Structural $f$-divergence: Tight universal bounds for cost function moments and gradients in parameterized quantum circuits

TL;DR

This paper introduces the structural f-divergence to analyze and bound the gradients and moments in parameterized quantum circuits, addressing the barren plateau problem in quantum machine learning.

Contribution

It develops a new divergence measure for probability distributions on quantum circuit parameters and derives bounds that inform the design of measures to mitigate barren plateaus.

Findings

Established bounds on gradient discrepancies and cost moments using the new divergence.

Derived necessary conditions for probability measures to avoid barren plateaus.

Provided sufficient conditions to suppress noise effects in quantum circuits.

Abstract

The barren plateau phenomenon, in which cost-function gradients of variational quantum algorithms vanish exponentially, remains a central obstacle for near-term quantum computing. Existing analyses typically depend on t-design or Haar-random assumptions and bound quantities at the level of unitary distributions, offering limited insight for designing probability measures on the parameter space of parameterized quantum circuits. In this paper, we introduce the structural $f$-divergence, a symmetric $f$-divergence-based measure between probability distributions on the parameter space. We establish analytically trade-off inequalities that bound the discrepancies in the expected gradient magnitude and in the cost-function moments between a distribution on PQC and a reference distribution; equality is attained by a minimal one-qubit, one-layer ansatz. As applications, we derive necessary conditions on probability measures for avoiding BPs and cost concentration, and sufficient conditions that suppress noise-induced deviations.