  Optimal realizations of quantum technology tasks lead to the necessity of a
detailed analytical study of the behavior of a $d$-level quantum system (qudit)
under a time-dependent Hamiltonian. In the present article, we introduce a new
general formalism describing the unitary evolution of a qudit $(d\geq2)$ in
terms of the Bloch-like vector space and specify how in a general case this
formalism is related to finding time-dependent parameters in the exponential
representation of the evolution operator under an arbitrary time-dependent
Hamiltonian. Applying this new general formalism to a qubit case $(d=2)$, we
specify the unitary evolution of a qubit via the evolution of a unit vector in
$\mathbb{R}^{4}$ and this allows us to derive the precise analytical expression
of the qubit unitary evolution operator for a wide class of nonstationary
Hamiltonians. This new analytical expression includes the qubit solutions known
in the literature only as particular cases.
