Spatial search by pseudo-Hermitian continuous-time quantum walks

We apply pseudo-Hermitian continuous-time quantum walks to the spatial search problem, derive exact analytical solutions for the resulting dynamics, and determine the first hitting time at which the target probability is maximal. In an idealized setting with a fixed coupling strength g, we show that the formal first-maximum time t = π/g is independent of the database size N. However, we emphasize that this does not imply a general complexity improvement: making t = π/g arbitrarily small requires a corresponding increase in the Hamiltonian matrix elements (energy scale), a fundamental resource trade-off. A fair assessment must also account for the measurement and postselection overhead required to extract the target probability and for the cost of implementing the evolution. Our results provide an analytically tractable non-Hermitian model that supports state preparation and parameter-controlled quantum dynamics.