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Challenging combinatorial optimization problems are ubiquitous in science andengineering. Several quantum methods for optimization have recently beendeveloped, 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 andcorresponding algorithmic primitives are expressed using a discrete quantumintermediate representation (DQIR) that is encoding-independent. This compactrepresentation often allows for more efficient problem compilation, automatedanalyses of different encoding choices, easier interpretability, more complexruntime procedures, and richer programmability, as compared to previousapproaches, which we demonstrate with a number of examples. Second, we performnumerical studies comparing several qubit encodings; the results exhibit anumber of preliminary trends that help guide the choice of encoding for aparticular set of hardware and a particular problem and algorithm. Our studyincludes problems related to graph coloring, the traveling salesperson problem,factory/machine scheduling, financial portfolio rebalancing, and integer linearprogramming. Third, we design low-depth graph-derived partial mixers (GDPMs) upto 16-level quantum variables, demonstrating that compact (binary) encodingsare more amenable to QAOA than previously understood. We expect this toolkit ofprogramming abstractions and low-level building blocks to aid in designingquantum algorithms for discrete combinatorial problems.

Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems