333333333 0.0413333333333333 0.003598136858263712 0.0035114743705089158 253.9999999999998 247.9999999999998
4 0.002 4000 0.09125 0.09075 0.007942064662371462 0.00789658974957197 365.0 363.0
5 0.0025 2000 0.2125 0.019710810011481006 425.0
6 0.003 2000 0.3495 0.3505 0.03520004381148534 0.03532372820929974 699.0 701.0
7 0.0035 2000 0.5385 0.545 0.062407102537387016 0.06351473218845116 1077.0 1090.0
8 0.004 2000 0.73 0.733 0.10336921268218224 0.10420368457269413 1460.0 1466.0

View File

@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,4000,0.09525,0.00830669189113975,381.0 0.001,4000,0.0955,0.008329530127408447,382.0
0.0015,2000,0.2055,0.018987611527110704,411.0 0.0015,2000,0.206,0.019039074473767514,412.0
0.002,2000,0.357,0.03613195793118629,714.0 0.002,2000,0.3575,0.0361944392516631,715.0
0.0025,2000,0.545,0.06351473218845116,1090.0 0.0025,2000,0.5435,0.06325784394063028,1087.0
0.003,2000,0.6935,0.09384489827464226,1387.0 0.003,2000,0.6975,0.09483632938531728,1395.0
0.0035,2000,0.832,0.1381286582468193,1664.0 0.0035,2000,0.8315,0.1379151915045972,1663.0
0.004,2000,0.93,0.19876821407276757,1860.0 0.004,2000,0.931,0.19972836446856335,1862.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 4000 0.09525 0.0955 0.00830669189113975 0.008329530127408447 381.0 382.0
3 0.0015 2000 0.2055 0.206 0.018987611527110704 0.019039074473767514 411.0 412.0
4 0.002 2000 0.357 0.3575 0.03613195793118629 0.0361944392516631 714.0 715.0
5 0.0025 2000 0.545 0.5435 0.06351473218845116 0.06325784394063028 1090.0 1087.0
6 0.003 2000 0.6935 0.6975 0.09384489827464226 0.09483632938531728 1387.0 1395.0
7 0.0035 2000 0.832 0.8315 0.1381286582468193 0.1379151915045972 1664.0 1663.0
8 0.004 2000 0.93 0.931 0.19876821407276757 0.19972836446856335 1860.0 1862.0

View File

@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,10000,0.0241,0.002030864734142268,241.0 0.001,10000,0.0235,0.00197974842986981,235.0
0.0015,4000,0.06575,0.00565157033403707,263.0 0.0015,4000,0.065,0.005585074297480008,260.0
0.002,2000,0.141,0.012585659483247746,282.0 0.002,2000,0.139,0.012394281484896852,278.0
0.0025,2000,0.2935,0.028537560287876573,587.0 0.0025,2000,0.29,0.028137416075114108,580.0
0.003,2000,0.4435,0.0476671526131055,887.0 0.003,2000,0.4435,0.0476671526131055,887.0
0.0035,2000,0.6345,0.08045296488086273,1269.0 0.0035,2000,0.6325,0.08003470274679836,1265.0
0.004,2000,0.8025,0.1264309108240147,1605.0 0.004,2000,0.8015,0.12606316890291858,1603.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 10000 0.0241 0.0235 0.002030864734142268 0.00197974842986981 241.0 235.0
3 0.0015 4000 0.06575 0.065 0.00565157033403707 0.005585074297480008 263.0 260.0
4 0.002 2000 0.141 0.139 0.012585659483247746 0.012394281484896852 282.0 278.0
5 0.0025 2000 0.2935 0.29 0.028537560287876573 0.028137416075114108 587.0 580.0
6 0.003 2000 0.4435 0.0476671526131055 887.0
7 0.0035 2000 0.6345 0.6325 0.08045296488086273 0.08003470274679836 1269.0 1265.0
8 0.004 2000 0.8025 0.8015 0.1264309108240147 0.12606316890291858 1605.0 1603.0

View File

@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,4000,0.07675,0.006632509273037823,307.0 0.001,4000,0.0755,0.0065205010146435205,302.0
0.0015,2000,0.158,0.014229068201835782,316.0 0.0015,2000,0.1605,0.014473306861186974,321.0
0.002,2000,0.2905,0.028194468681520868,581.0 0.002,2000,0.29,0.028137416075114108,580.0
0.0025,2000,0.4345,0.046393101582849816,869.0 0.0025,2000,0.4385,0.046957034683799304,877.0
0.003,2000,0.6165,0.07676176791425904,1233.0 0.003,2000,0.6175,0.07696262481687499,1235.0
0.0035,2000,0.7815,0.11904386007281431,1563.0 0.0035,2000,0.7815,0.11904386007281431,1563.0
0.004,2000,0.902,0.17598426223523023,1804.0 0.004,2000,0.9015,0.1756347320107573,1803.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 4000 0.07675 0.0755 0.006632509273037823 0.0065205010146435205 307.0 302.0
3 0.0015 2000 0.158 0.1605 0.014229068201835782 0.014473306861186974 316.0 321.0
4 0.002 2000 0.2905 0.29 0.028194468681520868 0.028137416075114108 581.0 580.0
5 0.0025 2000 0.4345 0.4385 0.046393101582849816 0.046957034683799304 869.0 877.0
6 0.003 2000 0.6165 0.6175 0.07676176791425904 0.07696262481687499 1233.0 1235.0
7 0.0035 2000 0.7815 0.11904386007281431 1563.0
8 0.004 2000 0.902 0.9015 0.17598426223523023 0.1756347320107573 1804.0 1803.0

View File

@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,6000,0.0351666666666666,0.0029788796512925098,210.99999999999963 0.001,6000,0.0356666666666666,0.0030219465890483077,213.99999999999957
0.0015,2000,0.102,0.008925364554660087,204.0 0.0015,2000,0.102,0.008925364554660087,204.0
0.002,2000,0.22,0.02049223417716306,440.0 0.002,2000,0.2205,0.02054457368926077,441.0
0.0025,2000,0.39,0.040354525526934304,780.0 0.0025,2000,0.389,0.040223524818782996,778.0
0.003,2000,0.5675,0.06746420864522562,1135.0 0.003,2000,0.567,0.06737441654139043,1134.0
0.0035,2000,0.7385,0.10575612450061989,1477.0 0.0035,2000,0.7395,0.10604159775255029,1479.0
0.004,2000,0.8585,0.15037026489320615,1717.0 0.004,2000,0.8585,0.15037026489320615,1717.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 6000 0.0351666666666666 0.0356666666666666 0.0029788796512925098 0.0030219465890483077 210.99999999999963 213.99999999999957
3 0.0015 2000 0.102 0.008925364554660087 204.0
4 0.002 2000 0.22 0.2205 0.02049223417716306 0.02054457368926077 440.0 441.0
5 0.0025 2000 0.39 0.389 0.040354525526934304 0.040223524818782996 780.0 778.0
6 0.003 2000 0.5675 0.567 0.06746420864522562 0.06737441654139043 1135.0 1134.0
7 0.0035 2000 0.7385 0.7395 0.10575612450061989 0.10604159775255029 1477.0 1479.0
8 0.004 2000 0.8585 0.15037026489320615 1717.0

