pith. sign in

arxiv: 2311.10433 · v4 · submitted 2023-11-17 · 🪐 quant-ph · cs.ET

Task Scheduling Optimization with Direct Constraints from a Tensor Network Perspective

Alejandro Mata Ali , I\~nigo Perez Delgado , Beatriz Garc\'ia Markaida , Aitor Moreno Fdez. de Leceta This is my paper

Pith reviewed 2026-05-24 05:59 UTC · model grok-4.3

classification 🪐 quant-ph cs.ET
keywords task scheduling optimizationtensor networksdirected constraintsquantum-inspiredcombinatorial optimizationindustrial applicationsexact algorithms
0
0 comments X

The pith

A tensor network constructs an exact solution for optimal task scheduling with directed constraints by minimizing execution cost.

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

The paper presents a method to solve task scheduling optimization using tensor networks. It builds a network that captures all possible task-machine assignments respecting constraints and selects the one with lowest total cost. Techniques such as problem preprocessing, logical constraint condensation, and computation reuse lower the complexity. The main algorithm contracts the network, while the iterative version adds constraints as needed and the genetic version hybrids with evolutionary methods. Simple implementations were tested to show the approach works.

Core claim

By representing the scheduling problem as a tensor network that encodes the valid assignments and their costs, contracting the network after applying preprocessing, condensation, and reuse techniques provides an exact and explicit solution for the minimum-cost task combination under directed constraints.

What carries the argument

The tensor network representation of the problem tensor, which is contracted to find the optimal solution after reductions in complexity.

If this is right

  • Exact optimal schedules are computed rather than approximated.
  • Computational effort is reduced through preprocessing and intermediate result reuse.
  • The iterative algorithm incorporates only necessary constraints for efficiency.
  • Hybrid genetic search can accelerate finding solutions in some cases.
  • The methods are implemented and available for further testing.

Where Pith is reading between the lines

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

  • Similar tensor network encodings might apply to other constraint satisfaction problems in operations research.
  • The quantum-inspired nature suggests potential for implementation on quantum hardware if tensor contractions can be mapped accordingly.
  • Scalability to very large industrial plants may require additional optimization beyond the described steps.
  • Benchmarking against classical solvers like integer programming would clarify practical advantages.

Load-bearing premise

After preprocessing and condensation, the tensor network remains small enough to contract in polynomial time for industrially relevant sizes.

What would settle it

If the method applied to a known small instance with a unique optimal schedule returns a non-optimal or invalid assignment, the claim of exactness would be falsified.

Figures

Figures reproduced from arXiv: 2311.10433 by Aitor Moreno Fdez. de Leceta, Alejandro Mata Ali, Beatriz Garc\'ia Markaida, I\~nigo Perez Delgado.

Figure 1
Figure 1. Figure 1: FIG. 1: Example of problem instance with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Initialization nodes. a) Initial superposition [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Name of the indexes for the different tensors [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Tensor network performing the optimization, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Scaling of the mean runtime of the normal algorithm with respect to different parameters. There are [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Scaling of the mean runtime of the iterative algorithm with respect to different parameters. There are [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Scaling of the mean number of required steps of the iterative algorithm with respect to different parameters. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Mean success rate of the genetic algorithm [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
read the original abstract

This work presents a novel method for task optimization in industrial plants using quantum-inspired tensor network technology. This method obtains the best possible combination of tasks on a set of machines with directed constraints while minimizing the total execution cost. With this method, an exact and explicit solution of the problem is provided. This algorithm constructs a tensor network representation of the tensor which provides the solution of the problem. This method is improved in order to reduce the computational complexity of the solution computation, using problem preprocessing, new techniques of condensation of logical constraints, optimization of the value determination technique with previously calculated results, reuse of intermediate computations, and iterative relations for constraints. Three algorithms for computation are presented: the main algorithm, the iterative algorithm which adds only the minimal amount of necessary constraints, and the genetic algorithm which combines the iterative algorithm with basic genetic algorithms. Finally, a simple version of both algorithms was implemented, and their performance was tested, all publicly available.

