State-space reduction techniques exploiting specific constraints for quantum search Application to a specific job scheduling problem
Rodolphe Griset, Ioannis Lavdas, Jiri Guth Jarkovsky
TL;DR
The paper addresses scheduling problems with time-window and resource constraints where plain quantum search on an unstructured space is impractical due to exponential growth. It introduces a state-space reduction technique inspired by discrete quantum walks to construct a reduced initial superposition that satisfies scheduling constraints, enabling more efficient subsequent quantum search for remaining constraints. The work provides encoding schemes, two sets of oracles (time-window/spacing and resource constraints), and a fixed-point amplitude amplification framework, plus a quantum-walk-based method to generate feasible paths, yielding a reduced search space of size $N_{red}$ contrasted with the full space $N$. Numerical emulations demonstrate notable speedups in convergence to feasible solutions and quantify the substantial depth/qubit challenges for industrial-scale instances, guiding directions toward near-term emulation and longer-term fault-tolerant implementations.
Abstract
Quantum search has emerged as one of the most promising fields in quantum computing. State-of-the-art quantum search algorithms enable the search for specific elements in a distribution by monotonically increasing the density of these elements until reaching a high density. This kind of algorithms demonstrate a theoretical quadratic speed-up on the number of queries compared to classical search algorithms in unstructured spaces. Unfortunately, the major part of the existing literature applies quantum search to problems which size grows exponnentialy with the input size without exploiting any specific problem structure, rendering this kind of approach not exploitable in real industrial problems. In contrast, this work proposes exploiting specific constraints of scheduling problems to build an initial superposition of states with size almost quadraticaly increasing as a function of the problem size. This state space reduction, inspired by the quantum walk algorithm, constructs a state superposition corresponding to all paths in a state-graph embedding spacing constraints between jobs. Our numerical results on quantum emulators highlights the potential of state space reduction approach, which could lead to more efficient quantum search processes by focusing on a smaller, more relevant, solution space.