Connecting ansatz expressibility to gradient magnitudes and barren plateaus

By Zoe Holmes, M. Cerezo, Patrick J. Coles. PRX Quantum, 2021

650 citations recorded by the source corpus.

Parameterized quantum circuits serve as ans\"{a}tze for solving variational problems and provide a flexible paradigm for programming near-term quantum computers. Ideally, such ans\"{a}tze should be highly expressive so that a close approximation of the desired solution can be accessed. On the other hand, the ansatz must also have sufficiently large gradients to allow for training. Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability. This is done by extending the well established barren plateau phenomenon, which holds for ans\"{a}tze that form exact 2-designs, to arbitrary ans\"{a}tze. Specifically, we calculate the variance in the cost gradient in terms of the expressibility of the ansatz, as measured by its distance from being a 2-design. Our resulting bounds indicate that highly expressive ans\"{a}tze exhibit flatter cost landscapes and therefore will be harder to train. Furthermore, we provide numerics illustrating the effect of expressiblity on gradient scalings, and we discuss the implications for designing strategies to avoid barren plateaus.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/fed98c8899e42cf86d914b7953ab373e950449b3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

Citations in the corpus

Challenges and opportunities in quantum machine learning (2022, background): “The barren plateau phenomenon was first studied in deep hardware-efficient QNNs [77], where they arise due to the high expressivity of the model [81].”
Barren plateaus in variational quantum computing (2024, positive): “In fact, equation (7) precludes any type of non-exponentially suppressed features in the landscape.”
Absence of Barren Plateaus in Quantum Convolutional Neural Networks (2020, background): “Similarly, when the two-qubit gates do not form 2-designs, then one can expect the variance of the gradient to be larger [64].”
Noise-induced barren plateaus in variational quantum algorithms (2020, background): “For example, it was proven that the gradient vanishes exponentially in n for randomly initialized, deep Hardware Efficient ansatzes [31, 32] and dissipative quantum neural networks [33], and also for shallow depth with global cost functions [34].”
Diagnosing Barren Plateaus with Tools from Quantum Optimal Control (2021, background): “In particular, one can derive an upper bound for the variance of partial derivatives in terms of the degree of expressibility on the invariant subspaces, in a spirit similar to that of [47].”