Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits

• URL: http://arxiv.org/abs/2001.00550v3
• Date: Sat, 20 Mar 2021 18:55:39 GMT
• Title: Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Authors: M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, Patrick J. Coles
• Abstract summary: Variational quantum algorithms (VQAs) optimize the parameters $vectheta$ of a parametrized quantum circuit. We prove two results, assuming $V(vectheta)$ is an alternating layered ansatz composed of blocks forming local 2-designs. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.
• Score: 0.755972004983746
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Variational quantum algorithms (VQAs) optimize the parameters $\vec{\theta}$ of a parametrized quantum circuit $V(\vec{\theta})$ 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 $V(\vec{\theta})$ 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 $V(\vec{\theta})$ 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 $V(\vec{\theta})$ is $\mathcal{O}(\log n)$. 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.

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