Quantum Search Approaches to Sampling-Based Motion Planning

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.

Parallel Quantum Rapidly-Exploring Random Trees

“…Besides our previous q-RRT algorithm, quantum computing has not, to the best of our knowledge, been applied to sampling-based planning algorithms. A further overview on how quantum computing has been applied to motion planning and robotics can be found in [1].…”

“…In [1], we introduced how quantum computing methods can be applied to sampling-based motion planning in two ways, a full path database search and an RRT-based single-state database search q-RRT. The Quantum Rapidly-Exploring Random Trees algorithm, q-RRT, uses Quantum Amplitude Amplification (QAA) to search databases of possible reachable states.…”

“…The Quantum Rapidly-Exploring Random Trees algorithm, q-RRT, uses Quantum Amplitude Amplification (QAA) to search databases of possible reachable states. A focus of our work [1] was in estimating solution likelihood (so QAA could be performed an optimal number of times) through the use of random square lattice environments and numerical simulations. We chose this approach over quantum counting in an effort to keep oracle efficiency high.…”

“…With respect to robotics and motion planning, quantum algorithms have also been found to increase speed and efficiency. The heart of quantum advantage is derived from quantum parallelism and the ability to perform simultaneous computations on superpositions of states, which motivated our work in [1]. To aid in sampling-based motion planning, the key efficiency bottleneck is the search for dynamically-reachable, obstacle-free states to connect to the search tree.…”

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In this paper, we present the Parallel Quantum Rapidly-Exploring Random Tree (Pq-RRT) algorithm, a parallel version of the Quantum Rapidly-Exploring Random Trees (q-RRT) algorithm [1]. Parallel Quantum RRT is a parallel quantum algorithm formulation of a sampling-based motion planner that uses Quantum Amplitude Amplification to search databases of reachable states for addition to a tree.

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