Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and limit sets

TL;DR

This paper investigates how various noise types, including non-unital noise like amplitude damping, induce barren plateaus and limit sets in variational quantum algorithms, extending understanding beyond unital noise effects.

Contribution

It generalizes the study of noise-induced barren plateaus to non-unital maps, introduces noise-induced limit sets, and extends the parameter shift rule to noisy quantum settings.

Findings

NIBPs exist for a broader class of noise maps including non-unital ones.

Noise-induced limit sets (NILS) are proven to occur under certain noise conditions.

Numerical simulations confirm the analytical bounds for depolarizing and amplitude damping noise.

Abstract

Variational quantum algorithms (VQAs) hold much promise but face the challenge of exponentially small gradients. Unmitigated, this barren plateau (BP) phenomenon leads to an exponential training overhead for VQAs. Perhaps the most pernicious are noise-induced barren plateaus (NIBPs), a type of unavoidable BP arising from open system effects, which have so far been shown to exist for unital noise maps. Here, we generalize the study of NIBPs to more general completely positive, trace-preserving maps, investigating the existence of NIBPs in the unital case and a class of non-unital maps we call Hilbert-Schmidt (HS)-contractive. The latter includes amplitude damping. We identify the associated phenomenon of noise-induced limit sets (NILS) of the VQA cost function and prove its existence for both unital and HS-contractive non-unital noise maps. Along the way, we extend the parameter shift rule of VQAs to the noisy setting. We provide rigorous bounds in terms of the relevant variables that give rise to NIBPs and NILSs, along with numerical simulations of the depolarizing and amplitude-damping maps that illustrate our analytical results.