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

0 major / 3 minor

Summary. The paper presents a tensor network approach to task scheduling optimization with directed constraints, claiming to deliver an exact solution for the optimal task-machine assignment that minimizes total execution cost. The method constructs a tensor network representation of the problem, then applies preprocessing, condensation of logical constraints, reuse of intermediate results, and iterative relations to reduce complexity. Three algorithms are described (main, iterative adding minimal constraints, and a genetic variant), with a simplified implementation and performance tests on small instances made publicly available.

Significance. If the exactness claim holds under the described reductions, the work offers a quantum-inspired exact method for a practical constrained optimization problem, with the public code and empirical checks on small cases providing a verifiable starting point. The explicit algorithms and acknowledgment of general TN hardness are strengths that distinguish it from purely heuristic approaches.

minor comments (3)
  1. The abstract states that the tensor network 'provides the solution of the problem' but does not name the specific tensor indices or contraction order; a brief illustrative example in §3 or §4 would clarify how the optimal assignment is recovered from the contracted tensor.
  2. The iterative algorithm is described as adding 'only the minimal amount of necessary constraints,' yet no pseudocode or precise stopping criterion is supplied; adding this would improve reproducibility.
  3. Performance figures are reported only for small instances; a short discussion of observed scaling with number of tasks/machines (even if preliminary) would help readers assess the practical reach of the condensation steps.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the recognition of its strengths in providing explicit algorithms and acknowledging TN hardness, and the recommendation for minor revision. No major comments were provided in the report, so we have no specific points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs a tensor-network representation of the scheduling problem, applies explicit preprocessing/condensation steps, and supplies three algorithms (main, iterative, genetic) whose correctness is verified on small instances with public code. No equations reduce a claimed result to a fitted parameter or self-citation by construction; the exactness claim rests on the explicit TN contraction after the listed reductions, which are independently described and not tautological. No load-bearing uniqueness theorem or ansatz is imported from prior self-work. The derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty pending full text.

pith-pipeline@v0.9.0 · 5706 in / 931 out tokens · 25337 ms · 2026-05-24T05:59:45.411905+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages · 4 internal anchors

  1. [1]

    The creation of the initial uniform superposition state in the qudits:|ψ0⟩ = P ⃗ x|⃗ x⟩

    work page

  2. [2]

    The application of an imaginary time evolution, so that the amplitude of a combination depends on its cost and the damping constant τ: |ψ1⟩ =P ⃗ xe−τ C(⃗ x) |⃗ x⟩

    work page

  3. [3]

    Rnr−1e−τ C(⃗ x) |⃗ x⟩

    The removal of states by applying con- straints by means of projectors: |ψ2⟩ =P ⃗ xR0R1 . . . Rnr−1e−τ C(⃗ x) |⃗ x⟩

    work page

  4. [4]

    From a classical point of view, it consists of the follow- ing:

    The measurement and extraction of the basis state with the maximum amplitude of the superposition. From a classical point of view, it consists of the follow- ing:

    work page

  5. [5]

    The creation of the initial represented tensor where all elements are equal, representing all possible combinations: ψ0 x0,x1,... = 1

    work page

  6. [6]

    = e−τ C(⃗ x)

    The application of an operation to assign to ev- ery possible combination element a value propor- tional to its cost, with a damping constant τ: ψ1 x0,x1,... = e−τ C(⃗ x)

    work page

  7. [7]

    = R0 x0,x1,...R1 x0,x1,

    The removal of elements of incompatible com- binations by means of projective operations: ψ2 x0,x1,... = R0 x0,x1,...R1 x0,x1,... . . . Rnr−1 x0,x1,...e−τ C(⃗ x)

    work page

  8. [8]

    This section is structured as follows

    The measurement and extraction of the maximum element position of the represented tensor. This section is structured as follows. First, Ssec. IIIA presents the initialization and evolution tensor layers. Second, Ssec. IIIB presents the creation of the layers of tensors that implement the constraints, and Ssec. IIIC presents the application of the contrain...

    work page

  9. [9]