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@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,8000,0.02775,0.0023424443475220658,222.0 0.001,8000,0.028,0.002363824632785727,224.0
0.0015,4000,0.072,0.006207614833215747,288.0 0.0015,4000,0.07175,0.00618530723400279,287.0
0.002,2000,0.1445,0.012921555968088194,289.0 0.002,2000,0.1445,0.012921555968088194,289.0
0.0025,2000,0.2795,0.026947740549572385,559.0 0.0025,2000,0.2805,0.027060355839749417,561.0
0.003,2000,0.441,0.04731136584915907,882.0 0.003,2000,0.442,0.04745350518327096,884.0
0.0035,2000,0.6185,0.07716396365395395,1237.0 0.0035,2000,0.62,0.0774668808446417,1240.0
0.004,2000,0.788,0.12125812649764789,1576.0 0.004,2000,0.788,0.12125812649764789,1576.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 8000 0.02775 0.028 0.0023424443475220658 0.002363824632785727 222.0 224.0
3 0.0015 4000 0.072 0.07175 0.006207614833215747 0.00618530723400279 288.0 287.0
4 0.002 2000 0.1445 0.012921555968088194 289.0
5 0.0025 2000 0.2795 0.2805 0.026947740549572385 0.027060355839749417 559.0 561.0
6 0.003 2000 0.441 0.442 0.04731136584915907 0.04745350518327096 882.0 884.0
7 0.0035 2000 0.6185 0.62 0.07716396365395395 0.0774668808446417 1237.0 1240.0
8 0.004 2000 0.788 0.12125812649764789 1576.0

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@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,6000,0.0446666666666666,0.003800672363933444,267.9999999999996 0.001,6000,0.0443333333333333,0.0037717109761096212,265.99999999999983
0.0015,2000,0.1105,0.00971052637237435,221.0 0.0015,2000,0.1105,0.00971052637237435,221.0
0.002,2000,0.212,0.019658957946067646,424.0 0.002,2000,0.212,0.019658957946067646,424.0
0.0025,2000,0.384,0.03957145487337721,768.0 0.0025,2000,0.383,0.03944162334187884,766.0
0.003,2000,0.5495,0.06429008164677474,1099.0 0.003,2000,0.55,0.0643766693499016,1100.0
0.0035,2000,0.734,0.10448375252924946,1468.0 0.0035,2000,0.7345,0.10462414862704472,1469.0
0.004,2000,0.8555,0.14888354449188368,1711.0 0.004,2000,0.8555,0.14888354449188368,1711.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 6000 0.0446666666666666 0.0443333333333333 0.003800672363933444 0.0037717109761096212 267.9999999999996 265.99999999999983
3 0.0015 2000 0.1105 0.00971052637237435 221.0
4 0.002 2000 0.212 0.019658957946067646 424.0
5 0.0025 2000 0.384 0.383 0.03957145487337721 0.03944162334187884 768.0 766.0
6 0.003 2000 0.5495 0.55 0.06429008164677474 0.0643766693499016 1099.0 1100.0
7 0.0035 2000 0.734 0.7345 0.10448375252924946 0.10462414862704472 1468.0 1469.0
8 0.004 2000 0.8555 0.14888354449188368 1711.0

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@@ -3,6 +3,6 @@ physical_p,num_trials,LER,LER_per_round,num_errors
0.0015,2000,0.202,0.018628199928893086,404.0 0.0015,2000,0.202,0.018628199928893086,404.0
0.002,2000,0.37,0.0377711386183186,740.0 0.002,2000,0.37,0.0377711386183186,740.0
0.0025,2000,0.512,0.05803452525767827,1024.0 0.0025,2000,0.512,0.05803452525767827,1024.0
0.003,2000,0.6965,0.0945873500188964,1393.0 0.003,2000,0.6975,0.09483632938531728,1395.0
0.0035,2000,0.834,0.13898839048277334,1668.0 0.0035,2000,0.834,0.13898839048277334,1668.0
0.004,2000,0.9305,0.19924670627472074,1861.0 0.004,2000,0.9305,0.19924670627472074,1861.0
1 physical_p num_trials LER LER_per_round num_errors
3 0.0015 2000 0.202 0.018628199928893086 404.0
4 0.002 2000 0.37 0.0377711386183186 740.0
5 0.0025 2000 0.512 0.05803452525767827 1024.0
6 0.003 2000 0.6965 0.6975 0.0945873500188964 0.09483632938531728 1393.0 1395.0
7 0.0035 2000 0.834 0.13898839048277334 1668.0
8 0.004 2000 0.9305 0.19924670627472074 1861.0

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@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,100000,0.00039,3.250581082292481e-05,39.0 0.001,4000,0.0665,0.005718115322006723,266.0
0.0015,84000,0.0023809523809523,0.00019862955014460226,199.9999999999932 0.0015,2000,0.163,0.014718213155383575,326.0
0.002,22000,0.0093181818181818,0.0007798513967101073,204.9999999999996 0.002,2000,0.3335,0.03324443774167962,667.0
0.0025,6000,0.0348333333333333,0.002950179723826829,208.9999999999998 0.0025,2000,0.535,0.06181659632516823,1070.0
0.003,2000,0.102,0.008925364554660087,204.0 0.003,2000,0.7185,0.10024721430418793,1437.0
0.0035,2000,0.2065,0.01909056713578361,413.0 0.0035,2000,0.849,0.14575703232266357,1698.0
0.004,2000,0.37,0.0377711386183186,740.0 0.004,2000,0.928,0.1968850543381443,1856.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 100000 4000 0.00039 0.0665 3.250581082292481e-05 0.005718115322006723 39.0 266.0
3 0.0015 84000 2000 0.0023809523809523 0.163 0.00019862955014460226 0.014718213155383575 199.9999999999932 326.0
4 0.002 22000 2000 0.0093181818181818 0.3335 0.0007798513967101073 0.03324443774167962 204.9999999999996 667.0
5 0.0025 6000 2000 0.0348333333333333 0.535 0.002950179723826829 0.06181659632516823 208.9999999999998 1070.0
6 0.003 2000 0.102 0.7185 0.008925364554660087 0.10024721430418793 204.0 1437.0
7 0.0035 2000 0.2065 0.849 0.01909056713578361 0.14575703232266357 413.0 1698.0
8 0.004 2000 0.37 0.928 0.0377711386183186 0.1968850543381443 740.0 1856.0

