Fourier Analysis of Parameterized Quantum Circuits and the Barren Plateau Problem

TL;DR

This paper explores how Fourier coefficients relate to the barren plateau problem in parameterized quantum circuits, revealing that their sum of squares diminishes exponentially with qubits, affecting gradient-based optimization.

Contribution

It introduces a novel Fourier analysis perspective on the barren plateau problem, showing that the sum of squared Fourier coefficients decreases exponentially, independent of initial distributions.

Findings

Sum of squares of Fourier coefficients diminishes exponentially with qubits.

The property leads to vanishing probabilities and expectations in quantum circuits.

The approach does not require explicit gradient variance or initial distribution assumptions.

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.