Many computational problems involve optimization over discrete variables with quadratic interactions. Known as discrete quadratic models (DQMs), these problems in general are NP-hard. Accordingly, there is increasing interest in encoding DQMs as quadratic unconstrained binary optimization (QUBO) models to allow their solution by quantum and quantum-inspired hardware with architectures and solution methods designed specifically for such problem types. However, converting DQMs to QUBO models often introduces invalid solutions to the solution space of the QUBO models. These solutions must be penalized by introducing appropriate constraints to the QUBO objective function that are weighted by a tunable penalty parameter to ensure that the global optimum is valid. However, selecting the strength of this parameter is non-trivial, given its influence on solution landscape structure. Here, we investigate the effects of choice of encoding and penalty strength on the structure of QUBO DQM solution landscapes and their optimization, focusing specifically on one-hot and domain-wall encodings.

翻译：许多计算问题涉及具有二次相互作用的离散变量优化。这些问题被称为离散二次模型（DQM），通常属于NP难问题。因此，将DQM编码为二次无约束二进制优化（QUBO）模型日益受到关注，以便利用为这类问题专门设计的架构和求解方法的量子及量子启发硬件进行求解。然而，将DQM转换为QUBO模型时，QUBO模型的解空间中常会引入无效解。必须通过在QUBO目标函数中引入由可调惩罚参数加权的适当约束来惩罚这些解，以确保全局最优解有效。然而，鉴于惩罚参数对解空间结构的影响，其强度的选择并非易事。本文针对独热编码与畴壁编码，重点探究了编码方式及惩罚强度对QUBO DQM解空间结构及其优化的影响。