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@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,100000,0.00012,1.0000550042188472e-05,12.0 0.001,4000,0.05825,0.004988800426538265,233.0
0.0015,100000,0.00088,7.33629277510639e-05,88.0 0.0015,2000,0.15,0.013451947011868914,300.0
0.002,38000,0.0052894736842105,0.0004418617234457356,200.999999999999 0.002,2000,0.305,0.0298652369138821,610.0
0.0025,10000,0.0219,0.0018435789073465436,219.0 0.0025,2000,0.503,0.056598927298699975,1006.0
0.003,4000,0.062,0.005319578163374583,248.0 0.003,2000,0.6875,0.0923797676224748,1375.0
0.0035,2000,0.131,0.01163282218493733,262.0 0.0035,2000,0.8255,0.13539790019792786,1651.0
0.004,2000,0.257,0.024451045518294245,514.0 0.004,2000,0.9,0.1745958147319816,1800.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 100000 4000 0.00012 0.05825 1.0000550042188472e-05 0.004988800426538265 12.0 233.0
3 0.0015 100000 2000 0.00088 0.15 7.33629277510639e-05 0.013451947011868914 88.0 300.0
4 0.002 38000 2000 0.0052894736842105 0.305 0.0004418617234457356 0.0298652369138821 200.999999999999 610.0
5 0.0025 10000 2000 0.0219 0.503 0.0018435789073465436 0.056598927298699975 219.0 1006.0
6 0.003 4000 2000 0.062 0.6875 0.005319578163374583 0.0923797676224748 248.0 1375.0
7 0.0035 2000 0.131 0.8255 0.01163282218493733 0.13539790019792786 262.0 1651.0
8 0.004 2000 0.257 0.9 0.024451045518294245 0.1745958147319816 514.0 1800.0

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@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,98000,0.0020510204081632,0.00017107925018833292,200.9999999999936 0.001,8000,0.02625,0.00221426836603833,210.0
0.0015,18000,0.0115,0.0009634219750452866,207.0 0.0015,4000,0.0735,0.006341576227078316,294.0
0.002,6000,0.0333333333333333,0.0028211423839156202,199.9999999999998 0.002,2000,0.1445,0.012921555968088194,289.0
0.0025,4000,0.0985,0.0086040412348235,394.0 0.0025,2000,0.3035,0.029690924608377744,607.0
0.003,2000,0.197,0.01811725341494619,394.0 0.003,2000,0.4855,0.05387442629125816,971.0
0.0035,2000,0.361,0.03663306068521166,722.0 0.0035,2000,0.65,0.08376754717420642,1300.0
0.004,2000,0.538,0.06232249368583764,1076.0 0.004,2000,0.7795,0.11837469015649149,1559.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 98000 8000 0.0020510204081632 0.02625 0.00017107925018833292 0.00221426836603833 200.9999999999936 210.0
3 0.0015 18000 4000 0.0115 0.0735 0.0009634219750452866 0.006341576227078316 207.0 294.0
4 0.002 6000 2000 0.0333333333333333 0.1445 0.0028211423839156202 0.012921555968088194 199.9999999999998 289.0
5 0.0025 4000 2000 0.0985 0.3035 0.0086040412348235 0.029690924608377744 394.0 607.0
6 0.003 2000 0.197 0.4855 0.01811725341494619 0.05387442629125816 394.0 971.0
7 0.0035 2000 0.361 0.65 0.03663306068521166 0.08376754717420642 722.0 1300.0
8 0.004 2000 0.538 0.7795 0.06232249368583764 0.11837469015649149 1076.0 1559.0

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@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,100000,0.00024,2.0002200337376763e-05,24.0 0.001,10000,0.0205,0.00172459796871538,205.0
0.0015,100000,0.00194,0.00016181059369868578,194.0 0.0015,4000,0.058,0.004966791530059078,232.0
0.002,22000,0.009090909090909,0.000760750779222108,199.999999999998 0.002,2000,0.141,0.012585659483247746,282.0
0.0025,6000,0.0346666666666666,0.002935833167148738,207.9999999999996 0.0025,2000,0.2655,0.02538599006454134,531.0
0.003,4000,0.09,0.007828420342483211,360.0 0.003,2000,0.4425,0.047524662435589615,885.0
0.0035,2000,0.187,0.01710405023748829,374.0 0.0035,2000,0.609,0.07527046293149453,1218.0
0.004,2000,0.3445,0.03458422416587814,689.0 0.004,2000,0.7395,0.10604159775255029,1479.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 100000 10000 0.00024 0.0205 2.0002200337376763e-05 0.00172459796871538 24.0 205.0
3 0.0015 100000 4000 0.00194 0.058 0.00016181059369868578 0.004966791530059078 194.0 232.0
4 0.002 22000 2000 0.009090909090909 0.141 0.000760750779222108 0.012585659483247746 199.999999999998 282.0
5 0.0025 6000 2000 0.0346666666666666 0.2655 0.002935833167148738 0.02538599006454134 207.9999999999996 531.0
6 0.003 4000 2000 0.09 0.4425 0.007828420342483211 0.047524662435589615 360.0 885.0
7 0.0035 2000 0.187 0.609 0.01710405023748829 0.07527046293149453 374.0 1218.0
8 0.004 2000 0.3445 0.7395 0.03458422416587814 0.10604159775255029 689.0 1479.0

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@@ -1,8 +1,8 @@
physical_p,num_trials,LER,LER_per_round,num_errors physical_p,num_trials,LER,LER_per_round,num_errors
0.001,100000,0.00116,9.671809923239572e-05,116.0 0.001,16000,0.01325,0.0011109294970204076,212.0
0.0015,30000,0.0067,0.0005600552572093687,201.0 0.0015,6000,0.0361666666666666,0.003065034000747535,216.99999999999957
0.002,10000,0.0216,0.0018180698558927233,216.0 0.002,4000,0.095,0.008283859438899643,380.0
0.0025,4000,0.07625,0.006587689298086841,305.0 0.0025,2000,0.1775,0.01615203373482954,355.0
0.003,2000,0.151,0.01354871960160986,302.0 0.003,2000,0.3365,0.03360781199094065,673.0
0.0035,2000,0.299,0.029170044726508193,598.0 0.0035,2000,0.4985,0.055890042576412724,997.0
0.004,2000,0.449,0.04845506933740451,898.0 0.004,2000,0.671,0.08847974243602208,1342.0
1 physical_p num_trials LER LER_per_round num_errors
2 0.001 100000 16000 0.00116 0.01325 9.671809923239572e-05 0.0011109294970204076 116.0 212.0
3 0.0015 30000 6000 0.0067 0.0361666666666666 0.0005600552572093687 0.003065034000747535 201.0 216.99999999999957
4 0.002 10000 4000 0.0216 0.095 0.0018180698558927233 0.008283859438899643 216.0 380.0
5 0.0025 4000 2000 0.07625 0.1775 0.006587689298086841 0.01615203373482954 305.0 355.0
6 0.003 2000 0.151 0.3365 0.01354871960160986 0.03360781199094065 302.0 673.0
7 0.0035 2000 0.299 0.4985 0.029170044726508193 0.055890042576412724 598.0 997.0
8 0.004 2000 0.449 0.671 0.04845506933740451 0.08847974243602208 898.0 1342.0

