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arxiv: 2509.08328 · v2 · submitted 2025-09-10 · 🪐 quant-ph

Towards solving industrial integer linear programs with Decoded Quantum Interferometry

Francesc Sabater , Ouns El Harzli , Geert-Jan Besjes , Marvin Erdmann , Johannes Klepsch , Jonas Hiltrop , Jean-Francois Bobier , Yudong Cao
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Carlos A. Riofrio
This is my paper

Pith reviewed 2026-05-18 18:32 UTC · model grok-4.3

classification 🪐 quant-ph
keywords decoded quantum interferometryinteger linear programmax-XORSATbelief propagationvehicle option pricingquantum circuitLDPC decoding
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The pith

Converting industrial integer linear programs to max-XORSAT enables decoded quantum interferometry with belief propagation to address vehicle pricing optimization.

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

The paper aims to show that decoded quantum interferometry can be used for solving an industrial integer linear program from the automotive sector by transforming it into a max-XORSAT problem. This matters to a sympathetic reader because it provides a concrete way to bring quantum methods to practical optimization tasks like setting prices for vehicle options and packages. The authors develop the full pipeline including the formulation, conversion, and a quantum circuit for the belief propagation decoder. They test it against standard solvers to assess its potential.

Core claim

By formulating the vehicle option-package pricing problem as an integer linear program and converting it to max-XORSAT instances, the authors implement decoded quantum interferometry using a belief propagation algorithm realized on a quantum circuit, offering a general method applicable to other industrially relevant integer linear programs.

What carries the argument

The transformation of the integer linear program into max-XORSAT followed by quantum-circuit belief propagation decoding within the decoded quantum interferometry framework.

If this is right

  • Industrial optimization problems expressed as integer linear programs become solvable via decoded quantum interferometry after conversion to max-XORSAT.
  • The belief propagation heuristic can be embedded in a quantum circuit to decode the solutions.
  • Effectiveness is demonstrated through direct comparison with the Gurobi solver and a random sampling baseline on the specific problem instances.

Where Pith is reading between the lines

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

  • If the conversion process is general and efficient, similar techniques could apply decoded quantum interferometry to a wider range of supply chain or configuration problems.
  • Performance advantages may appear in instances where the max-XORSAT structure aligns well with low-density parity-check code properties used in the decoding.

Load-bearing premise

The conversion from the vehicle option-package integer linear program to max-XORSAT instances must preserve enough of the original optimization structure for the belief propagation decoder to yield competitive solutions.

What would settle it

If benchmarks show that the decoded quantum interferometry solutions are no better than random sampling or consistently inferior to Gurobi outputs on the tested instances, the claim of practical applicability would be falsified.

Figures

Figures reproduced from arXiv: 2509.08328 by Carlos A. Riofrio, Francesc Sabater, Geert-Jan Besjes, Jean-Francois Bobier, Johannes Klepsch, Jonas Hiltrop, Marvin Erdmann, Ouns El Harzli, Yudong Cao.

Figure 1
Figure 1. Figure 1: Success rate of our implementation of the BP1 decoder for the zero code-word for [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit construction inspired by Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: First iteration of the quantum-circuit implementation of the Binary Belief Propagation [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Second iteration of the quantum-circuit implementation of the Binary Belief Propagation [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Final step of the quantum-circuit implementation of the BP1 algorithm for the [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic for building an encoding for integer addition (an “integer adder circuit”) using [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schematic for building an encoding for weighted integer addition (a “weighted integer adder”) [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Schematic illustration for the multiple integer adder circuit using the weighted integer adder [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Schematic illustration of how integer comparator computes the intermediate quantity [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Success rate of our implementation of a state-of-the-art, soft-decision, belief propagation [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Performance comparison of DQI for downscaled max-XORSAT instances derived from the [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Number of qubits required to implement the proposed DQI circuit for downscaled max [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Total number of gates required to implement the proposed DQI circuit after transpilation [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Total number of gates of each type required to implement the proposed DQI circuit after [PITH_FULL_IMAGE:figures/full_fig_p037_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Histogram of the number of satisfied constraints obtained from DQI sampling for different [PITH_FULL_IMAGE:figures/full_fig_p038_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Comparison between predicted and empirical average number of satisfied constraints across [PITH_FULL_IMAGE:figures/full_fig_p039_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Success rate of the Gauss-Jordan decoder for the zero codeword for [PITH_FULL_IMAGE:figures/full_fig_p044_17.png] view at source ↗
read the original abstract

