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REVIEW 3 major objections 3 minor 36 references

Combinatorial Optimization with Quantum Computers

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper provides a complete, repeatable recipe that transforms arbitrary combinatorial optimization problems into QUBO form and runs them on quantum annealers and gate-based machines.

desk verdict A solid, honest tutorial on QUBO/Ising encoding and QAOA; no new results, but the printed D-Wave code and Eq. (40) don't match the math and should be fixed before anyone uses it as a recipe. read the letter →

arxiv 2412.15778 v2 pith:QV2QTTPE submitted 2024-12-20 cs.ET quant-ph

Francisco Chicano , Gabiel Luque , Zakaria Abdelmoiz Dahi , Rodrigo Gil-Merino This is my paper · ORCID
classification cs.ETquant-ph MSC 90C2781P6890C09
keywords quantumoptimizationQUBOIsingmodelannealingQAOApseudo-Booleancombinatorialquadratization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper is a practical tutorial whose goal is to establish that a non-specialist can take a combinatorial optimization problem stated as a polynomial, a set of constraints, or a MAX-SAT instance and, by following a fixed sequence of steps, run it on either a quantum annealer or a gate-based quantum computer. The sequence is: encode solutions with binary variables, fold constraints into penalty terms, reduce all higher-order monomials to a quadratic pseudo-Boolean function (a QUBO), and translate that QUBO into an Ising Hamiltonian for an annealer or into a QAOA circuit for a gate-based machine. The paper demonstrates the recipe on a constrained integer polynomial and on a MAX-SAT instance, reporting sampled solutions from both hardware types. If the recipe holds, it gives researchers an entry point for experimenting with quantum optimization without first becoming quantum physicists.

What carries the argument

The machine that carries the tutorial is the QUBO encoding pipeline, a fixed sequence of algebraic transformations that lands a small combinatorial problem on quantum hardware; here QUBO is a quadratic unconstrained binary optimization model, a pseudo-Boolean function whose terms multiply at most two binary variables. The load-bearing pieces are the QUBO/Ising equivalence $x_j=(1-s_j)/2$ with its coefficient mapping, the global penalty rule $P=1+\sum_{S\subseteq[n],S\ne\emptyset}|a_S|$, the global quadratization identity $P(xy-2xz-2yx+3z)$ for replacing a product by a new variable, the local transformation $a(\sum_{j\in S}x_j-|S|+1)t$ for negative-coefficient monomials, and the QAOA ansatz built from $U(H_P,\gamma_k)U(H_M,\beta_k)$ with the diagonal Hamiltonian obtained by substituting $Z_j$ for each Ising variable. Each piece converts the previous step's output into the next machine's input, from polynomial objective to quadratic form to Ising Hamiltonian to parameterized quantum circuit.

What would settle it

Enumerate all assignments of a small problem with a known optimum, such as a five-variable MAX-SAT instance with a cubic objective term, after applying the tutorial's encoding, penalty, and quadratization steps; compare the QUBO's global minimum with the original problem's optimum. If any instance yields a minimum at an infeasible assignment or a different objective value, the recipe's correctness guarantee fails.

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Extended reading notes

Core claim

The paper's central claim is that the entire path from problem statement to quantum execution can be systematized. Integer variables are expanded into binary bits, with the most-significant bit adjusted for intervals that are not powers of two; categorical variables and permutations are encoded one-hot with sum-to-one constraints; and every constraint becomes an additive penalty whose constant is set to $P=1+\sum_{S\subseteq[n],S\ne\emptyset}|a_S|$, the sum of absolute monomial coefficients plus one, so no infeasible solution can outrank the best feasible solution. High-degree monomials are then quadratized, either globally by replacing a product of two variables with a new variable and adding the penalty $P(xy-2xz-2yx+3z)$, or locally by replacing a negative-coefficient monomial with a quadratic expression involving a new auxiliary variable. The resulting QUBO is translated to an Ising model for annealers, or converted via $x_j=(1-s_j)/2$ into a diagonal problem Hamiltonian whose QAOA circuit uses only $RZ$, $CNOT$, and $RX$ gates. The tutorial reports sampled solutions for both hardware types, with optimal assignments appearing most frequently, and concludes that the reader can reproduce these executions from the provided code.