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@@ -0,0 +1,8 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,10528,0.0354293313069908,0.0030015014232951387,372.99999999999915
128,68422,0.0029230364502645,0.00024391332035389457,199.99999999999764
256,100000,0.00184,0.00015346279666106355,184.0
512,100000,0.00122,0.00010172355962756452,122.0
1024,100000,0.00084,7.002696447200307e-05,84.0
2048,100000,0.00052,4.33436645435048e-05,51.99999999999999
4096,100000,0.00042,3.5006739308451884e-05,42.0
1 max_iter num_trials LER LER_per_round num_errors
2 32 10528 0.0354293313069908 0.0030015014232951387 372.99999999999915
3 128 68422 0.0029230364502645 0.00024391332035389457 199.99999999999764
4 256 100000 0.00184 0.00015346279666106355 184.0
5 512 100000 0.00122 0.00010172355962756452 122.0
6 1024 100000 0.00084 7.002696447200307e-05 84.0
7 2048 100000 0.00052 4.33436645435048e-05 51.99999999999999
8 4096 100000 0.00042 3.5006739308451884e-05 42.0

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@@ -0,0 +1,8 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.3575227963525836,0.03619728900635699,1882.0
128,5264,0.1118920972644376,0.00983977212107956,588.9999999999995
256,5264,0.0818768996960486,0.007093372523371166,430.99999999999983
512,5264,0.0645896656534954,0.005548714177492475,339.9999999999998
1024,5264,0.0524316109422492,0.004477957765848362,275.9999999999998
2048,5264,0.0442629179331307,0.0037655941776483237,233.0
4096,10528,0.0361892097264437,0.0030669771276242708,380.99999999999926
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.3575227963525836 0.03619728900635699 1882.0
3 128 5264 0.1118920972644376 0.00983977212107956 588.9999999999995
4 256 5264 0.0818768996960486 0.007093372523371166 430.99999999999983
5 512 5264 0.0645896656534954 0.005548714177492475 339.9999999999998
6 1024 5264 0.0524316109422492 0.004477957765848362 275.9999999999998
7 2048 5264 0.0442629179331307 0.0037655941776483237 233.0
8 4096 10528 0.0361892097264437 0.0030669771276242708 380.99999999999926

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@@ -0,0 +1,8 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.8651215805471124,0.15375677320993897,4554.0
128,5264,0.5987841945288754,0.0732807818579091,3152.0000000000005
256,5264,0.5184270516717325,0.05907464062706547,2729.0
512,5264,0.4591565349544073,0.04992921073125611,2417.0
1024,5264,0.4209726443768997,0.04451268995716784,2216.0
2048,5264,0.3865881458966565,0.03990837706831185,2035.0
4096,5264,0.3563829787234042,0.036054914686007855,1875.9999999999995
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.8651215805471124 0.15375677320993897 4554.0
3 128 5264 0.5987841945288754 0.0732807818579091 3152.0000000000005
4 256 5264 0.5184270516717325 0.05907464062706547 2729.0
5 512 5264 0.4591565349544073 0.04992921073125611 2417.0
6 1024 5264 0.4209726443768997 0.04451268995716784 2216.0
7 2048 5264 0.3865881458966565 0.03990837706831185 2035.0
8 4096 5264 0.3563829787234042 0.036054914686007855 1875.9999999999995

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@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.5083586626139818,0.05745079247636664,2676.0
128,5264,0.3879179331306991,0.04008199450020078,2042.0000000000002
256,5264,0.364741641337386,0.03710440702030626,1920.0
512,5264,0.3478343465045592,0.03499441487180499,1830.9999999999995
1024,5264,0.3345364741641337,0.033369810593094074,1760.9999999999998
1536,5264,0.2794452887537993,0.026941583361394406,1470.9999999999995
2048,5264,0.1073328267477203,0.009417167606857957,564.9999999999997
2560,5264,0.0381838905775075,0.00323907625680131,200.99999999999946
3072,5264,0.0381838905775075,0.00323907625680131,200.99999999999946
3584,5264,0.0381838905775075,0.00323907625680131,200.99999999999946
4096,5264,0.0381838905775075,0.00323907625680131,200.99999999999946
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.5083586626139818 0.05745079247636664 2676.0
3 128 5264 0.3879179331306991 0.04008199450020078 2042.0000000000002
4 256 5264 0.364741641337386 0.03710440702030626 1920.0
5 512 5264 0.3478343465045592 0.03499441487180499 1830.9999999999995
6 1024 5264 0.3345364741641337 0.033369810593094074 1760.9999999999998
7 1536 5264 0.2794452887537993 0.026941583361394406 1470.9999999999995
8 2048 5264 0.1073328267477203 0.009417167606857957 564.9999999999997
9 2560 5264 0.0381838905775075 0.00323907625680131 200.99999999999946
10 3072 5264 0.0381838905775075 0.00323907625680131 200.99999999999946
11 3584 5264 0.0381838905775075 0.00323907625680131 200.99999999999946
12 4096 5264 0.0381838905775075 0.00323907625680131 200.99999999999946

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.4316109422492401,0.04598806248392684,2272.0
128,5264,0.2809650455927052,0.0271127759679356,1479.0
256,5264,0.2515197568389057,0.02385343933427242,1323.9999999999995
512,5264,0.2386018237082066,0.022460490669208744,1255.9999999999995
1024,5264,0.2313829787234042,0.021691485903849506,1217.9999999999995
1536,5264,0.2211246200607902,0.02061000156097803,1163.9999999999995
2048,5264,0.1988981762917933,0.018310882399525497,1047.0
2560,5264,0.0797872340425532,0.006905245826546502,420.0
3072,10528,0.0241261398176291,0.0020330923404081602,253.99999999999918
3584,10528,0.0241261398176291,0.0020330923404081602,253.99999999999918
4096,10528,0.0241261398176291,0.0020330923404081602,253.99999999999918
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.4316109422492401 0.04598806248392684 2272.0
3 128 5264 0.2809650455927052 0.0271127759679356 1479.0
4 256 5264 0.2515197568389057 0.02385343933427242 1323.9999999999995
5 512 5264 0.2386018237082066 0.022460490669208744 1255.9999999999995
6 1024 5264 0.2313829787234042 0.021691485903849506 1217.9999999999995
7 1536 5264 0.2211246200607902 0.02061000156097803 1163.9999999999995
8 2048 5264 0.1988981762917933 0.018310882399525497 1047.0
9 2560 5264 0.0797872340425532 0.006905245826546502 420.0
10 3072 10528 0.0241261398176291 0.0020330923404081602 253.99999999999918
11 3584 10528 0.0241261398176291 0.0020330923404081602 253.99999999999918
12 4096 10528 0.0241261398176291 0.0020330923404081602 253.99999999999918