Optimization via decoded quantum interferometry (DQI) has recently gained a great deal of attention as a promising avenue for solving optimization problems using quantum computers. In this paper, we apply DQI to an industrial optimization problem in the automotive industry: the vehicle option-package pricing problem. Our main contributions are 1) formulating the industrial problem as an integer linear program (ILP), 2) converting the ILP into instances of max-XORSAT, and 3) developing a detailed quantum circuit implementation for belief propagation, a heuristic algorithm for decoding LDPC codes. Thus, we provide a full implementation of the DQI algorithm using Belief Propagation, which can be applied to any industrially relevant ILP by first transforming it into a max-XORSAT instance. We also evaluate the effectiveness of our implementation by benchmarking it against both Gurobi and a random sampling baseline.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper formulates the vehicle option-package pricing problem as an integer linear program (ILP), converts it to max-XORSAT instances, provides a detailed quantum circuit implementation of belief propagation for decoding in the Decoded Quantum Interferometry (DQI) framework, and benchmarks the approach against Gurobi and random sampling. It claims this pipeline yields a full implementation applicable to any industrially relevant ILP via the ILP-to-max-XORSAT transformation.

Significance. If the ILP-to-max-XORSAT mapping preserves objective equivalence and belief propagation decoding recovers competitive solutions, the work could demonstrate a practical route for applying DQI to industrial optimization. The explicit circuit implementation and intent to benchmark against a commercial solver are strengths; reproducible code or parameter-free derivations would further strengthen the contribution.

major comments (1)
  1. [formulation and conversion section] Formulation and conversion section: the central claim that the pipeline applies to any industrially relevant ILP and yields competitive solutions with Gurobi requires that low-energy assignments of the max-XORSAT instance correspond to high-quality (or feasible) solutions of the original ILP. No explicit bounds, objective-equivalence proof, or analysis of how auxiliary variables or integrality relaxation affect the objective are provided; without this, it is unclear whether BP decoding on the DQI circuit can recover useful ILP solutions rather than XORSAT-optimal but ILP-suboptimal assignments.
minor comments (2)
  1. [abstract] Abstract and results section: quantitative benchmarking details (instance sizes, number of trials, error bars, or specific objective values versus Gurobi) are referenced but not summarized; these should be stated explicitly to support the effectiveness claims.
  2. [quantum circuit implementation] Quantum circuit implementation: the description of the belief-propagation decoder circuit would benefit from a pseudocode listing or additional diagram to clarify gate counts and depth for the LDPC decoding step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight an important point for strengthening the clarity of our claims. We address the major comment below.

read point-by-point responses

  1. Referee: Formulation and conversion section: the central claim that the pipeline applies to any industrially relevant ILP and yields competitive solutions with Gurobi requires that low-energy assignments of the max-XORSAT instance correspond to high-quality (or feasible) solutions of the original ILP. No explicit bounds, objective-equivalence proof, or analysis of how auxiliary variables or integrality relaxation affect the objective are provided; without this, it is unclear whether BP decoding on the DQI circuit can recover useful ILP solutions rather than XORSAT-optimal but ILP-suboptimal assignments.

    Authors: We agree that the manuscript would benefit from a more explicit treatment of objective equivalence to fully support the claim of applicability to arbitrary ILPs. The ILP-to-max-XORSAT conversion encodes each linear constraint via auxiliary variables into XOR clauses while preserving the original objective through additional penalty clauses; feasible integer solutions of the ILP map directly to satisfying assignments of the XORSAT instance, with the objective value preserved up to an additive constant independent of the auxiliaries. No integrality relaxation is performed. We will add a dedicated subsection to the formulation section containing a formal equivalence argument, including bounds on the contribution of auxiliary variables and a proof that XORSAT-optimal assignments yield ILP-optimal or near-optimal solutions. This will clarify that belief-propagation decoding targets solutions that remain competitive for the original problem. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper presents a constructive pipeline: formulate the vehicle option-package problem as an ILP, apply a standard ILP-to-max-XORSAT conversion, then implement DQI decoding via belief propagation on the resulting instance. These steps are described as direct applications of existing transformations and heuristics rather than any self-referential fitting, redefinition of outputs as inputs, or load-bearing self-citations that collapse the claimed result back to the input data by construction. The central claim is an end-to-end implementation and benchmarking against Gurobi, which remains independent of the paper's own fitted values or prior self-referential theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the approach rests on standard properties of ILP-to-max-XORSAT reductions and the applicability of belief propagation to LDPC-style decoding; no explicit free parameters, new entities, or ad-hoc axioms are introduced in the high-level description.