Load-bearing premise

The load-bearing premise is that the paper's order-reduction rules and penalty-constant rule preserve the global optimum for every input problem, so that after transformation no infeasible solution is ever better than the true best feasible solution.

Editorial extensions

If this is right

  • A reader can follow the tutorial's encoding rules to represent bounded integer, categorical, and permutation variables in binary form, so the recipe covers many NP-hard combinatorial problems in a uniform way.
  • Because the penalty constant rule $P=1+\sum|a_S|$ is argued to separate feasible from infeasible solutions, constraints can be handled mechanically: square the transformed constraint and add a slack variable for inequalities.
  • Once the objective is a QUBO, the same object feeds both paradigms: it becomes an Ising Hamiltonian for quantum annealers and a diagonal problem Hamiltonian whose QAOA circuit is implemented with $RZ$, $CNOT$, and $RX$ gates.
  • The reported runs on small instances show optimal assignments as the most frequent samples on both kinds of hardware, which makes the workflow usable as a teaching and experimentation tool at current scales.
  • The paper's own discussion identifies penalty tuning, quadratization variable count, and transformation choice as open issues that determine whether the recipe scales beyond toy instances.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that applying the pipeline naively grows the qubit count with the sizes of the variable domains, the constraints, and the auxiliary quadratization variables; a natural consequence is that the recipe's near-term value is pedagogical and benchmarking rather than competitive on large instances.
  • A testable extension would be to automate the search over alternative quadratizations for each high-degree monomial and compare which choice yields the smallest penalty constant or the best annealer success probability, since the paper does not prove the local transformation is optimal.
  • The paper notes that low-depth QAOA expectation values can be evaluated classically; this suggests the gate-based workflow could be restructured to tune angles on a classical computer and use the quantum device only for final sampling.
  • A broader inference, in line with the paper's closing discussion, is that once encoding is automated the main open challenge shifts to algorithm design: better penalty rules and problem-structure-aware transformations would matter more than the initial mapping to QUBO.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This tutorial-style paper presents a practical introduction to solving combinatorial optimization problems on quantum annealers and gate-based quantum computers. It develops the standard pipeline: encoding variables as qubits (binary, integer, categorical, permutation), transforming constrained problems into unconstrained pseudo-Boolean objectives with penalties, reducing higher-order polynomials to QUBO via Rosenberg's global penalty or a local quadratization, mapping QUBO to Ising for D-Wave, and implementing the objective and mixer unitaries for QAOA on gate-based machines. The paper includes worked examples (MAX-SAT and constrained integer optimization), D-Wave Ocean code, Qiskit code, and a discussion of open questions.

Significance. If corrected, the paper would be a useful educational resource. Its strengths are the explicit step-by-step derivations, the closed-form penalty bound of Eq. (36), the correct QUBO/Ising conversion, and the QAOA circuit decomposition with CNOT/RZ and RX gates. The paper is honest about limitations, including open questions on penalty selection and the unclear practical speedup of QAOA. However, the tutorial's practical promise is currently compromised by mismatches between the printed formulas and the printed code and by an inconsistent optimization direction in the QAOA section; these affect reproducibility but are locally fixable.

