We derive an Itô-type formula for one-dimensional continuous-time quantum random walks (CTQRWs). Modeling the walk through a quantum stochastic differential equation driven by creation, annihilation, and gauge processes, we establish a quantum Itô formula for sufficiently smooth functions of the position operator. The result extends the classical Itô formula to a noncommutative framework and provides a useful analytical tool for studying moments, generators, and asymptotic behavior of quantum random walks. As an application, we compute explicit evolution equations for polynomial observables and discuss the connection with the associated Lindblad generator.

Keywords

Quantum random walks; quantum stochastic calculus; Itô formula; Hudson–Parthasarathy equation; noncommutative probability.