  The Schwinger model is one of the simplest gauge theories. It is known that a
topological term of the model leads to the infamous sign problem in the
classical Monte Carlo method. In contrast to this, recently, quantum computing
in Hamiltonian formalism has gained attention. In this work, we estimate the
resources needed for quantum computers to compute physical quantities that are
challenging to compute on classical computers. Specifically, we propose an
efficient implementation of block-encoding of the Schwinger model Hamiltonian.
Considering the structure of the Hamiltonian, this block-encoding with a
normalization factor of $\mathcal{O}(N^3)$ can be implemented using
$\mathcal{O}(N+\log^2(N/\varepsilon))$ T gates. As an end-to-end application,
we compute the vacuum persistence amplitude. As a result, we found that for a
system size $N=100$ and an additive error $\varepsilon=0.01$, with an evolution
time $t$ and a lattice spacing a satisfying $t/2a=10$, the vacuum persistence
amplitude can be calculated using about $10^{13}$ T gates. Our results provide
insights into predictions about the performance of quantum computers in the
FTQC and early FTQC era, clarifying the challenges in solving meaningful
problems within a realistic timeframe.