major comments (3)
  1. [5.2, Figure 4 and Eq. (39)] Figure 4 does not implement the QUBO derived in Eq. (39). Eq. (39) contains the Rosenberg penalty P(x2x3 - 2x2z - 2x3z + 3z), but line 5 of the code omits the '- 2 * Binary('x3') * Binary('z')' term. The consequence is not cosmetic: for the feasible assignment (x2=1, x3=1, z=1), the intended penalty is 0 while the printed code evaluates 7*(1 - 2 + 3) = 14, so the code minimizes a different objective than the one derived and discussed. Since the paper's stated goal is to let readers reproduce the examples, this code/equation mismatch must be fixed.
  2. [6, Eq. (42), Figures 8 and 10] Section 6 states that the classical optimizer should 'maximize the expected value of HP' in Eq. (42), where HP is built directly from the objective function f to be minimized (Example 17). The provided code in Figures 8 and 10 instead uses scipy.optimize.minimize on classical_obj_fn, which returns the average value of f over the samples. For the minimization problems treated in this paper, the expectation of HP should be minimized, not maximized. As written, the text tells the reader to do the opposite of what the correct code does; this needs to be reconciled, e.g., by changing 'maximize' to 'minimize' throughout Section 6 or by defining HP as the negation of the objective.
  3. [5.1, Eq. (40)] Equation (40) writes the local quadratization as '-36(x2 + x2 + x3 - 2)t', repeating x2; the correct expression, following the rule stated earlier in Section 5.1 for the monomial -36x1x2x3, is '-36(x1 + x2 + x3 - 2)t'. This typo appears in the final formula of the running example and would be copied verbatim by a reader.
minor comments (3)
  1. [5.1, Rosenberg penalty formula] The displayed formula 'P(xy - 2xz - 2yx + 3z)' should be 'P(xy - 2xz - 2yz + 3z)'. The following sentence and Eq. (39) use the correct form, but the displayed formula is wrong as printed.
  2. [8, Figures 8-10] Figure 8 uses '# Omitted' for build_ansatz and classical_obj_fn, even though these are defined in Figures 9 and 10. The listing should either include the definitions inline or explicitly say 'see Figures 9 and 10'; as printed, the code in Figure 8 is not runnable by itself.
  3. [5.1, local transformation paragraph] The sentence 'We only showed one example of local transformation that is valid when the coefficient of the monomial is positive in negative in minimization (or positive in maximization)' is garbled and should read 'negative in minimization (or positive in maximization)'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: tutorial derivation is self-contained, with only non-load-bearing self-citations in the discussion; the code/equation typos are correctness defects, not circular reasoning.

full rationale

The paper's derivation chain is a tutorial transformation pipeline: encode variables (Sec. 4.1), express the objective as a pseudo-Boolean polynomial (Sec. 4.2), incorporate constraints via penalties (Sec. 4.3), reduce order to QUBO using Rosenberg's global penalty [35] and Dattani's local transformation [16] (Sec. 5.1), convert to Ising form with standard identities (Eqs. 8-11), and build QAOA circuits from commuting Z terms with CNOT/RZ decompositions (Sec. 6). Each load-bearing mathematical step is either an elementary identity proven inline (e.g., the penalty constant bound P = ub(f)-lb(f)+1 = 1 + sum |a_S| in Eq. 36, which is derived from the coefficients of f and provably dominates any infeasible solution) or an externally cited result (Rosenberg 1975, Dattani 2019). The penalty constant is a worst-case bound, not a parameter fitted to the paper's own outputs, so no 'prediction' is forced by construction. The self-citations in the paper ([11], [12], [15]) appear only in the Discussion as pointers to related work and as a caveat about QAOA's classical evaluation; they do not carry the derivation of the tutorial's central recipes. The discrepancies between Eq. (39) and the D-Wave code in Figure 4 (missing -2*x3*z term) and the x2/x2 typo in Eq. (40) are real reproducibility defects that a reader must correct, but they are fidelity errors in presentation, not circular reasoning: the mathematical transformations do not presuppose the code or the experimental tables. I therefore find no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted to data; the penalty constant P is computed deterministically from the problem coefficients. The paper relies on standard quantum-computation postulates and cited theorems, with no invented entities.

assumptions (3)
  • domain assumption The adiabatic theorem guarantees that a sufficiently slow Hamiltonian evolution keeps the system in its instantaneous ground state.
    Section 2.2 uses this to justify quantum annealers returning low-energy states; cited to [9] and [25].
  • standard math The gate set {RX, RY, RZ, CNOT} is universal for quantum computation.
    Section 2.3 relies on this to construct QAOA circuits; cited to [33].
  • standard math The Born rule and projective measurement are valid for extracting classical outcomes from qubits.
    Section 2.3 defines measurement probabilities as |a_w|^2, a standard quantum postulate.

