[Paper Review] Quantum Algorithm to Solve a Maze: Converting the Maze Problem into a Search Problem

Niraj Kumar, Debabrata Goswami|arXiv (Cornell University)|Dec 15, 2013

Advanced Algebra and Logic3 references3 citations

TL;DR

This paper proposes a quantum algorithm to solve 2D perfect mazes by reformulating the problem as a quantum search task using Grover's algorithm. It encodes all possible paths as quantum states, uses a fitness operator to evaluate proximity to the goal, and iteratively applies Grover's search with a dynamic threshold oracle to converge toward the optimal path, achieving quadratic speedup over classical search methods.

ABSTRACT

We propose a different methodology towards approaching a Maze problem. We convert the problem into a Quantum Search Problem (QSP), and its solutions are sought for using the iterative Grover's Search Algorithm. Though the category of mazes we are looking at are of the NP complete class, we have redirected such a NP complete problem into a QSP. Our solution deals with two dimensional perfect mazes with no closed loops. We encode all possible individual paths from the starting point of the maze into a quantum register. A quantum fitness operator applied on the register encodes each individual with its fitness value. We propose an oracle design which marks all the individuals above a certain fitness value and use the Grover search algorithm to find one of the marked states. Iterating over this method, we approach towards the optimum solution.

Motivation & Objective

• To address the NP-complete maze-solving problem using quantum computation principles.
• To reformulate maze traversal as a quantum search problem (QSP) to leverage quantum advantages.
• To design a fitness-based oracle that marks paths closer to the goal, enabling iterative refinement.
• To implement an iterative Grover search with dynamic thresholding to approach the optimal path efficiently.
• To demonstrate that quantum algorithms can provide quadratic speedup for maze-solving compared to classical exhaustive search.

Proposed method

• Maze generation via the recursive backtracker algorithm to ensure no closed loops.
• Encoding all possible paths from start to end as superpositions in a quantum register using Hadamard gates.
• Defining a fitness function that quantifies how close a path is to the goal, stored in an entangled fitness register.
• Designing a quantum oracle that marks paths with fitness values above a dynamically updated threshold.
• Applying Grover's diffusion operator iteratively to amplify the amplitude of marked (high-fitness) states.
• Repeating the process with updated thresholds to progressively converge toward the optimal path.

Experimental results

Research questions

• RQ1 Can a quantum algorithm effectively solve a 2D perfect maze by transforming it into a quantum search problem?
• RQ2 How can a fitness function be defined to evaluate path quality in a quantum-encoding framework?
• RQ3 What is the role of the oracle in marking high-fitness paths and enabling iterative improvement?
• RQ4 To what extent does iterative Grover search with dynamic thresholding improve convergence to the optimal path?
• RQ5 What are the computational complexity bounds of this approach compared to classical exhaustive search?

Key findings

• The algorithm encodes all possible paths in a quantum superposition using a register of n qubits, resulting in a search space of size 4^n.
• The fitness operator evaluates path quality and entangles it with the path state, enabling measurement-based selection of high-quality paths.
• The oracle marks paths with fitness above a threshold, and Grover's algorithm amplifies the amplitude of these marked states in O(√k) steps, where k is the number of marked paths.
• Iterative application of Grover's search with updated thresholds allows convergence toward the optimal path in a finite number of steps, though not guaranteed in polynomial time due to Grover's quadratic speedup.
• The method achieves a quadratic speedup over classical exhaustive search, but does not provide exponential speedup, highlighting the current limitations of Grover's algorithm for NP-complete problems.
• Future exponential-speedup quantum search algorithms could potentially solve NP-complete problems like maze-solving in polynomial time.