Fourier Analysis of Parameterized Quantum Circuits and the Barren Plateau Problem

Abstract

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.

Figures (4)

• Figure 1: Hardware-efficient embedding $E^L(x)$ for four qubits. Larger $L$ increases encoder expressivity.
• Figure 2: QNN used in the experiments (four qubits). The expectation is $f(x,\bm{\theta})=\mathop{\mathrm{Tr}}\nolimits[U(\bm{\theta})^\dagger E^L(x)^\dagger \rho\,E^L(x)U(\bm{\theta}) Z^{\otimes 4}]$.
• Figure 3: (Upper) Mean of squared Fourier coefficients versus $n$ for depths $L\in\{5,10,15,20,25,30,35,40,45,50\}$. The horizontal line shows Eq. (\ref{['fm']}). (Bottom) Variance across 300 random seeds.
• Figure 4: Representative Fourier coefficients at $L=15$ and $n\in\{2,4,6\}$. Coefficients decay rapidly with $n$, indicating exponential suppression of spectral power.