axioms (1)
  • domain assumption Belief propagation serves as an effective heuristic decoder for the max-XORSAT instances obtained from the target ILP.
    The paper develops a quantum circuit for this decoding step without providing independent verification of its performance on the specific instances.

pith-pipeline@v0.9.0 · 5715 in / 1545 out tokens · 58696 ms · 2026-05-18T18:32:12.495930+00:00 · methodology

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Forward citations

Cited by 5 Pith papers

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

  1. Multivariate Decoded Quantum Interferometry for Weighted Optimization

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    The work develops multivariate DQI states for weighted Max-LINSAT over prime fields, derives closed-form asymptotic expressions for expectation values and concentration, provides an explicit single-decoder preparation...

  2. Multivariate Decoded Quantum Interferometry for Weighted Optimization

    quant-ph 2026-05 unverdicted novelty 7.0

    Multivariate DQI uses N-variable polynomials for weighted Max-LINSAT, derives closed-form asymptotics for expectation and concentration, provides a single-decoder preparation circuit, and shows outperformance over wei...

  3. From Constraint to Code: DQI-Kit -- A Software Framework for Decoded Quantum Interferometry

    quant-ph 2026-05 unverdicted novelty 6.0

    DQI-Kit automates encoding of objectives and constraints into Max-LINSAT instances and estimates expected DQI performance on the resulting problems.

  4. Hybrid Quantum-HPC Middleware Systems for Adaptive Resource, Workload and Task Management

    quant-ph 2026-04 unverdicted novelty 5.0

    The authors present Pilot-Quantum, a middleware for adaptive resource management in hybrid quantum-HPC systems, along with execution motifs and a performance modeling toolkit called Q-Dreamer.

  5. Quantum Decoding Algorithms: Quantum Speedups in Optimization

    quant-ph 2026-05 unverdicted novelty 1.0

    A review describing the Decoded Quantum Interferometry algorithm for quantum speedups in max-LINSAT optimization, with claimed superpolynomial advantage in the OPI problem.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · cited by 4 Pith papers

  1. [1]

    Jordan, Noah Shutty, Mary Wootters, Adam Zalcman, Alexander Schmidhuber, Robbie King, Sergei V

    Stephen P. Jordan, Noah Shutty, Mary Wootters, Adam Zalcman, Alexander Schmidhuber, Robbie King, Sergei V. Isakov, and Ryan Babbush. Optimization by decoded quantum interferometry, 2024. 40

    work page 2024

  2. [2]

    The quantum decoding problem

    Andr´e Chailloux and Jean-Pierre Tillich. The quantum decoding problem. In19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024), volume 306 of LIPIcs, pages 6:1–6:22. Schloss Dagstuhl – Leibniz-Zentrum f ¨ur Informatik, 2024

    work page 2024

  3. [3]

    Quantum oblivious lwe sampling and insecurity of standard model lattice-based snarks

    Thomas Debris-Alazard, Pouria Fallahpour, and Damien Stehl´e. Quantum oblivious lwe sampling and insecurity of standard model lattice-based snarks. InProceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), 2024

    work page 2024

  4. [4]

    Quantum reduction of finding short code vectors to the decoding problem.IEEE Transactions on Information Theory, 2024

    Thomas Debris-Alazard, Maxime Remaud, and Jean-Pierre Tillich. Quantum reduction of finding short code vectors to the decoding problem.IEEE Transactions on Information Theory, 2024

    work page 2024

  5. [5]

    Lwe with quantum amplitudes: Algorithm, hardness, and oblivious sampling, 2024

    Yilei Chen, Zihan Hu, Qipeng Liu, Han Luo, and Yaxin Tu. Lwe with quantum amplitudes: Algorithm, hardness, and oblivious sampling, 2024

    work page 2024

  6. [6]

    Quantum advantage from soft decoders, 2024

    Andr ´e Chailloux and Jean-Pierre Tillich. Quantum advantage from soft decoders, 2024

    work page 2024

  7. [7]

    Quantum algorithms for variants of average-case lattice problems via filtering.arXiv, 2021

    Yilei Chen, Qipeng Liu, and Mark Zhandry. Quantum algorithms for variants of average-case lattice problems via filtering.arXiv, 2021

    work page 2021

  8. [8]

    Obstacles to variational quantum optimization from symmetry protection.Physical Review Letters, 125(26):260505, 2020

    Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang. Obstacles to variational quantum optimization from symmetry protection.Physical Review Letters, 125(26):260505, 2020

    work page 2020

  9. [9]

    Classical algorithms and quantum limitations for maximum cut on high-girth graphs

    Boaz Barak and Kunal Marwaha. Classical algorithms and quantum limitations for maximum cut on high-girth graphs. InProceedings of ITCS 2022 (LIPIcs, Vol. 215), pages 14:1–14:24, 2022

    work page 2022

  10. [10]

    Concentration bounds for quantum states and limitations on the qaoa from polynomial approximations.Quantum, 7:999, 2023

    Anurag Anshu and Tony Metger. Concentration bounds for quantum states and limitations on the qaoa from polynomial approximations.Quantum, 7:999, 2023

    work page 2023

  11. [11]

    Performance and limitations of the qaoa at constant levels on large sparse hypergraphs and spin glass models

    Joao Basso, David Gamarnik, Song Mei, and Leo Zhou. Performance and limitations of the qaoa at constant levels on large sparse hypergraphs and spin glass models. In63rd IEEE Symposium on Foundations of Computer Science (FOCS), 2022

    work page 2022

  12. [12]

    Dreiling, John P

    Ruslan Shaydulin, Changhao Li, Shouvanik Chakrabarti, Matthew DeCross, Dylan Herman, Niraj Kumar, Jeffrey Larson, Danylo Lykov, Pierre Minssen, Yue Sun, Yuri Alexeev, Joan M. Dreiling, John P. Gaebler, Thomas M. Gatterman, Justin A. Gerber, Kevin Gilmore, Dan Gresh, Nathan Hewitt, Chandler V. Horst, Shaohan Hu, Jacob Johansen, Mitchell Matheny, Tanner Men...

    work page 2024

  13. [13]

    Quantum-walk speedup of backtracking algorithms.Theory of Computing, 14(15):1–24, 2018

    Ashley Montanaro. Quantum-walk speedup of backtracking algorithms.Theory of Computing, 14(15):1–24, 2018

    work page 2018

  14. [14]

    Knueven, James Ostrowski, and Jean-Paul Watson

    Barrett D. Knueven, James Ostrowski, and Jean-Paul Watson. On mixed-integer programming formulations for the unit commitment problem.INFORMS Journal on Computing, 2020

    work page 2020

  15. [15]

    Forrest, and William R

    Ranga Anbil, John J. Forrest, and William R. Pulleyblank. Column generation and the airline crew pairing problem. InDocumenta Mathematica, Extra Volume ICM 1998. 1998

    work page 1998

  16. [16]

    The uncapacitated facility location problem

    G´erard Cornu´ejols et al. The uncapacitated facility location problem. Technical report, Cornell University, 1983

    work page 1983

  17. [17]

    Gilmore and Ralph E

    Paul C. Gilmore and Ralph E. Gomory. A linear programming approach to the cutting-stock problem.Operations Research, 9(6):849–859, 1961. 41

    work page 1961

  18. [18]

    SIAM, 2002

    Paolo Toth and Daniele Vigo, editors.The Vehicle Routing Problem. SIAM, 2002

    work page 2002

  19. [19]

    Richard M. Karp. Reducibility among combinatorial problems. InComplexity of Computer Compu- tations. Plenum, 1972

    work page 1972

  20. [20]

    P-complete approximation problems.Journal of the ACM, 23(3):555–565, 1976

    Sartaj Sahni and Teofilo Gonz ´alez. P-complete approximation problems.Journal of the ACM, 23(3):555–565, 1976

    work page 1976

  21. [21]

    Robert E. Bixby. A brief history of linear and mixed-integer programming computation.Documenta Mathematica, pages 107–121, 2012

    work page 2012

  22. [22]

    Scip: Solving constraint integer programs.Mathematical Programming Computation, 1(1-2):1–41, 2009

