# Trajectory optimization using quantum computing

## TL;DR

This paper introduces a novel framework for trajectory optimization that discretizes both independent and dependent variables, enabling the use of classical and quantum search algorithms for efficient global solutions.

## Contribution

It presents a new discretization scheme that reduces computational costs and integrates quantum algorithms for trajectory optimization, demonstrating potential quadratic speed-up.

## Key findings

- Quantum search algorithms offer quadratic speed-up over classical methods.
- Discretization of both variables enables flexible and efficient optimization.
- Comparative analysis shows quantum methods are promising for complex trajectory problems.

## Abstract

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.