Statistical models of barren plateaus and anti-concentration of Pauli observables

TL;DR

This paper introduces statistical models for the main causes of barren plateaus in quantum neural networks, revealing that Pauli observables are highly anti-concentrated and independent in these regimes, which impacts optimization strategies.

Contribution

It presents simplified statistical models for sources of barren plateaus and analyzes the anti-concentration properties of Pauli observables within these models.

Findings

Non-local observables modeled by random Pauli operators lead to exponential barren plateaus.

Pauli observables in barren regimes are highly anti-concentrated and essentially independent.

These results suggest new approaches to quantum landscape optimization and warm-start strategies.

Abstract

We introduce statistical models for each of the three main sources of barren plateaus: non-locality of the observable, entanglement of the initial state, and circuit expressivity. For instance, non-local observables are modeled by random Pauli operators, which lead to barren plateaus with probability exponentially close to one. These models are complementary to the conventional deterministic ones, and often simpler to analyze. Using this framework, we show that in the barren plateau regime any two Pauli observables are anti-concentrated with high probability in the following sense. While each of the observables is localized in an exponentially small parameter subspace, these regions are essentially independent, so that their overlap is yet exponentially smaller than each subspace. This invites to rethink the structure of quantum landscapes with barren plateaus and approaches to their optimization, including warm-start strategies.