  Theoretical and applied studies of quantum walks are abundant in quantum
science and technology thanks to their relative simplicity and versatility.
Here we derive closed-form expressions for the probability distribution of
quantum walks on a line. The most general two-state coin operator and the most
general (pure) initial state are considered in the derivation. The general coin
operator includes the common choices of Hadamard, Grover, and Fourier coins.
The method of Fibonacci-Horner basis for the power decomposition of a matrix is
employed in the analysis. Moreover, we also consider mixed initial states and
derive closed-form expression for the probability distribution of the Quantum
walk on a line. To prove the accuracy of our derivations, we retrieve the
simulated probability distribution of Hadamard walk on a line using our
closed-form expressions. With a broader perspective in mind, we argue that our
approach has the potential to serve as a helpful mathematical tool in obtaining
precise analytical expressions for the time evolution of qubit-based systems in
a general context.
