Barren plateaus in quantum neural network training landscapes
Open accessAbstract
Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied.
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The expectation value of the gradient of the objective function corresponding to randomly initialized PQCs decays exponentially to zero as a function of the number of qubits.
“It was recently shown that the expectation value of the gradient of the objective function corresponding to randomly initialized PQCs (RPQC) decays exponentially to zero as a function of the number of qubits (McClean et al. , 2018).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as McClean et al., 2018
The Methods state that, under the paper’s RPQC assumptions, the expected gradient is exactly zero; it is the variance that decays exponentially with the number of qubits. The Numerical simulations section reports exponential decay in the measured expected value and its spread, but that does not establish an exponentially decaying theoretical expectation. (LLM, conf=0.88)
Experimental — accuracy not yet measuredThese quantum machine learning approaches need careful consideration of subtle issues that do not appear in classical algorithms and are strongly intertwined with the underlying hardware.
“Regardless of the conditions of their advantages, these approaches need careful consideration of subtle issues (see, e.g., Refs.[112-115]) that do not appear in classical algorithms and are, in particular, strongly intertwined with the underlying hardware.”
From Neutral atom quantum computing hardware: performance and end-user perspective (2023) · cites this paper as [115]
Variational-state quantum metrology variationally searches for the optimal probe state.
“Variational-state quantum metrology variationally searches for the optimal probe state [190-193].”
From Variational quantum algorithms (2021) · cites this paper as [192]
The Introduction describes optimizing parameterized quantum circuits, but discusses applications in simulation, optimization, machine learning, and error correction—not metrology. Searches for variational-state quantum metrology and optimal probe states found no passage explaining whether that method searches for an optimal probe state. (LLM, conf=0.73)
Experimental — accuracy not yet measuredTwo of the new entangling blocks are designed to implement number-conserving operators, more suitable for simulations of molecular systems, and can be implemented natively on superconducting processors with transmon qubits.
“Two of these new blocks are designed to implement number-conserving operators, more suitable for simulations of molecular systems, and can be implemented natively on superconducting processor with transmon qubits 340-342 .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [341]
When training is carried out classically, it may become hard due to Barren plateaus, especially for large data sets.
“However, the training phase, when carried out classically, may itself become hard due to the emergence of so-called Barren plateaus, especially for large data sets.[230,231]”
From Quantum Computing for Molecular Biology** (2023) · cites this paper as [230]
For standard binary encoding, the quantum model for a single input feature corresponds to a multi-dimensional Fourier series in the angles, with a frequency spectrum size of at most n.
“Our investigation here states that the quantum model for a single input feature corresponds to a multi-dimensional Fourier series in the angles, with a frequency spectrum size of at most n .”
From Effect of data encoding on the expressive power of variational quantum-machine-learning models (2021) · cites this paper as [4]
A generic expectation value f(θ) is a linear combination of different trigonometric functions weighted by 3^m scalar coefficients.
“Therefore, a generic expectation value f ( θ ) is a linear combination of different trigonometric functions weighted by 3 m scalar coefficients, a fact which was already noticed in Ref. [20].”
From Estimating the gradient and higher-order derivatives on quantum hardware (2021) · cites this paper as [20]
The expressiveness of the L-VQE circuit makes the parameters hard to optimize, both due to their high number and due to gradients vanishing as the number of qubits grows in some cases.
“At the same time, the expressiveness of L-VQE circuit makes the parameters hard to optimize, both due to their high number and due to the gradients vanishing as the number of qubits grows in some cases44,45.”
From Constrained quantum optimization for extractive summarization on a trapped-ion quantum computer (2022) · cites this paper as [44]
Random ansätze tend to lead to vanishing gradients when the number of qubits increases, meaning that it will be impossible to find a physically meaningful energy minimum.
“While hardware-efficient ansätze are attractive due to their ease of implementation in current devices, recent work by McClean et al. has provided analytical and numerical evidence that random ansätze tend to lead to vanishing gradients when the number of qubits increases [174], meaning that it will be impossible to find a physically meaningful energy minimum.”
From The prospects of quantum computing in computational molecular biology (2021) · cites this paper as [174]
The success of the VQE algorithm largely depends on the choice of ansatz, which determines the general form of the state.
“the success of the algorithm largely depends on this choice which determines the general form of the state”
From Quantum computing algorithms: getting closer to critical problems in computational biology (2022) · cites this paper as [103]
The gradient of the expectation value decays exponentially as a function of the number of qubits if the trial state is chosen randomly.
“the gradient of the expectation value decays exponentially as a function of the number of qubits if the trial state is chosen randomly [103]”
From Quantum computing algorithms: getting closer to critical problems in computational biology (2022) · cites this paper as [103]
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Quantum Computing for Molecular Biology**
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“Indeed, the success of the algorithm largely depends on this choice which determines the general form of the state, as it can be demonstrated that the gradient of the expectation value decays exponentially as a function of the number of qubits if the trial state is chosen randomly [103] (a vanishing gradient is proble…”
BackgroundQuantum variational optimization: The role of entanglement and problem hardness
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“https://doi.org/10.1038/s43588-026-01007-8 Furthermore, research indicates that the presence of barren pla-teaus is closely related to the two-design characteristics of circuits 48 and can be assessed through the gradient variance of the loss function 43 .”
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“Moreover, a prevalent issue in VQAs is the emergence of barren plateaus [26], [27].”
- Authors
- McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R., & Neven, H.
- Affiliations
- Google (United States) (US)
- Corresponding author
- McClean, Jarrod R.
- Boixo, Sergio
- Smelyanskiy, Vadim N.
- Venue
- Nature Communications
- Published
- 2018-11-16
- Volume
- 9
- Issue
- 1
- Pages
- 4812-4812
- ISSN
- 2041-1723 (ISSN-L)
- PMC
- PMC6240101
- OpenAlex
- W2794444783
- Cited by
- 2,329
- References
- 50
- Citations by year
- 2026: 551 · 2025: 494 · 2024: 365 · 2023: 330 · 2022: 256 · 2021: 218 · 2020: 76 · 2019: 35 · 2018: 4
- FWCI
- 46.92
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Locations
- Nature Communications (publishedVersion, cc-by)
- Europe PMC (PubMed Central) (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- barren plateaus, quantum neural networks, hybrid quantum-classical algorithms, variational quantum circuits, random circuits, gradient estimation, noisy intermediate-scale quantum computing, quantum machine learning
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