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  • Open Access

Estimates of Loss Function Concentration in Noisy Parametrized Quantum Circuits

Giulio Crognaletti1,2,3,*, Michele Grossi3,†, and Angelo Bassi1,2

  • *Contact author: giu.crognaletti@gmail.com
  • †Contact author: michele.grossi@cern.ch
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PRX Quantum 7, 020336 – Published 26 May, 2026

DOI: https://doi.org/10.1103/dw2h-ll6r

Abstract

Variational quantum computing offers a powerful framework with applications across diverse fields such as quantum chemistry, machine learning, and optimization. However, its scalability is hindered by the exponential concentration of the loss function, known as the barren plateau problem. While significant progress has been made and prior work has separately analyzed barren plateaus in unitary and noisy settings, their combined impact remains poorly understood, largely due to limitations in conventional Lie-algebraic approaches. In this work, we introduce an analytical framework based on non-negative matrix theory that enables the description of the variance in layered noisy quantum circuits with arbitrary noise channels. This approach enables the derivation of exact expressions in the deep-circuit regime, uncovering the complex interplay between unitary layers and noise. Notably, we identify a noise-induced absorption mechanism—a phenomenon absent in purely unitary dynamics—which provides new insight into how noise shapes circuit behavior. We further present a controlled convergence analysis, establishing general lower bounds on the variance of both deep and shallow circuits. This leads to a principled connection between noise resilience and the expressive capacity of parameterized quantum circuits, particularly under smart initialization strategies. Our theoretical results are supported by numerical simulations and illustrative applications.

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Popular Summary

Quantum computing is a promising approach that could one day revolutionize fields like chemistry, machine learning, and optimization. However, as parametrized quantum circuits become deeper, they often suffer from a phenomenon known as the barren plateaus vanishing gradient problem that makes training these circuits extremely difficult. While previous studies have explored this issue in either purely ideal (unitary) or noisy scenarios, the combined effect of both has remained elusive.

In this work, we develop a mathematical framework using tools from non-negative matrix theory to better understand how noise and quantum operations interact in deep circuits. Surprisingly, we discover a new effect noise-induced absorption that helps explain how noise can sometimes dominate circuit behavior in unexpected ways. Our analysis also shows when and how circuits can retain useful signal, even in the presence of noise, especially when using smart strategies to initialize them.

This study provides a deeper theoretical foundation for tackling barren plateaus in realistic quantum hardware and offers practical insights for designing more trainable quantum algorithms in the noisy, intermediate-scale quantum era.

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Outline

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PRX Quantum 7, 020336– Published 26 May, 2026
  • Received 21 May 2025
  • Revised 29 December 2025
  • Accepted 16 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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