In this work we develop tools to address combinatorial optimization problems with a cardinality constraint, in which only a subset of variables end up having nonzero values. Firstly, we introduce a new heuristic pruning method that iteratively discards variables through a hybrid quantum–classical optimization step. Secondly, we analyse the use of soft constraints in the form of ‘chemical potentials’ to control the number of non-zero variables. We illustrate the power of both techniques using the problem of index tracking, which aims to mimicking the performance of a financial index with a balanced subset of assets. We also compare the performance of different state-of-the-art quantum variational optimization algorithms in our pruning method.

在这项工作中，我们开发了工具来解决具有基数约束的组合优化问题，其中只有一部分变量最终具有非零值。首先，我们引入了一种新的启发式剪枝方法，该方法通过混合量子经典优化步骤迭代地丢弃变量。其次，我们分析了使用“化学势”形式的软约束来控制非零变量的数量。我们使用指数跟踪问题来说明这两种技术的威力，该问题旨在模拟具有平衡资产子集的金融指数的表现。我们还在我们的剪枝方法中比较了不同最先进的量子变分优化算法的性能。