Abstract
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 {mathcal{O}}(mathrm{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.
Highlights
Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C
While scaling results have been obtained for classical neural networks[45], very few such results exist for the trainability of parametrized quantum circuits, and more generally for quantum neural networks
Rigorous scaling results are urgently needed for VQAs, which many researchers believe will provide the path to quantum advantage with near-term quantum computers
Summary
Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. 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. VQAs employ a quantum computer to efficiently evaluate a cost function C, while a classical optimizer trains the parameters θ of a Parametrized Quantum. Pushing complexity onto classical computers, while only running short-depth quantum circuits, is an effective strategy for error mitigation on NISQ devices. As recent large-scale implementations for chemistry[7] and optimization[8] applications have shown, this ansatz leads to smaller errors due to hardware noise. One of the few known scaling results is that deep versions of randomly initialized hardware-efficient ansatzes lead to exponentially vanishing gradients[9]. Very little is known about the scaling of the gradient in such ansatzes for shallow depths, and it would be especially useful to have a converse bound that guarantees non-exponentially vanishing gradients for certain depths
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