Abstract
The next few years will be exciting as prototype universal quantum processors emerge, enabling the implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation and which have the potential to significantly expand the breadth of applications for which quantum computers have an established advantage. A leading candidate is Farhi et al.’s quantum approximate optimization algorithm, which alternates between applying a cost function based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the quantum alternating operator ansatz, is the consideration of general parameterized families of unitaries rather than only those corresponding to the time evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach, in the spirit of the quantum approximate optimization algorithm, to a wide variety of approximate optimization, exact optimization, and sampling problems. In addition to introducing the quantum alternating operator ansatz, we lay out design criteria for mixing operators, detail mappings for eight problems, and provide a compendium with brief descriptions of mappings for a diverse array of problems.
Highlights
Today, challenging computational problems arising in the practical world are frequently tackled by heuristic algorithms
A key question is: “What are good quantum heuristic algorithms to try?” A leading candidate is Farhi et al.’s quantum approximate optimization algorithm, a quantum gate-model meta-heuristic which alternates between applying unitaries drawn from two families, a cost function based unitary family UP (γ) = e−iγH f and a family of mixing unitaries UM ( β) = e−iβHB, for some fixed cost function based Hamiltonian H f and some fixed mixing Hamiltonian HB
One example might be for exact optimization of the problems considered here; for these problems, the worst case algorithmic complexity is exponential, but it is worth exploring whether quantum alternating operator ansatz (QAOA) might outperform classical heuristics in expanding the tractable range for some problems
Summary
Today, challenging computational problems arising in the practical world are frequently tackled by heuristic algorithms. As we shall see, expanding the design space of families of one-parameter mixing operators allowed enables the ansatz to support more efficiently implementable mixers than was possible in the original framework. After describing the framework for the ansatz, we show explicit mappings to quantum circuits and resource estimates for a diverse set of problems, designing phase separation and mixing operators appropriate for each problem. The general strategy is to incorporate the hard constraints as penalty terms in the cost function and convert the cost function to a cost Hamiltonian [2,3,4,5] This approach means that the algorithm must search a much larger space than if the evolution were confined to feasible configurations, making the search less efficient than if it were possible to constrain the evolution. For the benefit of the reader, we include a glossary of important terminology used in the paper, and a review of some useful elementary quantum operations as Appendices B and C, respectively
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