    These sets are the constraints that can be condensed in the next step

    Constraint grouping First, the constraints are grouped into sets that start and end on the same machines and have the same con- ditioned machine, as this is a condensation requirement. These sets are the constraints that can be condensed in the next step. In case the conditioned machine is before the first conditioning machine, all the constraints that ha...

    work page

  10. [10]

    This is done to avoid summing in the next step the compatible states differently accord- ing to their extremal relation to the constraint

    Constraint condensation Within each group, the condensation begins with the generation of subgroups of up toPk constraints, with k being the first (or last in the final extreme case) condi- tioning machine in the constraint, so that the condition- ing task on the first machine (or last in the final extreme case) is never repeated. This is done to avoid su...

    work page

  11. [11]

    However, in this case, the signal be- tween them indicates which constraint has been acti- vated

    Creating constraint layers In order to impose such restrictions, a set of constraint layers is created. However, in this case, the signal be- tween them indicates which constraint has been acti- vated. IfthefirstCtrlofthelayerfindsthatitsassociated machine is in the task that indicates the third constraint, it will tell the following tensor to verify if i...

    work page

  12. [12]

    This is because obviously these constraints will never be activated

    All the constraints in which the previous machine conditioned another machine and had to have a dif- ferent value than the one found are removed. This is because obviously these constraints will never be activated

    work page

  13. [13]

    If the previous machine conditions another machine and its value is the one found, the constraint is kept by eliminating the dependence on the previous machine, since it will always satisfy this part

    work page

  14. [14]

    This is because it will always be active

    If the only conditioner of a constraint is the previ- ous machine and has the obtained value, this con- straint is eliminated, and the task of the target ma- chine is forced to be the one imposed by the con- straint. This is because it will always be active. 9 This allows to determine the variable without con- tracting the tensor network

    work page

  15. [15]

    If a constraint has the previous machine as condi- tional and its value is the obtained, it disappears, since it is always satisfied

    work page

  16. [16]

    This is be- cause if that combination is found, the constraint would not be satisfied

    If a constraint has as conditioned the previous ma- chine with a value different from the one obtained, the constraint is replaced by one that makes the conditioned combination cannot occur. This is be- cause if that combination is found, the constraint would not be satisfied. For each variable determined, either by contraction or by this process, the pro...

    work page

  17. [17]

    The algorithm checks if the simple minimum, the optimal of the unconstrained problem, satisfies all the constraints. If it does, it finishes, meaning that this is the case where the global minimum of the unconstrained problem is the same as the global minimum of the constrained problem by the set of constraints. If not, it continues

    work page

  18. [18]

    If the result satisfies all the constraints, it finishes

    The first unmatched constraint is applied, also adding the constraints that are compatible with it (and also unmatched) to condense them, and tak- ing advantage to eliminate more combinations and iterations. If the result satisfies all the constraints, it finishes. If not, this step is repeated adding more unmatched constraints until the result satisfies ...

    work page

  19. [19]

    If it reaches the limit of iterations, that is, the num- ber of sets of condensed constraints allowed to be applied, it finishes with a negative result: no solu- tion that follows all constraints of the chosen set has been found. It is important to note that this is an algorithm for deter- mining the solution of the minimal set of constraints, not the min...

    work page

  20. [20]

    Random initialization of the population

    work page

  21. [21]

    Only a proportion of the best individuals are kept, using the cost function as the evaluator

    Computationoftheresultforeachindividual. Only a proportion of the best individuals are kept, using the cost function as the evaluator

    work page

  22. [22]

    Crosses pairs of parents from the previous genera- tion and corrects the associated constraints, remov- ing the ones that will never be activated

    With these individuals, a crossover is performed. Crosses pairs of parents from the previous genera- tion and corrects the associated constraints, remov- ing the ones that will never be activated

    work page

  23. [23]

    Mu- tations are performed by changing some tasks to perform in the machines

    Creation of a set of mutated individuals from the parents and correct the associated constraints. Mu- tations are performed by changing some tasks to perform in the machines

    work page

  24. [24]

