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@INPROCEEDINGS{Sensebat:1037283,
      author       = {Sensebat, Orkun and Willsch, Dennis and Bode, Mathis and
                      Michielsen, Kristel},
      title        = {{Q}uantum {A}nnealers and {P}artial {D}ifferential
                      {E}quations: {T}he {G}ate-{B}ased {E}ncoding {A}pproach},
      reportid     = {FZJ-2025-00611},
      year         = {2024},
      abstract     = {Representing Partial Differential Equations (PDEs) on
                      Quantum Annealers (QAs) is not obvious. The first step of
                      solving a PDE on a QA is to discretize it. We used finite
                      differences. Next, a Quadratic Unconstrained Binary
                      Optimization (QUBO) model is obtained that is equivalent to
                      an Ising model. Finally, the QUBO coefficients are computed
                      using an encoding in terms of the qubits to represent the
                      variables. The conventional choice for this encoding is a
                      binary encoding. A challenge is that the accessible size of
                      systems is constrained due to the limited number of qubits.
                      In other words, it becomes impossible to achieve a non-zero
                      convergence probability. We addressed this challenge by
                      developing a novel encoding, the gate-based encoding (GBE),
                      to replace the conventionally used binary encoding. This
                      aims to overcome the problem of exponentially scaling
                      coefficients of binary encoding. GBE utilizes the structure
                      of the discretized operators to construct a circuit that
                      results in QUBO weights of comparative magnitude for each
                      bit in the final bit-string, thus overcoming the
                      exponentially rising weight given to the leading digits in
                      the conventionally used binary encoding. GBE allows to
                      successfully solve problems with larger encoding lenths for
                      all problem sizes and solvers, e.g., 21 vs. 17 for the
                      hybrid solver with our test setup. This is also emphasized
                      by the significantly higher success probability for all
                      encoding lengths compared to binary encoding. The cost of
                      GBE is the higher number of variables},
      month         = {May},
      date          = {2024-05-12},
      organization  = {ISC High Performance 2024, Hamburg
                       (Germany), 12 May 2024 - 16 May 2024},
      subtyp        = {After Call},
      cin          = {JSC},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {5111 - Domain-Specific Simulation $\&$ Data Life Cycle Labs
                      (SDLs) and Research Groups (POF4-511)},
      pid          = {G:(DE-HGF)POF4-5111},
      typ          = {PUB:(DE-HGF)24},
      url          = {https://juser.fz-juelich.de/record/1037283},
}