% IMPORTANT: The following is UTF-8 encoded.  This means that in the presence
% of non-ASCII characters, it will not work with BibTeX 0.99 or older.
% Instead, you should use an up-to-date BibTeX implementation like “bibtex8” or
% “biber”.

@ARTICLE{MontanezBarrera:1025138,
      author       = {Montanez-Barrera, Jhon Alejandro and Willsch, Dennis and
                      Maldonado-Romo, A. and Michielsen, Kristel},
      title        = {{U}nbalanced penalization: a new approach to encode
                      inequality constraints of combinatorial problems for quantum
                      optimization algorithms},
      journal      = {Quantum science and technology},
      volume       = {9},
      number       = {2},
      issn         = {2058-9565},
      address      = {Philadelphia, PA},
      publisher    = {IOP Publishing},
      reportid     = {FZJ-2024-02715},
      pages        = {025022 -},
      year         = {2024},
      abstract     = {Solving combinatorial optimization problems of the kind
                      that can be codified by quadratic unconstrained binary
                      optimization (QUBO) is a promising application of quantum
                      computation. Some problems of this class suitable for
                      practical applications such as the traveling salesman
                      problem (TSP), the bin packing problem (BPP), or the
                      knapsack problem (KP) have inequality constraints that
                      require a particular cost function encoding. The common
                      approach is the use of slack variables to represent the
                      inequality constraints in the cost function. However, the
                      use of slack variables considerably increases the number of
                      qubits and operations required to solve these problems using
                      quantum devices. In this work, we present an alternative
                      method that does not require extra slack variables and
                      consists of using an unbalanced penalization function to
                      represent the inequality constraints in the QUBO. This
                      function is characterized by larger penalization when the
                      inequality constraint is not achieved than when it is. We
                      evaluate our approach on the TSP, BPP, and KP, successfully
                      encoding the optimal solution of the original optimization
                      problem near the ground state cost Hamiltonian.
                      Additionally, we employ D-Wave Advantage and D-Wave hybrid
                      solvers to solve the BPP, surpassing the performance of the
                      slack variables approach by achieving solutions for up to 29
                      items, whereas the slack variables approach only handles up
                      to 11 items. This new approach can be used to solve
                      combinatorial problems with inequality constraints with a
                      reduced number of resources compared to the slack variables
                      approach using quantum annealing or variational quantum
                      algorithms.},
      cin          = {JSC},
      ddc          = {530},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {5111 - Domain-Specific Simulation $\&$ Data Life Cycle Labs
                      (SDLs) and Research Groups (POF4-511) / BMBF 13N16149 -
                      QSolid (BMBF-13N16149)},
      pid          = {G:(DE-HGF)POF4-5111 / G:(DE-Juel1)BMBF-13N16149},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:001195462400001},
      doi          = {10.1088/2058-9565/ad35e4},
      url          = {https://juser.fz-juelich.de/record/1025138},
}