Reducing circuit resources in Grover’s algorithm via constraint-aware initialization

Abstract

Abstract Grover’s search algorithm provides a quadratic speedup over classical brute-force methods for unstructured search problems in terms of query complexity and is widely used as a versatile subroutine in numerous quantum algorithms, including those for combinatorial problems with large search spaces. For such problems, it is natural to reduce the effective search space by incorporating problem constraints at the initialization step, which in Grover’s algorithm can be achieved by preparing structured initial states that encode constraint information. In this work, we present a systematic framework with a simple preprocessing procedure for constraint-aware initialization in Grover’s algorithm, focusing on problems with linear constraints. The proposed framework incorporates cardinality constraints and parity constraints extracted from more general linear constraints. While such structured initial states can reduce the number of oracle queries required to obtain a solution, their preparation incurs additional circuit-level costs. We therefore offer a conservative circuit-level resource analysis, showing that the resulting constraint-aware initialization can improve resource efficiency in terms of gate counts and circuit depth. The proposed framework is further illustrated numerically using the exact-cover problem as a representative application. Overall, our results indicate that this approach serves as a practical baseline for achieving more resource-efficient implementations of Grover’s algorithm compared to the standard uniform initialization.