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.3903875379939209,0.040405346212274096,2054.9999999999995
128,5264,0.2405015197568389,0.02266397103267348,1266.0
256,5264,0.2159954407294833,0.020074146889335953,1137.0
512,5264,0.2002279635258358,0.018446781884206676,1053.9999999999995
1024,5264,0.1973784194528875,0.018155821680927553,1038.9999999999998
1536,5264,0.1890197568389057,0.017307768448815875,994.9999999999997
2048,5264,0.1903495440729483,0.01744214849069614,1001.9999999999998
2560,5264,0.1724924012158054,0.01565426054958796,907.9999999999995
3072,5264,0.1073328267477203,0.009417167606857957,564.9999999999997
3584,10528,0.0251709726443769,0.002122176652162411,265.0
4096,10528,0.020991641337386,0.0017663630187246815,220.9999999999998
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.3903875379939209 0.040405346212274096 2054.9999999999995
3 128 5264 0.2405015197568389 0.02266397103267348 1266.0
4 256 5264 0.2159954407294833 0.020074146889335953 1137.0
5 512 5264 0.2002279635258358 0.018446781884206676 1053.9999999999995
6 1024 5264 0.1973784194528875 0.018155821680927553 1038.9999999999998
7 1536 5264 0.1890197568389057 0.017307768448815875 994.9999999999997
8 2048 5264 0.1903495440729483 0.01744214849069614 1001.9999999999998
9 2560 5264 0.1724924012158054 0.01565426054958796 907.9999999999995
10 3072 5264 0.1073328267477203 0.009417167606857957 564.9999999999997
11 3584 10528 0.0251709726443769 0.002122176652162411 265.0
12 4096 10528 0.020991641337386 0.0017663630187246815 220.9999999999998

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.6259498480243161,0.07867931965075214,3295.0
128,5264,0.5583206686930091,0.06583070229097776,2939.0
256,5264,0.5526215805471124,0.06483211115106036,2908.9999999999995
512,5264,0.5391337386018237,0.06251446315798959,2838.0
1024,5264,0.5098784194528876,0.05769393705396664,2684.0000000000005
1536,5264,0.4791033434650456,0.05289972185893943,2522.0
2048,5264,0.2674772036474164,0.025604890903806354,1408.0
2560,5264,0.0921352583586626,0.008022635037369774,484.9999999999999
3072,5264,0.0921352583586626,0.008022635037369774,484.9999999999999
3584,5264,0.0921352583586626,0.008022635037369774,484.9999999999999
4096,5264,0.0921352583586626,0.008022635037369774,484.9999999999999
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.6259498480243161 0.07867931965075214 3295.0
3 128 5264 0.5583206686930091 0.06583070229097776 2939.0
4 256 5264 0.5526215805471124 0.06483211115106036 2908.9999999999995
5 512 5264 0.5391337386018237 0.06251446315798959 2838.0
6 1024 5264 0.5098784194528876 0.05769393705396664 2684.0000000000005
7 1536 5264 0.4791033434650456 0.05289972185893943 2522.0
8 2048 5264 0.2674772036474164 0.025604890903806354 1408.0
9 2560 5264 0.0921352583586626 0.008022635037369774 484.9999999999999
10 3072 5264 0.0921352583586626 0.008022635037369774 484.9999999999999
11 3584 5264 0.0921352583586626 0.008022635037369774 484.9999999999999
12 4096 5264 0.0921352583586626 0.008022635037369774 484.9999999999999

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.4373100303951368,0.046788885074059094,2302.0
128,5264,0.315919452887538,0.031144661244724814,1663.0
256,5264,0.2980623100303951,0.029061892102767328,1568.9999999999998
512,5264,0.2870440729483283,0.027800879955706126,1511.0
1024,5264,0.2840045592705167,0.027456158538678666,1495.0
1536,5264,0.2767857142857143,0.026642791153438816,1457.0
2048,5264,0.2562689969604863,0.024371098470341246,1348.9999999999998
2560,5264,0.1187310030395136,0.0104774227315102,624.9999999999997
3072,10528,0.0359992401215805,0.00305060376648969,378.9999999999995
3584,10528,0.0359992401215805,0.00305060376648969,378.9999999999995
4096,10528,0.0359992401215805,0.00305060376648969,378.9999999999995
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.4373100303951368 0.046788885074059094 2302.0
3 128 5264 0.315919452887538 0.031144661244724814 1663.0
4 256 5264 0.2980623100303951 0.029061892102767328 1568.9999999999998
5 512 5264 0.2870440729483283 0.027800879955706126 1511.0
6 1024 5264 0.2840045592705167 0.027456158538678666 1495.0
7 1536 5264 0.2767857142857143 0.026642791153438816 1457.0
8 2048 5264 0.2562689969604863 0.024371098470341246 1348.9999999999998
9 2560 5264 0.1187310030395136 0.0104774227315102 624.9999999999997
10 3072 10528 0.0359992401215805 0.00305060376648969 378.9999999999995
11 3584 10528 0.0359992401215805 0.00305060376648969 378.9999999999995
12 4096 10528 0.0359992401215805 0.00305060376648969 378.9999999999995

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.3936170212765957,0.0408300073999609,2072.0
128,5264,0.2572188449848024,0.024474993793634048,1353.9999999999998
256,5264,0.236322188449848,0.022216927636340866,1244.0
512,5264,0.2203647416413373,0.020530411940441584,1159.9999999999995
1024,5264,0.2203647416413373,0.020530411940441584,1159.9999999999995
1536,5264,0.2159954407294833,0.020074146889335953,1137.0
2048,5264,0.2078267477203647,0.01922734698807005,1093.9999999999998
2560,5264,0.1778115501519757,0.01618309464777068,936.0
3072,5264,0.1225303951367781,0.010833635736165581,644.9999999999999
3584,10528,0.0296352583586626,0.0025037982429529926,311.9999999999999
4096,10528,0.0237462006079027,0.0020007197211502348,249.9999999999996
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.3936170212765957 0.0408300073999609 2072.0
3 128 5264 0.2572188449848024 0.024474993793634048 1353.9999999999998
4 256 5264 0.236322188449848 0.022216927636340866 1244.0
5 512 5264 0.2203647416413373 0.020530411940441584 1159.9999999999995
6 1024 5264 0.2203647416413373 0.020530411940441584 1159.9999999999995
7 1536 5264 0.2159954407294833 0.020074146889335953 1137.0
8 2048 5264 0.2078267477203647 0.01922734698807005 1093.9999999999998
9 2560 5264 0.1778115501519757 0.01618309464777068 936.0
10 3072 5264 0.1225303951367781 0.010833635736165581 644.9999999999999
11 3584 10528 0.0296352583586626 0.0025037982429529926 311.9999999999999
12 4096 10528 0.0237462006079027 0.0020007197211502348 249.9999999999996

