A Quantum Collocation Approach to One-Dimensional Boundary Value Problems with Coherent Amplitude Amplification
We propose a quantum collocation framework for approximating solutions of one-dimensional linear and nonlinear boundary value problems. The method formulates the search for admissible solutions as a residual-based quantum search over a discretized ansatz space, where candidate solutions are evaluated through residual conditions imposed at collocation points. A residual-threshold oracle is constructed that acts jointly on spatial and parameter registers. This joint oracle structure leads to amplification dynamics that decompose into a coherent superposition of spatially conditioned amplitude-amplification processes rather than a single global amplification mechanism. We derive the corresponding amplification geometry and show that the success probability is governed by a weighted combination of spatially dependent amplification angles. Furthermore, we prove that the reversible residual oracle can be implemented with gate complexity polynomial in the logarithm of the number of collocation points, while retaining the quadratic search acceleration associated with amplitude amplification in the parameter space. We analyze how the spatially dependent oracle structure influences the amplification dynamics and corresponding success probabilities. Furthermore, we investigate how discretization, ansatz expressivity, oracle tolerance, and finite-precision effects influence both approximation quality and amplification behavior. Numerical experiments validate the theoretical predictions and illustrate the resulting search dynamics across different discretization and precision regimes.