    Check for repeat individuals for their removal

    work page

  25. [25]

    Addition of new randomly created individuals to make up for the missing ones, to have the number of individuals it should have

    work page

  26. [26]

    This is performed to make each in- dividual subproblem representative with respect to the main problem

    Addition of new randomly selected constraints for the new generation individuals, until they have a fixed amount. This is performed to make each in- dividual subproblem representative with respect to the main problem

    work page

  27. [27]

    The chromosome crossover is performed by exchang- ing, within the same machine, the possible active tasks a number of times, chosen at random between the two individuals

    Repetition of steps 2 → 6 until the convergence criteria are met or the fixed maximum number of generations is reached. The chromosome crossover is performed by exchang- ing, within the same machine, the possible active tasks a number of times, chosen at random between the two individuals. The constraint correction is applied to each individual each time ...

    work page 2022

  28. [28]

    Liang, P

    Z. Liang, P. Zhong, M. Liu, C. Zhang, and Z. Zhang, A computationalefficientoptimizationofflowshopschedul- ing problems, Scientific Reports12, 845 (2022)

    work page 2022

  29. [29]

    B. A and A. X. M, A simple model to optimize general flow-shop scheduling problems with known break down time and weights of jobs, Procedia Engineering38, 191 (2012), iNTERNATIONAL CONFERENCE ON MOD- ELLING OPTIMIZATION AND COMPUTING

    work page 2012

  30. [30]

    L. Li, S. Liang, Z. Zhu, C. Ding, H. Zha, and B. Wu, Learning to optimize permutation flow shop scheduling via graph-based imitation learning, Proceedings of the AAAI Conference on Artificial Intelligence 38, 20185 (2024)

    work page 2024

  31. [31]

    Pelikan, D

    M. Pelikan, D. Goldberg, and F. Lobo, A survey of op- timization by building and using probabilistic models, in Proceedings of the 2000 American Control Confer- ence. ACC (IEEE Cat. No.00CH36334), Vol. 5 (2000) pp. 3289–3293 vol.5

    work page 2000

  32. [32]

    Karaboga and B

    D. Karaboga and B. Basturk, Artificial bee colony (abc) optimization algorithm for solving constrained optimiza- tion problems, in Foundations of Fuzzy Logic and Soft Computing,editedbyP.Melin, O.Castillo, L.T.Aguilar, J. Kacprzyk, and W. Pedrycz (Springer Berlin Heidel- berg, Berlin, Heidelberg, 2007) pp. 789–798

    work page 2007

  33. [33]

    A Quantum Approximate Optimization Algorithm

    E. Farhi, J. Goldstone, and S. Gutmann, A quan- tum approximate optimization algorithm (2014), arXiv:1411.4028 [quant-ph]. 13 4 6 8 10 12 14 16 18 20 Number of Machines 1.0 1.5 2.0 2.5 3.0 3.5 4.0Number of Steps 4 tasks 6 tasks 8 tasks 10 tasks 12 tasks 14 tasks 16 tasks 18 tasks 20 tasks (a) Steps against number of machines m, for different number of tas...

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  34. [34]

    Tilly, H

    J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson, The variational quantum eigensolver: A re- viewofmethodsand bestpractices, Physics Reports 986, 1 (2022), the Variational Quantum Eigensolver: a review of methods and best practices

    work page 2022

  35. [35]

    Hen and F

    I. Hen and F. M. Spedalieri, Quantum annealing for con- strainedoptimization,Phys.Rev.Appl. 5,034007(2016)

    work page 2016

  36. [36]

    Aramon, G

    M. Aramon, G. Rosenberg, E. Valiante, T. Miyazawa, H. Tamura, and H. G. Katzgraber, Physics-inspired optimization for quadratic unconstrained problems using a digital annealer, Frontiers in Physics 7, 10.3389/fphy.2019.00048 (2019)

    work page doi:10.3389/fphy.2019.00048 2019

  37. [37]

    Tensor Networks in a Nutshell

    J. Biamonte and V. Bergholm, Tensor networks in a nut- shell (2017), arXiv:1708.00006 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  38. [38]

