Introduction to QUDO, Tensor QUDO and HOBO formulations: Qudits, Equivalences, Knapsack Problem, Traveling Salesman Problem and Combinatorial Games

Alejandro Mata Ali

arxiv: 2508.01958 · v2 · submitted 2025-03-31 · 💻 cs.ET · quant-ph

Introduction to QUDO, Tensor QUDO and HOBO formulations: Qudits, Equivalences, Knapsack Problem, Traveling Salesman Problem and Combinatorial Games

Alejandro Mata Ali This is my paper

Pith reviewed 2026-05-22 21:37 UTC · model grok-4.3

classification 💻 cs.ET quant-ph

keywords QUDOT-QUDOHOBOcombinatorial optimizationknapsack problemtraveling salesman problemquditsquantum optimization

0 comments

The pith

QUDO, T-QUDO and HOBO formulations encode combinatorial problems using d-ary variables and higher-order terms with explicit mappings to each other.

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

The paper reviews and introduces Quadratic Unconstrained D-ary Optimization (QUDO), Tensor Quadratic Unconstrained D-ary Optimization (T-QUDO) and Higher-Order Unconstrained Binary Optimization (HOBO) as formulations for combinatorial optimization. It supplies explicit encodings that relate these to one another and to standard binary quadratic forms, along with concrete examples on the knapsack problem, traveling salesman problem and several combinatorial games. The stated purpose is to open these problems to quantum or quantum-inspired solvers that accept the more general variable types or interaction orders. Limitations of the formulations are also discussed.

Core claim

We present a brief review and introduction to Quadratic Unconstrained D-ary Optimization (QUDO), Tensor Quadratic Unconstrained D-ary Optimization (T-QUDO) and Higher-Order Unconstrained Binary Optimization (HOBO) formulations for combinatorial optimization problems. We also show explicit encodings between these formulations and discuss their limitations, with examples for the knapsack problem, traveling salesman problem and combinatorial games.

What carries the argument

Explicit encodings between QUDO, T-QUDO and HOBO that map d-ary or higher-order unconstrained problems onto each other and onto binary quadratic instances.

Load-bearing premise

New quantum or quantum-inspired algorithms will be able to run the more general QUDO, T-QUDO or HOBO instances even when they cannot run the equivalent QUBO instances.

What would settle it

A new solver that accepts QUDO or HOBO inputs but produces identical performance and solution quality when the same problem is first reduced to its QUBO equivalent.

Figures

Figures reproduced from arXiv: 2508.01958 by Alejandro Mata Ali.

**Figure 1.** Figure 1: Hashiwokakero with a 5 × 6 grid. a) Problem instance, b) Solution. As the variables to describe a solution naturally have three possible values, it is susceptible to be treated with a QUDO formulation. To properly formulate the problem in a simple way, we will redefine it in the following way. We have a set of N nodes, with Di the objective number of connections of the i-th node with other ones, and Ci is… view at source ↗

**Figure 2.** Figure 2: N-Queens solution with a 8 × 8 chessboard. 1. Column condition: if two queens are from different rows, their column must be different. So, if i ̸= j, then xi ̸= xj : Ci,j,a,a = 1, ∀a ∈ [0, N − 1] ∧ i ̸= j. (39) 2. Diagonal condition: if two queens are from different rows, separated by k positions, their column cannot be the same separated by k positions in the same direction. So, if j = i + k, then xj ̸= x… view at source ↗

**Figure 3.** Figure 3: a) Kakuro problem for a 6 × 6 board, b) solution for a 6 × 6 Kakuro. P i ′ ,j′∈Cjk xi ′j ′ = S c jk ∀j, k, so the constraint term is Cs = N X−1 i=0 X k   X i ′ ,j′∈Rik xi ′j ′ − S r ik   2 + N X−1 j=0 X k   X i ′ ,j′∈Cjk xi ′j ′ − S c jk   2 . (42) This is a QUDO term. The second restriction, the non-repetition one, is a QUDO term with compares each pair of variables in each region. Their non-zero … view at source ↗

**Figure 4.** Figure 4: a) Ishin no heya problem for a 5 × 5 board, b) Solution for the 5 × 5 Ishin no heya. We use variables xij as the value in the cell of the i-th row and j-th column. The first constraint is the sum of all the cells in a region. This is implemented with a QUDO term Cs = X k   X i ′ ,j′∈Rk xi ′j ′ − Sk   2 . (46) The repetition condition is implemented with a T-QUDO term Cr = N−X 1,N−1 i,j=0   N X−1 i ′>… view at source ↗