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Cite this review

Pith. "Pith review of Combinatorial Optimization with Quantum Computers." pith.science (2026). https://pith.science/paper/QV2QTTPE

@misc{pith2026241215778,
  author       = {Pith},
  title        = {Pith review of: Combinatorial Optimization with Quantum Computers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QV2QTTPE}},
  note         = {Machine review of arXiv:2412.15778}
}
read the original abstract

Quantum computers leverage the principles of quantum mechanics to do computation with a potential advantage over classical computers. While a single classical computer transforms one particular binary input into an output after applying one operator to the input, a quantum computer can apply the operator to a superposition of binary strings to provide a superposition of binary outputs, doing computation apparently in parallel. This feature allows quantum computers to speed up the computation compared to classical algorithms. Unsurprisingly, quantum algorithms have been proposed to solve optimization problems in quantum computers. Furthermore, a family of quantum machines called quantum annealers are specially designed to solve optimization problems. In this paper, we provide an introduction to quantum optimization from a practical point of view. We introduce the reader to the use of quantum annealers and quantum gate-based machines to solve optimization problems.

Figures

Figures reproduced from arXiv: 2412.15778 by the authors.

Figure 1
Figure 1. The Bloch sphere representation of a qubit state. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. The solutions in our problem can be classified as feasible and infeasible. In [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Graph topology of D-Wave machines. Source: D-Wave official documen [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Code using D-Wave Ocean SDK to solve Example 15. [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Workflow of QAOA we will use the only non-trivial diagonal quantum gate we have seen: the Pauli￾Z gate. We already saw in Section 2 that Zj |xj ⟩ = (−1)xj |xj ⟩. Thus, we only need to express f(x) in terms of sj = (−1)xj variables, which take values in {−1, 1}. This is…
Figure 6
Figure 6. Figure 6: Circuit for U(HP , γk) where HP = −2Z1 − 5 2 Z2 − 9 2 Z1 ⊗ Z2 ⊗ Z3. previous expression are the terms e −iβkXj , which are developed as follows e −iβkXj =  cos βk −isin βk −isin βk cos βk  = RXj (2βk), (51) where RXj (2βk) is the RX gate with angle 2βk for qubit j. E…
Figure 7
Figure 7. Figure 7: Circuit for U(HM, βk) for the ansatz required to optimize the problem in Examples 17 and 18. ⊓⊔ Qiskit is the IBM SDK to simulate and run quantum circuits in IBM gate￾based quantum computers. We show in [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Main code of the QAOA implementation in Qiskit of the program to [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: Code to build the ansatz of QAOA for Example 10. [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]
Figure 10
Figure 10. Figure 10: Code to sample the ansatz and compute the average objective function [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]

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Works this paper leans on

36 extracted references · 26 canonical work pages

  1. [1]

    Reviews of Mod- ern Physics 90(1), 015002 (2018)

    Albash, T., Lidar, D.A.: Adiabatic quantum computation. Reviews of Mod- ern Physics 90(1), 015002 (2018)

    work page 2018

  2. [2]

    Princeton University Press (2006)

    Applegate, D.L., Bixby, R.E., Chv´ atal, V., Cook, W.J.: The Traveling Sales- man Problem: A Computational Study. Princeton University Press (2006)

    work page 2006

  3. [3]

    In: C´ aceres, L.P., V´ erel, S

    Ayodele, M.: Penalty weights in QUBO formulations: Permutation prob- lems. In: C´ aceres, L.P., V´ erel, S. (eds.) Evolutionary Computation in Com- binatorial Optimization - 22nd European Conference, EvoCOP 2022, Held as Part of EvoStar 2022, Madrid, Spain, April 20-22, 2022, Proceedings. Lec- ture Notes in Computer Science, vol. 13222, pp. 159–174. Sprin...

    work page doi:10.1007/978-3-031-04148-8 2022

  4. [5]

    Physical review A 52(5), 3457 (1995)

    Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Physical review A 52(5), 3457 (1995)

    1995

  5. [6]

    In: 2020 IEEE International Con- ference on Quantum Computing and Engineering (QCE)

    B¨ artschi, A., Eidenbenz, S.: Grover mixers for QAOA: Shifting complexity from mixer design to state preparation. In: 2020 IEEE International Con- ference on Quantum Computing and Engineering (QCE). pp. 72–82. IEEE (2020)

    work page 2020

  6. [7]

    1356–1362

    Battiti, R.: Maximum satisfiability problemMaximum Satisfiabil- ity Problem, pp. 1356–1362. Springer US, Boston, MA (2001). https://doi.org/10.1007/0-306-48332-7 277

    doi:10.1007/0-306-48332-7 2001

  7. [8]

    Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.V.: Strengths and weaknesses of quantum computing. SIAM J. Comput. 26(5), 1510–1523 (1997). https://doi.org/10.1137/S0097539796300933

    doi:10.1137/s0097539796300933 1997

  8. [9]

    Zeitschrift f¨ ur Physik51(3), 165–180 (1928)

    Born, M., Fock, V.: Beweis des adiabatensatzes. Zeitschrift f¨ ur Physik51(3), 165–180 (1928)

    work page 1928

Show all 36 references
  1. [10]

    Theory and Algorithms

    C ¸ ela, E.: The Quadratic Assignment Problem. Theory and Algorithms. Springer (1998)

    1998

  2. [11]

    In: Silva, S., Paquete, L

    Chicano, F., Dahi, Z.A., Luque, G.: An efficient QAOA via a polynomial QPU-needless approach. In: Silva, S., Paquete, L. (eds.) Companion Pro- ceedings of the Conference on Genetic and Evolutionary Computation, GECCO 2023, Companion Volume, Lisbon, Portugal, July 15-19, 2023. ...

    2023

  3. [12]

    In: Affenzeller, M., Win- kler, S.M., Kononova, A.V., Trautmann, H., Tusar, T., Machado, P., B¨ ack, 30 F

    Chicano, F., Whitley, D., Ochoa, G., Tin´ os, R.: Generalizing and unifying gray-box combinatorial optimization operators. In: Affenzeller, M., Win- kler, S.M., Kononova, A.V., Trautmann, H., Tusar, T., Machado, P., B¨ ack, 30 F. Chicano, G. Luque, Z. Dahi, and R. Gil-Merino T...

    2024

  4. [13]

    Journal of the American Chemical Society 146(1), 1053–1061 (2024)

    Chicco, S., Allodi, G., Chiesa, A., Garlatti, E., Buch, C.D., Santini, P., De Renzi, R., Piligkos, S., Carretta, S.: Proof-of-concept quantum simulator based on molecular spin qudits. Journal of the American Chemical Society 146(1), 1053–1061 (2024). https://doi.org/10.1021/ja...

    2024 doi

  5. [14]

    Nature 453(7198), 1031–1042 (2008)

    Clarke, J., Wilhelm, F.K.: Superconducting quantum bits. Nature 453(7198), 1031–1042 (2008)

    2008

  6. [15]

    In: Affenzeller, M., Winkler, S.M., Kononova, A.V., Trautmann, H., Tusar, T., Machado, P., B¨ ack, T

    Dahi, Z.A., Chicano, F., Luque, G., Derbel, B., Alba, E.: Scalable quan- tum approximate optimiser for pseudo-boolean multi-objective optimisa- tion. In: Affenzeller, M., Winkler, S.M., Kononova, A.V., Trautmann, H., Tusar, T., Machado, P., B¨ ack, T. (eds.) Parallel Problem S...

    2024

  7. [16]

    CoRR abs/1901.04405 (2019), https://arxiv.org/abs/1901.04405

    Dattani, N.: Quadratization in discrete optimization and quantum mechan- ics. CoRR abs/1901.04405 (2019), https://arxiv.org/abs/1901.04405

    2019 arXiv

  8. [17]

    CoRRabs/1910.13583 (2019), https: //arxiv.org/abs/1910.13583

    Dattani, N., Chau, H.T.: All 4-variable functions can be perfectly quadra- tized with only 1 auxiliary variable. CoRRabs/1910.13583 (2019), https: //arxiv.org/abs/1910.13583

    2019 arXiv

  9. [18]

    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Physical review 47(10), 777 (1935)

    1935

  10. [19]

    Johnston, Nic Harrigan, M.G.S.: Programming Quantum Comput- ers

    Eric R. Johnston, Nic Harrigan, M.G.S.: Programming Quantum Comput- ers. O’Reilly Media Inc. (2019)

    2019

  11. [20]

    Farhi, E., Goldstone, J., Gutmann, S.: A quantum approximate optimiza- tion algorithm (2014), https://arxiv.org/abs/1411.4028

    2014 arXiv

  12. [21]

    International Journal of Theoretical Physics 21(6/7), 467–488 (1982)