    Sozykin, A

    K. Sozykin, A. Chertkov, R. Schutski, A.-H. Phan, A. S. CICHOCKI, and I. Oseledets, Ttopt: A maximum vol- ume quantized tensor train-based optimization and its application to reinforcement learning, in Advances in Neural Information Processing Systems, Vol. 35, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associates,...

    work page 2022

  39. [39]

    Alcazar, M

    J. Alcazar, M. Ghazi Vakili, C. B. Kalayci, and A. Perdomo-Ortiz, Enhancing combinatorial optimiza- tion with classical and quantum generative models, Na- ture Communications 15, 2761 (2024)

    work page 2024

  40. [40]

    T. Hao, X. Huang, C. Jia, and C. Peng, A quantum- inspired tensor network algorithm for constrained combi- natorial optimization problems, Frontiers in Physics10, 10.3389/fphy.2022.906590 (2022)

    work page doi:10.3389/fphy.2022.906590 2022

  41. [41]

    A. M. Ali, Explicit solution equation for every combina- torial problem via tensor networks: Melocoton (2025), arXiv:2502.05981 [cs.ET]

    work page arXiv 2025

  42. [42]

    A. M. Ali, I. P. Delgado, M. R. Roura, and A. M. F. de Leceta, Polynomial-time solver of tridiagonal qubo, qudo and tensor qudo problems with tensor networks (2025), arXiv:2309.10509 [quant-ph]

    work page arXiv 2025

  43. [43]

    Nedbálek and A

    L. Nedbálek and A. Novák, Bottleneck identification in resource-constrained project scheduling via constraint re- laxation, in Proceedings of the 14th International Con- ference on Operations Research and Enterprise Systems (SCITEPRESS - Science and Technology Publications,

    work page

  44. [44]

    H. Ding, C. Zhuang, and J. Liu, Extensions of the resource-constrained project scheduling problem, Au- tomation in Construction153, 104958 (2023)

    work page 2023

  45. [45]

    Ku and J

    W.-Y. Ku and J. C. Beck, Mixed integer programming models for job shop scheduling: A computational analy- sis, Computers & Operations Research73, 165 (2016)

    work page 2016

  46. [46]

    Ullman, Np-complete scheduling problems, Journal of Computer and System Sciences10, 384 (1975)

    J. Ullman, Np-complete scheduling problems, Journal of Computer and System Sciences10, 384 (1975)

    work page 1975

  47. [47]

    van Bevern, R

    R. van Bevern, R. Bredereck, L. Bulteau, C. Ko- musiewicz, N. Talmon, and G. J. Woeginger, Precedence- constrained scheduling problems parameterized by par- tial order width, inDiscrete Optimization and Operations Research, edited by Y. Kochetov, M. Khachay, V. Beres- nev, E. Nurminski, and P. Pardalos (Springer Interna- tional Publishing, Cham, 2016) pp. 105–120

    work page 2016

  48. [48]

    Blazewicz, J

    J. Blazewicz, J. Lenstra, and A. Kan, Scheduling sub- ject to resource constraints: classification and complex- ity, Discrete Applied Mathematics5, 11 (1983)

    work page 1983

  49. [49]

    Ganian, T

    R. Ganian, T. Hamm, and G. Mescoff, The complexity landscape of resource-constrained scheduling, inProceed- ings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI’20 (2021). 14

    work page 2021

  50. [50]

    Dridi, S

    O. Dridi, S. Krichen, and A. Guitouni, Solving resource- constrained project scheduling problem by a genetic lo- cal search approach, in2013 5th International Confer- ence on Modeling, Simulation and Applied Optimization (ICMSAO) (2013) pp. 1–5

    work page 2013

  51. [51]

    Pezzella, G

    F. Pezzella, G. Morganti, and G. Ciaschetti, A genetic algorithm for the flexible job-shop scheduling problem, Computers & Operations Research35, 3202 (2008), part Special Issue: Search-based Software Engineering

    work page 2008

  52. [52]

