  Motivated by recent progress of quantum technologies, we study a discretized
quantum adiabatic process for a one-dimensional free fermion system described
by a variational wave function, i.e., a parametrized quantum circuit. The wave
function is composed of $M$ layers of two elementary sets of time-evolution
operators, each set being decomposed into commutable local operators. The
evolution time of each time-evolution operator is treated as a variational
parameter so as to minimize the expectation value of the energy. We show that
the exact ground state is reached by applying the layers of time-evolution
operators as many as a quarter of the system size. This is the minimum number
$M_B$ of layers set by the limit of speed, i.e., the Lieb-Robinson bound, for
propagating quantum entanglement via the local time-evolution operators.
Quantities such as the energy $E$ and the entanglement entropy $S$ of the
optimized variational wave function with $M < M_B$ are independent of the
system size $L$ but fall into some universal functions of $M$. The development
of the entanglement in these ansatz is further manifested in the progressive
propagation of single-particle orbitals in the variational wave function. We
also find that the optimized variational parameters show a systematic structure
that provides the optimum scheduling function in the quantum adiabatic process.
We also investigate the imaginary-time evolution of this variational wave
function, where the causality relation is absent due to the non-unitarity of
the imaginary-time evolution operators, thus the norm of the wave function
being no longer conserved. We find that the convergence to the exact ground
state is exponentially fast, despite that the system is at the critical point,
suggesting that implementation of the non-unitary imaginary-time evolution in a
quantum circuit is highly promising to further shallow the circuit depth.
