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

Abstract

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.

Qubit Efficient Quantum Algorithms for the Vehicle Routing Problem on Noisy Intermediate‐Scale Quantum Processors

“…[17] For typical industrial applications, the vehicle routing problem involves numerous vehicles and possible routes, and enumerating through all possible solutions to find the optimal one is computationally unfeasible. Recent advancements in quantum computing have allowed for tools to be developed allowing near-term quantum methods to find approximate solutions to QUBO problems, [18,19] such as Quantum annealing [20,21] and variational methods such as the Quantum Approximate Optimization Algorithm (QAOA) [22][23][24][25][26][27] or hardware efficient parameterized quantum circuits. [28][29][30][31][32] However, the limited capability of state-of-the-art hardware within the era of Noisy Intermediate Scale Quantum (NISQ) devices poses a challenge when applying quantum algorithms to problem sizes typical of applied industry problems.…”