    Della Croce, R

    F. Della Croce, R. Tadei, and G. Volta, A genetic algo- rithm for the job shop problem, Computers & Operations Research 22, 15 (1995), genetic Algorithms

    work page 1995

  53. [53]

    Murata, H

    T. Murata, H. Ishibuchi, and H. Tanaka, Genetic algo- rithms for flowshop scheduling problems, Computers & Industrial Engineering 30, 1061 (1996)

    work page 1996

  54. [54]

    Frieze and J

    A. Frieze and J. Yadegar, A new integer programming formulation for the permutation flowshop problem, Eu- ropean Journal of Operational Research40, 90 (1989)

    work page 1989

  55. [55]

    Zhang, P

    C. Zhang, P. Li, Z. Guan, and Y. Rao, A tabu search algorithm with a new neighborhood structure for the job shop scheduling problem, Computers & Operations Re- search 34, 3229 (2007)

    work page 2007

  56. [56]

    Dell’Amico and M

    M. Dell’Amico and M. Trubian, Applying tabu search to the job-shop scheduling problem, Annals of Operations Research 41, 231 (1993)

    work page 1993

  57. [57]

    Avci, An effective iterated local search algorithm for the distributed no-wait flowshop scheduling problem, Engineering Applications of Artificial Intelligence 120, 105921 (2023)

    M. Avci, An effective iterated local search algorithm for the distributed no-wait flowshop scheduling problem, Engineering Applications of Artificial Intelligence 120, 105921 (2023)

    work page 2023

  58. [58]

    Ishigaki and S

    A. Ishigaki and S. Takaki, Iterated local search algorithm for flexible job shop scheduling, in2017 6th IIAI Interna- tional Congress on Advanced Applied Informatics (IIAI- AAI) (2017) pp. 947–952

    work page 2017

  59. [59]

    L. F. Pérez Armas, S. Creemers, and S. Deleplanque, Solving the resource constrained project scheduling prob- lem with quantum annealing, Scientific Reports 14, 16784 (2024)

    work page 2024

  60. [60]

    Schworm, X

    P. Schworm, X. Wu, M. Klar, M. Glatt, and J. C. Aurich, Multi-objective quantum annealing approach for solving flexible job shop scheduling in manufacturing, Journal of Manufacturing Systems 72, 142 (2024)

    work page 2024

  61. [61]

    Carugno, M

    C. Carugno, M. Ferrari Dacrema, and P. Cremonesi, Evaluating the job shop scheduling problem on a d-wave quantum annealer, Scientific Reports12, 6539 (2022)

    work page 2022

  62. [62]

    Schworm, X

    P. Schworm, X. Wu, M. Glatt, and J. C. Aurich, Solv- ing flexible job shop scheduling problems in manufactur- ing with quantum annealing, Production Engineering17, 105 (2023)

    work page 2023

  63. [63]

    Kurowski, T

    K. Kurowski, T. Pecyna, M. Slysz, R. Różycki, G. Waligóra, and J. Węglarz, Application of quantum approximate optimization algorithm to job shop schedul- ing problem, European Journal of Operational Research 310, 518 (2023)

    work page 2023

  64. [64]

    L. K. Grover, A fast quantum mechanical algorithm for database search (1996), arXiv:quant-ph/9605043 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  65. [65]

    TensorNetwork: A Library for Physics and Machine Learning

    C. Roberts, A. Milsted, M. Ganahl, A. Zalcman, B. Fontaine, Y. Zou, J. Hidary, G. Vidal, and S. Le- ichenauer, Tensornetwork: A library for physics and ma- chine learning (2019), arXiv:1905.01330 [physics.comp- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  66. [66]

    G. Bai, Y. Yang, and G. Chiribella, Quantum compres- sion of tensor network states, New Journal of Physics22, 043015 (2020)

    work page 2020

  67. [67]

    Lubasch, J

    M. Lubasch, J. I. Cirac, and M.-C. Bañuls, Unifying pro- jected entangled pair state contractions, New Journal of Physics 16, 033014 (2014)

    work page 2014