    Tobias Achterberg. Scip: Solving constraint integer programs.Mathematical Programming Computation, 1(1-2):1–41, 2009

    work page 2009

  23. [23]

    Solving MAXSAT by solving a sequence of simpler SAT instances

    Jessica Davies and Fahiem Bacchus. Solving MAXSAT by solving a sequence of simpler SAT instances. InPrinciples and Practice of Constraint Programming (CP 2011), LNCS. Springer, 2011

    work page 2011

  24. [24]

    Liffiton, Jordi Planes, and Jo˜ao Marques-Silva

    Ant´onio Morgado, Federico Heras, Mark H. Liffiton, Jordi Planes, and Jo˜ao Marques-Silva. Iterative and core-guided maxsat solving: A survey and assessment.Constraints, 18(4):478–534, 2013

    work page 2013

  25. [25]

    Translating pseudo-boolean constraints into SAT.Journal on Satisfiability, Boolean Modeling and Computation, 2:1–26, 2006

    Niklas Een and Niklas S¨orensson. Translating pseudo-boolean constraints into SAT.Journal on Satisfiability, Boolean Modeling and Computation, 2:1–26, 2006

    work page 2006

  26. [26]

    Towards an optimal CNF encoding of boolean cardinality constraints

    Carsten Sinz. Towards an optimal CNF encoding of boolean cardinality constraints. InCP 2005, volume 3709 ofLNCS, pages 827–831. Springer, 2005

    work page 2005

  27. [27]

    Efficient CNF encoding of boolean cardinality constraints

    Olivier Bailleux and Yacine Boufkhad. Efficient CNF encoding of boolean cardinality constraints. InCP 2003, volume 2833 ofLNCS, pages 108–122. Springer, 2003

    work page 2003

  28. [28]

    Manquinho

    Olivier Roussel and Vasco M. Manquinho. Pseudo-Boolean and cardinality constraints. InHandbook of Satisfiability, pages 695–733. IOS Press, 2009

    work page 2009

  29. [29]

    Quantum computing in the nisq era and beyond.Quantum, 2:79, 2018

    John Preskill. Quantum computing in the nisq era and beyond.Quantum, 2:79, 2018

    work page 2018

  30. [30]

    How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits.Quantum, 5:433, 2021

    Craig Gidney and Martin Eker˚a. How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits.Quantum, 5:433, 2021

    work page 2048

  31. [31]

    Craig Gidney and Austin G. Fowler. Efficient magic state factories with a catalyzed |ccz> to 2|t> transformation.Quantum, 3:135, 2019

    work page 2019

  32. [32]

    Berry, Nathan Wiebe, Jarrod R

    Ryan Babbush, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod R. McClean, Alexandru Paler, Austin G. Fowler, and Hartmut Neven. Encoding electronic spectra in quantum circuits with linear t complexity.Phys. Rev. X, 8:041015, 2018

    work page 2018

  33. [33]

    Quantum computing enhanced computational catalysis.Phys

    Valentin von Burg, Guglielmo Mazzola, et al. Quantum computing enhanced computational catalysis.Phys. Rev. Research, 3:033055, 2021

    work page 2021

  34. [34]

    Ross and Peter Selinger

    Neil J. Ross and Peter Selinger. Optimal ancilla-free clifford+t approximation of z-rotations. Quantum Information and Computation, 16(11-12):901–953, 2016

    work page 2016

  35. [35]

    Cross, Lev S

    Andrew W. Cross, Lev S. Bishop, John A. Smolin, Jay M. Gambetta, et al. Openqasm 3: A broader and deeper quantum assembly language. arXiv:2104.14722, 2022

    work page arXiv 2022

  36. [36]

    Qir specification: Representing quantum programs within llvm ir

    QIR Alliance. Qir specification: Representing quantum programs within llvm ir. https:// github.com/qir-alliance/qir-spec, 2021. 42

    work page 2021

  37. [37]

    Quantum circuit design for decoded quantum interferometry.arXiv preprint arXiv:2504.18334, 2025

    Natchapol Patamawisut, Naphan Benchasattabuse, Michal Hajdu ˇsek, and Rodney Van Meter. Quantum circuit design for decoded quantum interferometry.arXiv preprint arXiv:2504.18334, 2025

    work page arXiv 2025

  38. [38]

