Abstract
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.
Highlights
While quantum hardware is rapidly reaching the stage at which it can outperform classical supercomputers [1], we remain in the noisy intermediate-scale quantum (NISQ) era, in which the available devices are relatively small and prone to errors [2]
Variational quantum algorithms (VQAs) have gathered attention as a computational strategy that is well suited to the constraints imposed by NISQ devices [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]
Central to the success of VQAs is the construction of a parametrized quantum circuit, which serves as an ansatz with which to explore the space of solutions to the target problem
Summary
While quantum hardware is rapidly reaching the stage at which it can outperform classical supercomputers [1], we remain in the noisy intermediate-scale quantum (NISQ) era, in which the available devices are relatively small and prone to errors [2]. We demonstrate that this is the case by analytically relating the trainability of an ansatz to its expressibility This is done by extending the barren plateau phenomenon introduced in Ref. Highly expressive ansatze, which are generically used for many problems, as they can access a much larger space [Fig. 1(c)], are shown to lead to small gradients and can have trainability issues. Since our analytic bounds are upper bounds, they leave open the questions of how reducing the expressibility of an ansatz changes the cost landscape and how reducing the expressibility can be used to avoid the barren plateau phenomenon To address these questions, we provide extensive numerics studying the effect that tuning the expressibility of an ansatz may have on the scaling of gradient magnitudes. We find that strongly correlating parameters [37] and/or initializing close to the solution (and restricting the ansatz to explore the region close to the initialization [38]) are the most effective approaches to avoid exponentially vanishing cost gradients
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