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与工程领域广泛存在。近年来，针对不同设定（包括精确求解与近似求解），研究者开发了多种量子优化方法。本文围绕该研究领域，具有三个明确目标。首先，我们提出一种直观的方法，用于综合与分析离散（即基于整数）优化问题。该方法通过一种编码无关的离散量子中间表示（DQIR）来表达问题及相应的算法原语。与以往方法相比，这种紧凑表示通常能实现更高效的问题编译、编码选择的自动分析、更易解释性、更复杂的运行时程序以及更丰富的可编程性——我们通过若干示例予以证明。其次，我们开展了数值研究，比较了多种量子比特编码方案。结果揭示了若干初步趋势，有助于针对特定硬件、特定问题及算法选择编码方案。我们的研究涵盖了图着色问题、旅行商问题、工厂/机器调度问题、金融投资组合再平衡问题以及整数线性规划问题。第三，我们设计了基于低深度图衍生部分混合器（GDPM），支持最多16级量子变量，证明紧凑（二进制）编码比先前认知更适用于量子近似优化算法（QAOA）。我们预期，这套编程抽象与底层构建模块将有助于设计面向离散组合问题的量子算法。