Quantum Search Approaches to Sampling-Based Motion Planning

TL;DR

This paper introduces quantum algorithms for sampling-based motion planning, demonstrating potential quadratic speedups over classical methods in complex environments by leveraging quantum superpositions and amplitude amplification.

Contribution

It formulates quantum versions of existing motion planning algorithms, analyzes their probabilistic completeness, and provides numerical estimates and comparisons with classical algorithms.

Findings

Quadratic speedup demonstrated in 2D dense environments

Quantum algorithms create superpositions of paths and connections

Analysis of oracle call errors on algorithm performance

Abstract

In this paper, we present a novel formulation of traditional sampling-based motion planners as database-oracle structures that can be solved via quantum search algorithms. We consider two complementary scenarios: for simpler sparse environments, we formulate the Quantum Full Path Search Algorithm (q-FPS), which creates a superposition of full random path solutions, manipulates probability amplitudes with Quantum Amplitude Amplification (QAA), and quantum measures a single obstacle free full path solution. For dense unstructured environments, we formulate the Quantum Rapidly Exploring Random Tree algorithm, q-RRT, that creates quantum superpositions of possible parent-child connections, manipulates probability amplitudes with QAA, and quantum measures a single reachable state, which is added to a tree. As performance depends on the number of oracle calls and the probability of measuring good quantum states, we quantify how these errors factor into the probabilistic completeness properties of the algorithm. We then numerically estimate the expected number of database solutions to provide an approximation of the optimal number of oracle calls in the algorithm. We compare the q-RRT algorithm with a classical implementation and verify quadratic run-time speedup in the largest connected component of a 2D dense random lattice. We conclude by evaluating a proposed approach to limit the expected number of database solutions and thus limit the optimal number of oracle calls to a given number.