    Decoded quantum interferometry under noise.arXiv preprint arXiv:2508.10725, 2025

    Kaifeng Bu, Weichen Gu, Dax Enshan Koh, and Xiang Li. Decoded quantum interferometry under noise.arXiv preprint arXiv:2508.10725, 2025

    work page arXiv 2025

  39. [39]

    Short-depth circuits for dicke state preparation

    Andreas Bartschi and Stephan Eidenbenz. Short-depth circuits for dicke state preparation. In2022 IEEE International Conference on Quantum Computing and Engineering (QCE), page 87–96. IEEE, September 2022

    work page 2022

  40. [40]

    Quantum state preparation using an exact cnot synthesis formulation

    Hanyu Wang, Jason Cong, and Giovanni De Micheli. Quantum state preparation using an exact cnot synthesis formulation. In2024 Design, Automation & Test in Europe Conference & Exhibition, pages 1–6, 2024

    work page 2024

  41. [41]

    Gallager

    R. Gallager. Low-density parity-check codes.IRE Transactions on Information Theory, 8(1):21–28, 1962

    work page 1962

  42. [42]

    Cambridge University Press, USA, 2008

    Tom Richardson and Ruediger Urbanke.Modern Coding Theory. Cambridge University Press, USA, 2008

    work page 2008

  43. [43]

    Typical performance of gallager-type error-correcting codes.Phys

    Yoshiyuki Kabashima, Tatsuto Murayama, and David Saad. Typical performance of gallager-type error-correcting codes.Phys. Rev. Lett., 84:1355–1358, Feb 2000

    work page 2000

  44. [44]

    D.J.C. MacKay. Good error-correcting codes based on very sparse matrices.IEEE Transactions on Information Theory, 45(2):399–431, 1999

    work page 1999

  45. [45]

    Quantum circuit design for decoded quantum interferometry, 2025

    Natchapol Patamawisut, Naphan Benchasattabuse, Michal Hajdu ˇsek, and Rodney Van Meter. Quantum circuit design for decoded quantum interferometry, 2025

    work page 2025

  46. [46]

    Høyer and R

    P. Høyer and R. ˇSpalek. Quantum fan-out is powerful.Theory of Computing, 1(5):81–103, 2005

    work page 2005

  47. [47]

    Quantum networks for concentrating entanglement.Journal of Physics A: Mathematical and General, 34(35):6939–6948, August 2001

    Phillip Kaye and Michele Mosca. Quantum networks for concentrating entanglement.Journal of Physics A: Mathematical and General, 34(35):6939–6948, August 2001

    work page 2001

  48. [48]

    Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D

    Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J. Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D. Nation, Lev S. Bishop, Andrew W. Cross, Blake R. Johnson, and Jay M. Gambetta. Quantum computing with qiskit, 2024

    work page 2024

  49. [49]

    Ising formulations of many np problems.Frontiers in Physics, 2, 2014

    Andrew Lucas. Ising formulations of many np problems.Frontiers in Physics, 2, 2014

    work page 2014

  50. [50]

    Sat-based maxsat algorithms.Artificial Intelligence, 196(2):77–105, 2013

    Carlos Ans´otegui, Mar´ıa Luisa Bonet, and Jordi Levy. Sat-based maxsat algorithms.Artificial Intelligence, 196(2):77–105, 2013

    work page 2013

  51. [51]

    Gurobi Optimizer Reference Manual, 2025

    Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual, 2025

    work page 2025

  52. [52]

    Belief propagation decoding of quantum channels by passing quantum messages

    Joseph M Renes. Belief propagation decoding of quantum channels by passing quantum messages. New Journal of Physics, 19(7):072001, July 2017

    work page 2017

  53. [53]

    Christophe Piveteau and Joseph M. Renes. Quantum message-passing algorithm for optimal and efficient decoding.Quantum, 6:784, August 2022

    work page 2022

  54. [54]

    Rectangular arrays with fixed margins

    Persi Diaconis and Anil Gangolli. Rectangular arrays with fixed margins. In David Aldous, Persi Diaconis, Joel Spencer, and J. Michael Steele, editors,Discrete Probability and Algorithms, pages 15–41, New York, NY, 1995. Springer New York. 43 10 Appendix 10.1 Benchmarking a Gauss-Jordan decoder In this section, we test an implementation of a Gauss-Jordan ...

    work page 1995