We present a framework wherein the trajectory optimization problem (or a problem involving calculus of variations) is formulated as a search problem in a discrete space. A distinctive feature of our work is the treatment of discretization of the optimization problem wherein we discretize not only independent variables (such as time) but also dependent variables. Our discretization scheme enables a reduction in computational cost through selection of coarse-grained states. It further facilitates the solution of the trajectory optimization problem via classical discrete search algorithms including deterministic and stochastic methods for obtaining a global optimum. This framework also allows us to efficiently use quantum computational algorithms for global trajectory optimization. We demonstrate that the discrete search problem can be solved by a variety of techniques including a deterministic exhaustive search in the physical space or the coefficient space, a randomized search algorithm, a quantum search algorithm or by employing a combination of randomized and quantum search algorithms depending on the nature of the problem. We illustrate our methods by solving some canonical problems in trajectory optimization. We also present a comparative study of the performances of different methods in solving our example problems. Finally, we make a case for using quantum search algorithms as they offer a quadratic speed-up in comparison to the traditional non-quantum algorithms.Keywords Trajectory optimization · calculus of variations · global optimization · quantum computation · randomized search algorithm · Brachistochrone problem Mathematics Subject Classification (2000) MSC 49M25 · 81P68 IntroductionThe goal of trajectory optimization is to find a path or trajectory that optimizes a given quantity of interest or any other objective function associated with a certain performance measure, under a set of given constraints on the dynamics of the system. Trajectory optimization problems appear naturally in many practical situations, especially in aerospace applications. Trajectory optimization problems are important,

No abstract

We give a short proof of Cayley's tree formula for counting the number of different labeled trees on n vertices. The following nonlinear recursive relation for the number of labeled trees on n vertices is deduced from a combinatorial argument,and then it is proved that T n = n n−2 , which gives yet another proof of the celebrated Cayley's tree formula.

Quantum state preparation involving a uniform superposition over a non-empty subset of n-qubit computational basis states is an important and challenging step in many quantum computation algorithms and applications. In this work, we address the problem of preparation of a uniform superposition state of thewhere M denotes the number of distinct states in the superposition state and 2 ≤ M ≤ 2 n . We show that the superposition state |Ψ ⟩ can be efficiently prepared with a gate complexity and circuit depth of only O(log 2 M) for all M. This demonstrates an exponential reduction in gate complexity in comparison to other existing approaches in the literature for the general case of this problem.Another advantage of the proposed approach is that it requires only n = ⌈log 2 M⌉ qubits. Furthermore, neither ancilla qubits nor any quantum gates with multiple controls are needed in our approach for creating the uniform superposition state |Ψ ⟩. It is also shown that a broad class of nonuniform superposition states that involve a mixture of uniform superposition states can also be efficiently created with the same circuit configuration that is used for creating the uniform superposition state |Ψ ⟩ described earlier, but with modified parameters.

No abstract