Cost function dependent barren plateaus in shallow parametrized quantum circuits

By M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, Patrick J. Coles. Nature Communications, 2021

1,279 citations recorded by the source corpus.

Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is O(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{O}}(\mathrm{log}\,n)$$\end{document}. Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation. Parametrised quantum circuits are a promising hybrid classical-quantum approach, but rigorous results on their effective capabilities are rare. Here, the authors explore the feasibility of training depending on the type of cost functions, showing that local ones are less prone to the barren plateau problem.

Subfields: Artificial Intelligence, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/ff744c5fd2628eea16654da81c5b1e5080e51d35

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Citations in the corpus

Quantum variational algorithms are swamped with traps (2022, neutral): “For one, VQAs tend to exhibit barren plataeus when they are deep; namely, gradients of deep variational quantum circuits vanish exponentially with the problem size in many settings 7,8,10 .”
Absence of Barren Plateaus in Quantum Convolutional Neural Networks (2020, methodology): “Finally, let us remark that as shown in [36], if W A and W B form independent 2 -designs, then (cid:104) ∂ µ C (cid:105) = 0 and FIG.”
Diagnosing Barren Plateaus with Tools from Quantum Optimal Control (2021, methodology): “For instance, if the cost function is global, the system can still exhibit barren plateaus even with trivial ansatzes that do not a have exponentially growing dimension of the DLA [49].”
Challenges and opportunities in quantum machine learning (2022, background): “Simply deﬁning a loss function based on a global observable (i.e., observables measuring all qubits) leads to barren plateaus even for shallow circuits with sharp priors [78], while local observables (those comparing quantum states at the single-qubit level) avoid this issue [78, 87].”
Variational Quantum Linear Solver (2019, background): “[38].”
Barren plateaus in variational quantum computing (2024, positive): “…do not form designs, one can instead assume that the local gates are sampled from a local group, thus mapping the variance evaluation to a Markov chain-like process which can be analysed analytically 48–60 , numerically via Monte Carlo 61 and tensor networks 62,63 , or studied via XZ calculus 64 .”