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.9587765957446808,0.23335321086836192,5047.0
128,5264,0.944338905775076,0.21392742431016676,4971.0
256,5264,0.9456686930091184,0.2155098179872279,4977.999999999999
512,5264,0.9439589665653496,0.2134816768834944,4969.0
1024,5264,0.9283814589665652,0.19724049628061424,4886.999999999999
1536,5264,0.910144376899696,0.18192063487808974,4790.999999999999
2048,5264,0.8763297872340425,0.15985272004411766,4613.0
2560,5264,0.8284574468085106,0.13662860463433912,4361.0
3072,5264,0.8284574468085106,0.13662860463433912,4361.0
3584,5264,0.8284574468085106,0.13662860463433912,4361.0
4096,5264,0.8284574468085106,0.13662860463433912,4361.0
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.9587765957446808 0.23335321086836192 5047.0
3 128 5264 0.944338905775076 0.21392742431016676 4971.0
4 256 5264 0.9456686930091184 0.2155098179872279 4977.999999999999
5 512 5264 0.9439589665653496 0.2134816768834944 4969.0
6 1024 5264 0.9283814589665652 0.19724049628061424 4886.999999999999
7 1536 5264 0.910144376899696 0.18192063487808974 4790.999999999999
8 2048 5264 0.8763297872340425 0.15985272004411766 4613.0
9 2560 5264 0.8284574468085106 0.13662860463433912 4361.0
10 3072 5264 0.8284574468085106 0.13662860463433912 4361.0
11 3584 5264 0.8284574468085106 0.13662860463433912 4361.0
12 4096 5264 0.8284574468085106 0.13662860463433912 4361.0

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.559080547112462,0.06596473909805234,2943.0
128,5264,0.4678951367781155,0.05121800243006214,2463.0
256,5264,0.4532674772036474,0.049071400643614704,2386.0
512,5264,0.4502279635258359,0.048631968564752825,2370.0
1024,5264,0.4262917933130699,0.045247241414411166,2244.0
1536,5264,0.418693009118541,0.04419977411271481,2204.0
2048,5264,0.4046352583586626,0.042294622550035466,2130.0
2560,5264,0.2661474164133738,0.025457607509746905,1400.9999999999998
3072,5264,0.0773176291793313,0.0066834185002044855,406.99999999999994
3584,5264,0.0773176291793313,0.0066834185002044855,406.99999999999994
4096,5264,0.0773176291793313,0.0066834185002044855,406.99999999999994
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.559080547112462 0.06596473909805234 2943.0
3 128 5264 0.4678951367781155 0.05121800243006214 2463.0
4 256 5264 0.4532674772036474 0.049071400643614704 2386.0
5 512 5264 0.4502279635258359 0.048631968564752825 2370.0
6 1024 5264 0.4262917933130699 0.045247241414411166 2244.0
7 1536 5264 0.418693009118541 0.04419977411271481 2204.0
8 2048 5264 0.4046352583586626 0.042294622550035466 2130.0
9 2560 5264 0.2661474164133738 0.025457607509746905 1400.9999999999998
10 3072 5264 0.0773176291793313 0.0066834185002044855 406.99999999999994
11 3584 5264 0.0773176291793313 0.0066834185002044855 406.99999999999994
12 4096 5264 0.0773176291793313 0.0066834185002044855 406.99999999999994

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.4071048632218845,0.04262630471820339,2143.0
128,5264,0.281724924012158,0.02719849668814711,1482.9999999999995
256,5264,0.2604483282674772,0.024829151186504417,1371.0
512,5264,0.2496200607902735,0.023647218339284293,1313.9999999999995
1024,5264,0.25,0.023688424222606863,1316.0
1536,5264,0.244870820668693,0.0231337521290782,1289.0
2048,5264,0.2420212765957446,0.022827091512257702,1273.9999999999995
2560,5264,0.2384118541033434,0.022440168236137703,1254.9999999999995
3072,5264,0.163563829787234,0.014773539998592322,860.9999999999998
3584,5264,0.0400835866261398,0.003403284629997727,210.9999999999999
4096,10528,0.0310600303951367,0.0026259311852924183,326.9999999999992
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.4071048632218845 0.04262630471820339 2143.0
3 128 5264 0.281724924012158 0.02719849668814711 1482.9999999999995
4 256 5264 0.2604483282674772 0.024829151186504417 1371.0
5 512 5264 0.2496200607902735 0.023647218339284293 1313.9999999999995
6 1024 5264 0.25 0.023688424222606863 1316.0
7 1536 5264 0.244870820668693 0.0231337521290782 1289.0
8 2048 5264 0.2420212765957446 0.022827091512257702 1273.9999999999995
9 2560 5264 0.2384118541033434 0.022440168236137703 1254.9999999999995
10 3072 5264 0.163563829787234 0.014773539998592322 860.9999999999998
11 3584 5264 0.0400835866261398 0.003403284629997727 210.9999999999999
12 4096 10528 0.0310600303951367 0.0026259311852924183 326.9999999999992

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.3850683890577507,0.03971037910801656,2026.9999999999998
128,5264,0.3187689969604863,0.03148161973097752,1677.9999999999998
256,5264,0.3077507598784194,0.03018579579258507,1619.9999999999995
512,5264,0.2995820668693009,0.029237246714529652,1576.9999999999998
1024,5264,0.2291033434650456,0.021450017503444885,1206.0
1536,5264,0.3221884498480243,0.03188767957701344,1696.0
2048,5264,0.3472644376899696,0.03492416878291982,1828.0
2560,5264,0.5324848024316109,0.061394752832159005,2802.9999999999995
3072,5264,0.5324848024316109,0.061394752832159005,2802.9999999999995
3584,5264,0.5324848024316109,0.061394752832159005,2802.9999999999995
4096,5264,0.5324848024316109,0.061394752832159005,2802.9999999999995
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.3850683890577507 0.03971037910801656 2026.9999999999998
3 128 5264 0.3187689969604863 0.03148161973097752 1677.9999999999998
4 256 5264 0.3077507598784194 0.03018579579258507 1619.9999999999995
5 512 5264 0.2995820668693009 0.029237246714529652 1576.9999999999998
6 1024 5264 0.2291033434650456 0.021450017503444885 1206.0
7 1536 5264 0.3221884498480243 0.03188767957701344 1696.0
8 2048 5264 0.3472644376899696 0.03492416878291982 1828.0
9 2560 5264 0.5324848024316109 0.061394752832159005 2802.9999999999995
10 3072 5264 0.5324848024316109 0.061394752832159005 2802.9999999999995
11 3584 5264 0.5324848024316109 0.061394752832159005 2802.9999999999995
12 4096 5264 0.5324848024316109 0.061394752832159005 2802.9999999999995

