Abstract
Many challenging scheduling, planning, and resource allocation problems come with real-world input data and hard problem constraints, and reduce to optimizing a cost function over a combinatorially defined feasible set, such as colorings of a graph. Toward tackling such problems with quantum computers using quantum approximate optimization algorithms, we present novel efficient quantum alternating operator ansatz (QAOA) constructions for optimization problems over proper colorings of chordal graphs. As our primary application, we consider the flight-gate assignment problem , where flights are assigned to airport gates as to minimize the total transit time of all passengers, and feasible assignments correspond to proper graph colorings of a conflict graph derived instancewise from the input data. We leverage ideas from classical algorithms and graph theory to show our constructions have the desirable properties of restricting quantum state evolution to the feasible subspace, and satisfying a particular reachability condition for most problem parameter regimes. Using classical preprocessing we show that we can always find and construct a suitable initial quantum (superposition) state efficiently. We show our constructions in detail, including explicit decompositions to a universal set of basic quantum gates, which we use to bound the required resource scaling as low-degree polynomials of the input parameters. In particular, we derive novel QAOA mixing operators and show that their implementation cost is commensurate with that of the QAOA phase operator for flight-gate assignment. A number of quantum circuit diagrams are included such that our constructions may be used as a template toward development and implementation of quantum gate-model approaches for a wider variety of potentially impactful real-world applications.
Highlights
Improving our ability to satisfactorily solve challenging realworld combinatorial optimization problems, in particular those related to operational planning and scheduling, is a broad and promising area for potential quantum advantages
For each subcircuit employed in our constructions, we show a circuit decomposition into basic quantum gates from which we derive estimates of the overall resources required
For each component of our ansatz, we showed explicit decompositions into basic quantum gates, with resulting resource estimates that scale as low-degree polynomials in the problem size
Summary
Improving our ability to satisfactorily solve challenging realworld combinatorial optimization problems, in particular those related to operational planning and scheduling, is a broad and promising area for potential quantum advantages. Often the set of feasible assignments for such problems corresponds to proper colorings of a derived problem graph, such that any scheduling conflicts are avoided, over which we seek to minimize a cost function that incorporates the problem input data. We show explicit constructions of QAOA circuits for optimization problems over the proper colorings of an important class of graphs, interval graphs, common in scheduling and assignment problems. The last contribution to the cost function is the total transfer passenger transit time ctrans(x) := ni jtαβ xiαx jβ i≤ j α≤β determined by the time tαβ it takes to travel between gates α and β, and the number of passengers ni j arriving from flight i and departing with flight j or vice versa.
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