    Feynman, R.P.: Simulating physics with computers. International Journal of Theoretical Physics 21(6/7), 467–488 (1982)

    1982

  13. [22]

    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman (1979)

    1979

  14. [24]

    Hoefler, T., H¨ aner, T., Troyer, M.: Disentangling hype from practicality: On realistically achieving quantum advantage. Commun. ACM 66(5), 82– 87 (2023). https://doi.org/10.1145/3571725

    2023 doi

  15. [25]

    Journal of Mathematical Physics 48(10) (2007) Combinatorial Optimization with Quantum Computers 31

    Jansen, S., Ruskai, M.B., Seiler, R.: Bounds for the adiabatic approxima- tion with applications to quantum computation. Journal of Mathematical Physics 48(10) (2007) Combinatorial Optimization with Quantum Computers 31

    2007

  16. [26]

    Wiley (2011)

    Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley (2011)

    2011

  17. [27]

    Physical Review E 58(5), 5355 (1998)

    Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse ising model. Physical Review E 58(5), 5355 (1998)

    1998

  18. [28]

    science 220(4598), 671–680 (1983)

    Kirkpatrick, S., Gelatt Jr, C.D., Vecchi, M.P.: Optimization by simulated annealing. science 220(4598), 671–680 (1983)

    1983

  19. [29]

    Kochenberger, G.A., Hao, J., Glover, F.W., Lewis, M.W., L¨ u, Z., Wang, H., Wang, Y.: The unconstrained binary quadratic program- ming problem: a survey. J. Comb. Optim. 28(1), 58–81 (2014). https://doi.org/10.1007/S10878-014-9734-0

    2014 doi

  20. [30]

    Liu, T., Li, Z.W., Dinneen, M.J.: Graph minor embedding for adiabatic quantum computing. Tech. rep., The University of Auckland (2021),https: //hdl.handle.net/2292/58011

    2021

  21. [31]

    npj Quantum Informa- tion 2(1), 1–8 (2016)

    Montanaro, A.: Quantum algorithms: an overview. npj Quantum Informa- tion 2(1), 1–8 (2016)

    2016

  22. [32]

    In: Fieldsend, J.E., Wagner, M

    Moraglio, A., Georgescu, S., Sadowski, P.: AutoQubo: data-driven auto- matic QUBO generation. In: Fieldsend, J.E., Wagner, M. (eds.) GECCO ’22: Genetic and Evolutionary Computation Conference, Companion Vol- ume, Boston, Massachusetts, USA, July 9 - 13, 2022. pp. 2232–2239. AC...

    2022

  23. [33]

    Cambridge university press (2010)

    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum informa- tion. Cambridge university press (2010)

    2010

  24. [34]

    In: Silva, S., Paquete, L

    Pauckert, J., Ayodele, M., Garc ´ ıa, M.D., Georgescu, S., Parizy, M.: Auto- QUBO v2: Towards efficient and effective QUBO formulations for ising ma- chines. In: Silva, S., Paquete, L. (eds.) Companion Proceedings of the Con- ference on Genetic and Evolutionary Computation, GE...

    2023

  25. [35]

    Cahiers Centre Etudes Rech

    Rosenberg, I.G.: Reduction of bivalent maximization to the quadratic case. Cahiers Centre Etudes Rech. Oper. 17, 71–74 (1975)

    1975

  26. [36]

    Theory and practice

    Weintraub, S.H.: Jordan canonical form. Theory and practice. Morgan & Claypool (2009)

    2009

  27. [37]

    Quantum Information Pro- cessing 21(4), 141 (2022)

    Willsch, D., Willsch, M., Gonzalez Calaza, C.D., Jin, F., De Raedt, H., Svensson, M., Michielsen, K.: Benchmarking advantage and d-wave 2000q quantum annealers with exact cover problems. Quantum Information Pro- cessing 21(4), 141 (2022)

    2022

  28. [38]

    In: Silva, S., Paquete, L

    Zielinski, S., N¨ ußlein, J., Stein, J., Gabor, T., Linnhoff-Popien, C., Feld, S.: Influence of different 3SAT-to-QUBO transformations on the solu- tion quality of quantum annealing: A benchmark study. In: Silva, S., Paquete, L. (eds.) Companion Proceedings of the Conference o...

    2023

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