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.2822948328267477,0.02726284179870908,1486.0
128,5264,0.2177051671732522,0.020252407127563266,1145.9999999999995
256,5264,0.2017477203647416,0.018602349442960464,1061.9999999999998
512,5264,0.1979483282674772,0.0182139378855688,1042.0
1024,5264,0.1631838905775076,0.014736254004048988,859.0
1536,5264,0.1800911854103343,0.016410698472618335,947.9999999999998
2048,5264,0.2974924012158054,0.02899622383582867,1565.9999999999995
2560,5264,0.3451747720364742,0.03466707988978823,1817.0
3072,5264,0.4967705167173252,0.055619147396563595,2615.0
3584,5264,0.4967705167173252,0.055619147396563595,2615.0
4096,5264,0.4967705167173252,0.055619147396563595,2615.0
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.2822948328267477 0.02726284179870908 1486.0
3 128 5264 0.2177051671732522 0.020252407127563266 1145.9999999999995
4 256 5264 0.2017477203647416 0.018602349442960464 1061.9999999999998
5 512 5264 0.1979483282674772 0.0182139378855688 1042.0
6 1024 5264 0.1631838905775076 0.014736254004048988 859.0
7 1536 5264 0.1800911854103343 0.016410698472618335 947.9999999999998
8 2048 5264 0.2974924012158054 0.02899622383582867 1565.9999999999995
9 2560 5264 0.3451747720364742 0.03466707988978823 1817.0
10 3072 5264 0.4967705167173252 0.055619147396563595 2615.0
11 3584 5264 0.4967705167173252 0.055619147396563595 2615.0
12 4096 5264 0.4967705167173252 0.055619147396563595 2615.0

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.2412613981762918,0.02274549383452562,1270.0
128,5264,0.184080547112462,0.016810404644448607,968.9999999999999
256,5264,0.175531914893617,0.01595606856502496,923.9999999999999
512,5264,0.1647036474164133,0.014885491207228485,866.9999999999997
1024,5264,0.1409574468085106,0.012581583360087745,741.9999999999998
1536,5264,0.116451367781155,0.010264370061932926,612.9999999999999
2048,5264,0.1852203647416413,0.016924935456301582,974.9999999999998
2560,5264,0.2931231003039514,0.02849438331179399,1543.0
3072,5264,0.3098404255319149,0.030430095656652667,1631.0
3584,5264,0.40330547112462,0.04211654648973906,2122.9999999999995
4096,5264,0.4549772036474164,0.049319566283534066,2395.0
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.2412613981762918 0.02274549383452562 1270.0
3 128 5264 0.184080547112462 0.016810404644448607 968.9999999999999
4 256 5264 0.175531914893617 0.01595606856502496 923.9999999999999
5 512 5264 0.1647036474164133 0.014885491207228485 866.9999999999997
6 1024 5264 0.1409574468085106 0.012581583360087745 741.9999999999998
7 1536 5264 0.116451367781155 0.010264370061932926 612.9999999999999
8 2048 5264 0.1852203647416413 0.016924935456301582 974.9999999999998
9 2560 5264 0.2931231003039514 0.02849438331179399 1543.0
10 3072 5264 0.3098404255319149 0.030430095656652667 1631.0
11 3584 5264 0.40330547112462 0.04211654648973906 2122.9999999999995
12 4096 5264 0.4549772036474164 0.049319566283534066 2395.0

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@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.5953647416413373,0.0726251577026028,3134.0
128,5264,0.5431231003039514,0.06319341818259305,2859.0
256,5264,0.5303951367781155,0.06104585785147554,2791.9999999999995
512,5264,0.5311550151975684,0.06117256378844538,2796.0
1024,5264,0.5026595744680851,0.056545094798418516,2646.0
1536,5264,0.4513677811550152,0.048796494007125824,2376.0
2048,5264,0.2623480243161094,0.02503814186021658,1380.9999999999998
2560,5264,0.297112462006079,0.02895247211574592,1563.9999999999998
3072,5264,0.297112462006079,0.02895247211574592,1563.9999999999998
3584,5264,0.297112462006079,0.02895247211574592,1563.9999999999998
4096,5264,0.297112462006079,0.02895247211574592,1563.9999999999998
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.5953647416413373 0.0726251577026028 3134.0
3 128 5264 0.5431231003039514 0.06319341818259305 2859.0
4 256 5264 0.5303951367781155 0.06104585785147554 2791.9999999999995
5 512 5264 0.5311550151975684 0.06117256378844538 2796.0
6 1024 5264 0.5026595744680851 0.056545094798418516 2646.0
7 1536 5264 0.4513677811550152 0.048796494007125824 2376.0
8 2048 5264 0.2623480243161094 0.02503814186021658 1380.9999999999998
9 2560 5264 0.297112462006079 0.02895247211574592 1563.9999999999998
10 3072 5264 0.297112462006079 0.02895247211574592 1563.9999999999998
11 3584 5264 0.297112462006079 0.02895247211574592 1563.9999999999998
12 4096 5264 0.297112462006079 0.02895247211574592 1563.9999999999998

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@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.3765197568389057,0.03860492869161691,1981.9999999999995
128,5264,0.2946428571428571,0.02866861322375136,1550.9999999999998
256,5264,0.2767857142857143,0.026642791153438816,1457.0
512,5264,0.2712765957446808,0.02602705514677528,1427.9999999999998
1024,5264,0.2665273556231003,0.025499663499135905,1403.0
1536,5264,0.2562689969604863,0.024371098470341246,1348.9999999999998
2048,5264,0.1667933130699088,0.015091099283021325,877.9999999999999
2560,5264,0.1476063829787234,0.013220733692390962,776.9999999999999
3072,5264,0.2625379939209726,0.025059068054687383,1381.9999999999998
3584,5264,0.2625379939209726,0.025059068054687383,1381.9999999999998
4096,5264,0.2625379939209726,0.025059068054687383,1381.9999999999998
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.3765197568389057 0.03860492869161691 1981.9999999999995
3 128 5264 0.2946428571428571 0.02866861322375136 1550.9999999999998
4 256 5264 0.2767857142857143 0.026642791153438816 1457.0
5 512 5264 0.2712765957446808 0.02602705514677528 1427.9999999999998
6 1024 5264 0.2665273556231003 0.025499663499135905 1403.0
7 1536 5264 0.2562689969604863 0.024371098470341246 1348.9999999999998
8 2048 5264 0.1667933130699088 0.015091099283021325 877.9999999999999
9 2560 5264 0.1476063829787234 0.013220733692390962 776.9999999999999
10 3072 5264 0.2625379939209726 0.025059068054687383 1381.9999999999998
11 3584 5264 0.2625379939209726 0.025059068054687383 1381.9999999999998
12 4096 5264 0.2625379939209726 0.025059068054687383 1381.9999999999998

