  We show the relationship between the Fourier coefficients and the barren
plateau problem emerging in parameterized quantum circuits. In particular, the
sum of squares of the Fourier coefficients is exponentially restricted
concerning the qubits under the barren plateau condition. Throughout theory and
numerical experiments, we introduce that this property leads to the vanishing
of a probability and an expectation formed by parameterized quantum circuits.
The traditional barren plateau problem requires the variance of gradient,
whereas our idea does not explicitly need a statistic. Therefore, it is not
required to specify the kind of initial probability distribution.
