Challenging combinatorial optimization problems are ubiquitous in science and engineering. Several quantum methods for optimization have recently been developed, in different settings including both exact and approximate solvers. Addressing this field of research, this manuscript has three distinct purposes. First, we present an intuitive method for synthesizing and analyzing discrete (i.e., integer-based) optimization problems, wherein the problem and corresponding algorithmic primitives are expressed using a discrete quantum intermediate representation (DQIR) that is encoding-independent. This compact representation often allows for more efficient problem compilation, automated analyses of different encoding choices, easier interpretability, more complex runtime procedures, and richer programmability, as compared to previous approaches, which we demonstrate with a number of examples. Second, we perform numerical studies comparing several qubit encodings; the results exhibit a number of preliminary trends that help guide the choice of encoding for a particular set of hardware and a particular problem and algorithm. Our study includes problems related to graph coloring, the traveling salesperson problem, factory/machine scheduling, financial portfolio rebalancing, and integer linear programming. Third, we design low-depth graph-derived partial mixers (GDPMs) up to 16-level quantum variables, demonstrating that compact (binary) encodings are more amenable to QAOA than previously understood. We expect this toolkit of programming abstractions and low-level building blocks to aid in designing quantum algorithms for discrete combinatorial problems.

翻译：具有挑战性的组合优化问题在科学和工程领域无处不在。近年来，针对不同求解场景（包括精确求解器和近似求解器），研究者已开发出多种量子优化方法。本文聚焦该研究领域，具有三个明确目的。首先，我们提出一种直观的离散（即基于整数）优化问题综合与分析方法，其中问题及相应算法原语采用编码无关的离散量子中间表示（DQIR）进行表达。相较此前方法，这种紧凑表示能够实现更高效的问题编译、编码选择的自动化分析、更易理解的解释性、更复杂的运行时流程以及更强的可编程性，我们通过多个实例对此加以验证。其次，我们对多种量子比特编码方案开展数值研究。结果表明若干初步趋势，有助于针对特定硬件、特定问题及算法选择最优编码方案。我们的研究涵盖图着色问题、旅行商问题、工厂/机器调度、金融投资组合再平衡及整数线性规划。第三，我们设计出低深度图衍生部分混合器（GDPM），支持高达16级量子变量，证明紧凑（二进制）编码对量子近似优化算法（QAOA）的适用性较先前理解更为显著。我们预期，这套编程抽象与底层构建模块的工具包将有助于设计面向离散组合问题的量子算法。