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@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.3183890577507599,0.031436617365659836,1676.0
128,5264,0.2232142857142857,0.020829240593055243,1175.0
256,5264,0.2104863221884498,0.01950216695467022,1107.9999999999998
512,5264,0.2068768996960486,0.019129401936932466,1089.0
1024,5264,0.1962386018237082,0.018039702664670587,1033.0
1536,5264,0.1876899696048632,0.017173590239868086,987.9999999999999
2048,5264,0.1386778115501519,0.01236348971224388,729.9999999999997
2560,5264,0.133548632218845,0.011874707191563583,703.0000000000001
3072,5264,0.130129179331307,0.011550323574472054,685.0
3584,5264,0.2042173252279635,0.018855726724976818,1074.9999999999998
4096,5264,0.2444908814589665,0.02309280281892645,1286.9999999999995
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.3183890577507599 0.031436617365659836 1676.0
3 128 5264 0.2232142857142857 0.020829240593055243 1175.0
4 256 5264 0.2104863221884498 0.01950216695467022 1107.9999999999998
5 512 5264 0.2068768996960486 0.019129401936932466 1089.0
6 1024 5264 0.1962386018237082 0.018039702664670587 1033.0
7 1536 5264 0.1876899696048632 0.017173590239868086 987.9999999999999
8 2048 5264 0.1386778115501519 0.01236348971224388 729.9999999999997
9 2560 5264 0.133548632218845 0.011874707191563583 703.0000000000001
10 3072 5264 0.130129179331307 0.011550323574472054 685.0
11 3584 5264 0.2042173252279635 0.018855726724976818 1074.9999999999998
12 4096 5264 0.2444908814589665 0.02309280281892645 1286.9999999999995

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@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.9587765957446808,0.23335321086836192,5047.0
128,5264,0.944338905775076,0.21392742431016676,4971.0
256,5264,0.9456686930091184,0.2155098179872279,4977.999999999999
512,5264,0.9439589665653496,0.2134816768834944,4969.0
1024,5264,0.9283814589665652,0.19724049628061424,4886.999999999999
1536,5264,0.910144376899696,0.18192063487808974,4790.999999999999
2048,5264,0.8763297872340425,0.15985272004411766,4613.0
2560,5264,0.8284574468085106,0.13662860463433912,4361.0
3072,5264,0.8284574468085106,0.13662860463433912,4361.0
3584,5264,0.8284574468085106,0.13662860463433912,4361.0
4096,5264,0.8284574468085106,0.13662860463433912,4361.0
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.9587765957446808 0.23335321086836192 5047.0
3 128 5264 0.944338905775076 0.21392742431016676 4971.0
4 256 5264 0.9456686930091184 0.2155098179872279 4977.999999999999
5 512 5264 0.9439589665653496 0.2134816768834944 4969.0
6 1024 5264 0.9283814589665652 0.19724049628061424 4886.999999999999
7 1536 5264 0.910144376899696 0.18192063487808974 4790.999999999999
8 2048 5264 0.8763297872340425 0.15985272004411766 4613.0
9 2560 5264 0.8284574468085106 0.13662860463433912 4361.0
10 3072 5264 0.8284574468085106 0.13662860463433912 4361.0
11 3584 5264 0.8284574468085106 0.13662860463433912 4361.0
12 4096 5264 0.8284574468085106 0.13662860463433912 4361.0

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@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.5298252279635258,0.060950951675050935,2789.0
128,5264,0.4523176291793313,0.04893383806510265,2381.0
256,5264,0.4544072948328267,0.049236765210533884,2392.0
512,5264,0.4450987841945288,0.047895451992423954,2342.9999999999995
1024,5264,0.4236322188449848,0.04487918892421794,2230.0
1536,5264,0.4116641337386018,0.04324198215552311,2166.9999999999995
2048,5264,0.3958966565349544,0.0411310182640684,2084.0
2560,5264,0.2315729483282674,0.021711637892541602,1218.9999999999995
3072,5264,0.1852203647416413,0.016924935456301582,974.9999999999998
3584,5264,0.1852203647416413,0.016924935456301582,974.9999999999998
4096,5264,0.1852203647416413,0.016924935456301582,974.9999999999998
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.5298252279635258 0.060950951675050935 2789.0
3 128 5264 0.4523176291793313 0.04893383806510265 2381.0
4 256 5264 0.4544072948328267 0.049236765210533884 2392.0
5 512 5264 0.4450987841945288 0.047895451992423954 2342.9999999999995
6 1024 5264 0.4236322188449848 0.04487918892421794 2230.0
7 1536 5264 0.4116641337386018 0.04324198215552311 2166.9999999999995
8 2048 5264 0.3958966565349544 0.0411310182640684 2084.0
9 2560 5264 0.2315729483282674 0.021711637892541602 1218.9999999999995
10 3072 5264 0.1852203647416413 0.016924935456301582 974.9999999999998
11 3584 5264 0.1852203647416413 0.016924935456301582 974.9999999999998
12 4096 5264 0.1852203647416413 0.016924935456301582 974.9999999999998

View File

@@ -0,0 +1,12 @@
max_iter,num_trials,LER,LER_per_round,num_errors
32,5264,0.3782294832826747,0.038824902532404226,1990.9999999999995
128,5264,0.2661474164133738,0.025457607509746905,1400.9999999999998
256,5264,0.2513297872340425,0.023832795657756978,1322.9999999999995
512,5264,0.2416413373860182,0.022786283304920496,1271.9999999999998
1024,5264,0.2353723404255319,0.022115639689234468,1239.0
1536,5264,0.2344224924012158,0.022014467014769612,1234.0
2048,5264,0.2349924012158054,0.022075156802095708,1236.9999999999995
2560,5264,0.1914893617021276,0.017557492508172845,1007.9999999999997
3072,5264,0.1274696048632219,0.011298832322427566,671.0000000000001
3584,5264,0.1289893617021276,0.011442455421834419,678.9999999999997
4096,5264,0.1508358662613981,0.013532828789684093,793.9999999999997
1 max_iter num_trials LER LER_per_round num_errors
2 32 5264 0.3782294832826747 0.038824902532404226 1990.9999999999995
3 128 5264 0.2661474164133738 0.025457607509746905 1400.9999999999998
4 256 5264 0.2513297872340425 0.023832795657756978 1322.9999999999995
5 512 5264 0.2416413373860182 0.022786283304920496 1271.9999999999998
6 1024 5264 0.2353723404255319 0.022115639689234468 1239.0
7 1536 5264 0.2344224924012158 0.022014467014769612 1234.0
8 2048 5264 0.2349924012158054 0.022075156802095708 1236.9999999999995
9 2560 5264 0.1914893617021276 0.017557492508172845 1007.9999999999997
10 3072 5264 0.1274696048632219 0.011298832322427566 671.0000000000001
11 3584 5264 0.1289893617021276 0.011442455421834419 678.9999999999997
12 4096 5264 0.1508358662613981 0.013532828789684093 793.9999999999997