**Figure 5.** Figure 5: a) Initial configuration of the peg solitaire, b) movement of the green ball across the red ball, [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

read the original abstract

In this paper, we present a brief review and introduction to Quadratic Unconstrained D-ary Optimization (QUDO), Tensor Quadratic Unconstrained D-ary Optimization (T-QUDO) and Higher-Order Unconstrained Binary Optimization (HOBO) formulations for combinatorial optimization problems. We also show explicit encodings between these formulations and discuss their limitations. To help their understanding, we make some examples for the knapsack problem, traveling salesman problem and different combinatorial games. The games chosen to exemplify are: Hashiwokakero, N-Queens, Kakuro, Inshi no Heya, and Peg Solitaire. Although some of these games have already been formulated in a QUBO formulation, we are going to approach them with more general formulations, allowing their execution in new quantum or quantum-inspired optimization algorithms. This can be an easier way to introduce these more complicated formulations for harder problems.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

This is a straightforward tutorial-style review of QUDO/T-QUDO/HOBO with explicit encodings and game examples, but the key claim about enabling new solvers lacks any supporting references or analysis.

read the letter

Colleague, This paper is an introductory review of QUDO, T-QUDO, and HOBO formulations for combinatorial optimization, complete with mappings to each other and examples on standard problems plus some puzzle games. It does well at laying out the encodings explicitly and showing how to apply them to knapsack, TSP, and games like N-Queens and Peg Solitaire. If you are looking for a clear walk-through of how these more general forms work on concrete instances, the examples here are straightforward and could be a useful reference. The section on limitations also gives a balanced view without overclaiming. The soft spot is in the motivation for why these formulations matter beyond QUBO. The text suggests they allow use on new quantum or quantum-inspired algorithms that the binary quadratic versions cannot reach, but it provides no references to such algorithms or any argument showing that the higher-order or d-ary structure survives reduction or offers a real advantage. Without that, the practical difference remains unshown. The work is honest in its framing as a review and introduction rather than a new method or large demonstration. There are no invented claims or circular arguments visible. This is the kind of paper that might help a graduate student or practitioner getting started with these tools, but it is not aimed at experts who already know the reductions. It does not contain the kind of new result or rigorous analysis that would make it a strong candidate for peer review in a research journal. I would recommend against sending it for refereeing and suggest it stays as an arXiv note or is expanded with actual solver comparisons if the authors want to pursue publication.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a review and introduction to Quadratic Unconstrained D-ary Optimization (QUDO), Tensor QUDO (T-QUDO), and Higher-Order Unconstrained Binary Optimization (HOBO) formulations for combinatorial optimization problems. It provides explicit encodings between these formulations, discusses limitations, and illustrates the approaches with examples for the knapsack problem, traveling salesman problem, and combinatorial games (Hashiwokakero, N-Queens, Kakuro, Inshi no Heya, and Peg Solitaire), arguing that the more general formulations enable execution on new quantum or quantum-inspired algorithms.

Significance. If the encodings and example formulations are correctly derived, the work could provide a useful entry point for researchers exploring higher-arity and higher-order optimization beyond standard QUBO, particularly in quantum-inspired solvers. The explicit inter-formulation encodings represent a potential strength if they are shown to be direct and free of auxiliary parameters.

major comments (1)

[Abstract] Abstract: the central motivation that QUDO/T-QUDO/HOBO 'allow their execution in new quantum or quantum-inspired optimization algorithms' (and provide 'an easier way to introduce these more complicated formulations') is not supported by any references to, or descriptions of, algorithms whose native input is one of these forms but which cannot accept or efficiently process the equivalent QUBO after standard reduction. This assumption is load-bearing for the claimed advantage over existing QUBO approaches.

minor comments (1)

The abstract states that 'some of these games have already been formulated in a QUBO formulation' but provides no citations to those prior works, making it difficult to assess the novelty of the new encodings.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for identifying a key issue in the abstract's motivation. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the central motivation that QUDO/T-QUDO/HOBO 'allow their execution in new quantum or quantum-inspired optimization algorithms' (and provide 'an easier way to introduce these more complicated formulations') is not supported by any references to, or descriptions of, algorithms whose native input is one of these forms but which cannot accept or efficiently process the equivalent QUBO after standard reduction. This assumption is load-bearing for the claimed advantage over existing QUBO approaches.

Authors: We agree that the abstract's claim regarding execution on new algorithms is not supported by specific references or descriptions in the manuscript, and that this weakens the stated advantage. The paper's primary contributions are the review of the formulations, the explicit inter-formulation encodings, the discussion of limitations, and the example problem encodings. We will revise the abstract to remove or qualify the unsubstantiated claim about new algorithms and instead emphasize the value of the encodings and examples as an accessible introduction to these more general formulations. No new algorithmic descriptions will be added, as they fall outside the paper's scope. revision: yes

Circularity Check

0 steps flagged

No circularity: review paper with explicit encodings and no self-referential derivations

full rationale

The paper is a review and introduction that presents explicit encodings between QUDO/T-QUDO/HOBO and QUBO formulations along with problem examples. No derivation chain is claimed that reduces by construction to its own inputs, no fitted parameters are renamed as predictions, and no load-bearing self-citations or uniqueness theorems are invoked. The motivation about enabling new algorithms is stated as a benefit of the formulations rather than a result derived from the paper's own equations or prior self-work. This matches the default expectation of a non-circular review.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms or invented entities are stated. The work rests on the unexamined assumption that prior QUBO literature and the new d-ary encodings are compatible with unspecified new solvers.

pith-pipeline@v0.9.0 · 5691 in / 1129 out tokens · 23825 ms · 2026-05-22T21:37:08.136622+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 3 internal anchors

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A QUBO Formulation for the Generalized LinkedIn Queens and Takuzu/Tango Game

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John Wiley & Sons, Inc., 1990

Silvano Martello and Paolo Toth.Knapsack problems: algorithms and computer im- plementations. John Wiley & Sons, Inc., 1990

work page 1990

[2] [2]

Applegate, Robert E

David L. Applegate, Robert E. Bixby, Vašek Chvátal, and William J. Cook. 2007

work page 2007

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A Quantum Approximate Optimization Algorithm

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate opti- mization algorithm, 2014. URLhttps://arxiv.org/abs/1411.4028

work page internal anchor Pith review Pith/arXiv arXiv 2014

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A tutorial on formulating and using qubo models

Fred Glover, Gary Kochenberger, and Yu Du. A tutorial on formulating and using qubo models, 2019. URLhttps://arxiv.org/abs/1811.11538

work page arXiv 2019

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Qudits and High-Dimensional Quantum Computing,

Yuchen Wang, Zixuan Hu, Barry C. Sanders, and Sabre Kais. Qudits and high- dimensional quantum computing. Frontiers in Physics , 8, November 2020. ISSN 2296-424X. DOI: 10.3389/fphy.2020.589504. URL http://dx.doi.org/10.3389/ fphy.2020.589504

work page doi:10.3389/fphy.2020.589504 2020

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Quantum approximate optimization algorithm for qudit systems

Yannick Deller, Sebastian Schmitt, Maciej Lewenstein, Steve Lenk, Marika Federer, Fred Jendrzejewski, Philipp Hauke, and Valentin Kasper. Quantum approximate optimization algorithm for qudit systems. Physical Review A , 107(6), June 2023. ISSN 2469-9934. DOI: 10.1103/physreva.107.062410. URLhttp://dx.doi.org/10. 1103/PhysRevA.107.062410

work page doi:10.1103/physreva.107.062410 2023

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doi:10.1007/s11128-022-03670-y

Krzysztof Domino, Akash Kundu, Özlem Salehi, and Krzysztof Krawiec. Quadratic and higher-order unconstrained binary optimization of railway rescheduling for quan- tum computing. Quantum Information Processing , 21(9):337, Sep 2022. ISSN 1573-1332. DOI: 10.1007/s11128-022-03670-y. URL https://doi.org/10.1007/ s11128-022-03670-y

work page doi:10.1007/s11128-022-03670-y 2022

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Almudever, and Sebastian Feld

Zoe Verchere, Sourour Elloumi, and Andrea Simonetto. Optimizing Variational Cir- cuits for Higher-Order Binary Optimization . In2023 IEEE International Conference on Quantum Computing and Engineering (QCE) , pages 19–25, Los Alamitos, CA, USA, September 2023. IEEE Computer Society. DOI: 10.1109/QCE57702.2023.00011. URL https://doi.ieeecomputersociety.org/...

work page doi:10.1109/qce57702.2023.00011 2023

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Beyond qubo and hobo formulations, solving the travelling salesman problem on a quantum boson sampler, 2024

Daniel Goldsmith and Joe Day-Evans. Beyond qubo and hobo formulations, solving the travelling salesman problem on a quantum boson sampler, 2024. URLhttps: //arxiv.org/abs/2406.14252

work page arXiv 2024

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Hashiwokakero is np-complete

