离散优化量子算法中的编码权衡与设计工具包：着色、路由、调度及其他问题
Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems
AI总结：
本文提出编码无关的离散量子中间表示DQIR和低深度图导出部分混合器GDPM，通过数值实验比较多种量子比特编码，为离散组合优化问题的量子算法设计提供编码选择指导和编程工具包。
摘要：
具有挑战性的组合优化问题在科学与工程中无处不在。近期已发展出多种用于优化的量子方法，涵盖精确求解器和近似求解器等不同设置。针对这一研究领域，本文有三个不同的目的。第一，我们提出一种直观的方法来合成与分析离散（即基于整数的）优化问题，其中问题及相应的算法原语使用一种与编码无关的离散量子中间表示（DQIR）来表达。与先前方法相比，这种紧凑表示通常能够实现更高效的问题编译、对不同编码选择的自动化分析、更易解释性、更复杂的运行时程序以及更丰富的可编程性，我们通过若干示例对此进行了演示。第二，我们进行了数值研究，比较了几种量子比特编码；结果展现出若干初步趋势，有助于针对特定硬件集合以及特定问题和算法指导编码选择。我们的研究包括与图着色、旅行商问题、工厂/机器调度、金融投资组合再平衡以及整数线性规划相关的问题。第三，我们设计了低深度图导出部分混合器（GDPM），支持高达16能级的量子变量，证明紧凑（二进制）编码比先前理解的更适用于QAOA。我们期望这一由编程抽象和底层构建块组成的工具包能够帮助设计用于离散组合问题的量子算法。
英文摘要：
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.
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