Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and limit sets
1Department of Physics and Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, CA 90089, USA
2Departments of Electrical Engineering, Chemistry, Physics & Astronomy, and Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, CA 90089, USA
| Published: | 2025-01-30, volume 9, page 1617 |
| Editor: | Kunal Sharma |
| Eprint: | arXiv:2402.08721v6 |
| Doi: | https://doi.org/10.22331/q-2025-01-30-1617 |
| Citation: | Quantum 9, 1617 (2025). |
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Abstract
Variational quantum algorithms (VQAs) hold much promise but face the challenge of exponentially small gradients. Unmitigated, this barren plateau (BP) phenomenon leads to an exponential training overhead for VQAs. Perhaps the most pernicious are noise-induced barren plateaus (NIBPs), a type of unavoidable BP arising from open system effects, which have so far been shown to exist for unital noise maps. Here, we generalize the study of NIBPs to more general completely positive, trace-preserving maps, investigating the existence of NIBPs in the unital case and a class of non-unital maps we call Hilbert-Schmidt (HS)-contractive. The latter includes amplitude damping. We identify the associated phenomenon of noise-induced limit sets (NILS) of the VQA cost function and prove its existence for both unital and HS-contractive non-unital noise maps. Along the way, we extend the parameter shift rule of VQAs to the noisy setting. We provide rigorous bounds in terms of the relevant variables that give rise to NIBPs and NILSs, along with numerical simulations of the depolarizing and amplitude-damping maps that illustrate our analytical results.
Featured image: We study variational quantum algorithms (VQA) under unital and non-unital noise. We show that while unital noise always induces noise-induced barren plateaus (NIBPs), non-unital noise gives rise to a different phenomenon we term noise-induced limit sets (NILS).
Popular summary
We explore NIBPs beyond unital noise, investigating a broad class of noise maps, including a type we call Hilbert-Schmidt (HS)-contractive non-unital maps. These maps encompass physically realistic noise models such as amplitude damping. Our study provides key insights into how noise fundamentally impacts the trainability of VQAs. Specifically, we identify a behavior we term "noise-induced limit sets" (NILS), where noise pushes the cost function toward a range of values rather than a single value, disrupting training in unexpected ways.
Our findings are supported by theory and numerical simulations, shedding light on the interplay between noise, circuit depth, and trainability. We find that while unital noise always induces NIBPs, HS-contractive non-unital noise does not necessarily lead to barren plateaus, suggesting that some noise types may be less detrimental to VQAs than previously thought.
These results enhance our theoretical understanding of noise in quantum algorithms and point toward new directions for designing noise-resilient quantum architectures. By highlighting the critical role of noise type in determining the feasibility of VQA training, our work illuminates pathways for optimizing quantum algorithms in the noisy intermediate-scale quantum era and beyond.
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