Daniel Andersson. Hashiwokakero is np-complete. Information Pro- cessing Letters , 109(19):1145–1146, 2009. ISSN 0020-0190. DOI: https://doi.org/10.1016/j.ipl.2009.07.017. URL https://www.sciencedirect. com/science/article/pii/S002001900900235X

work page doi:10.1016/j.ipl.2009.07.017 2009

[11] [11]

A novel approach for solving the n-queen prob- lem using a non-sequential conflict resolution algorithm

Omid Moghimi and Amin Amini. A novel approach for solving the n-queen prob- lem using a non-sequential conflict resolution algorithm. Electronics, 13(20), 2024. ISSN 2079-9292. DOI: 10.3390/electronics13204065. URL https://www.mdpi.com/ 2079-9292/13/20/4065

work page doi:10.3390/electronics13204065 2024

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A survey of known results and research areas for n-queens

Jordan Bell and Brett Stevens. A survey of known results and research areas for n-queens. Discrete Mathematics , 309(1):1–31, 2009. ISSN 0012-365X. DOI: https://doi.org/10.1016/j.disc.2007.12.043. URLhttps://www.sciencedirect.com/ science/article/pii/S0012365X07010394. Accepted in Quantum 9999-99-99, click title to verify. Published under CC-BY 4.0. 17

work page doi:10.1016/j.disc.2007.12.043 2009

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Comparing python, go, and c++ on the n-queens problem, 2020

Pascal Fua and Krzysztof Lis. Comparing python, go, and c++ on the n-queens problem, 2020. URL https://arxiv.org/abs/2001.02491

work page arXiv 2020

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Panigrahi

Santhosh G S, Piyush Joshi, Ayan Barui, and Prasanta K. Panigrahi. A quantum approach to solve n-queens problem, 2023. URL https://arxiv.org/abs/2312. 16312

work page 2023

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An incremental approach to the n-queen prob- lem with polynomial time

Bouneb Zine El Abidine. An incremental approach to the n-queen prob- lem with polynomial time. Journal of King Saud University - Com- puter and Information Sciences , 35(3):1–7, 2023. ISSN 1319-1578. DOI: https://doi.org/10.1016/j.jksuci.2023.02.002. URL https://www.sciencedirect. com/science/article/pii/S1319157823000319

work page doi:10.1016/j.jksuci.2023.02.002 2023

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Optimization of n-queens solvers on graphics processors

Tao Zhang, Wei Shu, and Min-You Wu. Optimization of n-queens solvers on graphics processors. In Olivier Temam, Pen-Chung Yew, and Binyu Zang, editors,Advanced Parallel Processing Technologies, pages 142–156, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg. ISBN 978-3-642-24151-2

work page 2011

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Benchmark Instances and Branch-and-Cut Algorithm for the Hashiwokakero Puzzle

Leandro C. Coelho, Gilbert Laporte, Arinei Lindbeck, and Thibaut Vidal. Benchmark instances and branch-and-cut algorithm for the hashiwokakero puzzle, 2019. URL https://arxiv.org/abs/1905.00973

work page internal anchor Pith review Pith/arXiv arXiv 2019

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Armagan Tarim

Christopher Jefferson, Angela Miguel, Ian Miguel, and S. Armagan Tarim. Modelling andsolvingenglishpegsolitaire. Computers & Operations Research, 33(10):2935–2959,

work page

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DOI: https://doi.org/10.1016/j.cor.2005.01.018

ISSN 0305-0548. DOI: https://doi.org/10.1016/j.cor.2005.01.018. URLhttps: //www.sciencedirect.com/science/article/pii/S0305054805000195. Part Spe- cial Issue: Constraint Programming

work page doi:10.1016/j.cor.2005.01.018 2005

[20] [20]

A QUBO Formulation for the Generalized LinkedIn Queens and Takuzu/Tango Game

AlejandroMataAliandEdgarMencia. Aquboformulationforthegeneralizedlinkedin queens and takuzu/tango game, 2024. URLhttps://arxiv.org/abs/2410.06429

work page internal anchor Pith review Pith/arXiv arXiv 2024

[21] [21]

J A Montañez-Barrera, Dennis Willsch, A Maldonado-Romo, and Kristel Michielsen. Unbalanced penalization: a new approach to encode inequality constraints of combi- natorial problems for quantum optimization algorithms.Quantum Science and Tech- nology, 9(2):025022, April 2024. ISSN 2058-9565. DOI: 10.1088/2058-9565/ad35e4. URL http://dx.doi.org/10.1088/2058...

work page doi:10.1088/2058-9565/ad35e4 2024