  We propose an efficient circuit structure of variational quantum circuit
\textit{Ans\"{a}tze} used for the variational quantum eigensolver (VQE)
algorithm in calculating gapped topological phases on the currently feasible
noisy intermediate-scale quantum computers. An efficient circuit
\textit{Ansatz} should include two layers: the initialization layer and the
variational layer. In the initialization layer, a fixed depth circuit state
with a compatible entanglement structure to the target topological phase is
constructed. The circuit state is further adjusted subsequently to capture the
details of the local correlations, which is dictated with the Hamiltonian, in
the parametrized variational layer. Based on this strategy, we design a circuit
\textit{Ansatz} to investigate the symmetry-protected topological Haldane phase
in a \textit{non-exactly} solvable alternating spin-$1/2$ Heisenberg chain by
VQE calculations. Main characterizations of the Haldane phase, including the
long-ranged string order, the four-fold nearly degenerate ground states
associated with four different localized edge mode patterns for the system with
open boundaries, and the two-fold degeneracy of the entanglement spectrum, are
all observed for the optimized shallow circuit state with only one depth
variational layer both in numerical simulations and on real quantum computers.
We further demonstrate that the computational capacity (i.e., expressibility)
of this quantum circuit \textit{Ansatz} is determined not by the system size
but only by the intrinsic correlation length of the system, thus implying that
the scalable VQE calculation is possible.
