Challenge Submission: An Investigation of Barren Plateaus

[poig]

Closes #63

This pull request provides a complete, working solution to the QOSF Monthly Challenge on "Escaping the Barren Plateau." It implements all coding tasks in a robust, stable, and theoretically sound manner, designed to guide users through the discovery of all three primary types of barren plateaus.

Summary of Changes

This PR completes the notebook by implementing three core experimental tasks and one advanced challenge. The code is structured to be both educational and challenging, requiring the user to complete a central run_experiment function.

The core of the challenge is a controlled experiment that directly compares the three main scenarios on a level playing field. We measure the gradient variance vs. the number of qubits to demonstrate the nature of the training landscape.

• Task 1 (Trainable Baseline): Implements a shallow circuit with a local cost function. This establishes a baseline for a trainable landscape, which correctly shows a high, non-vanishing gradient variance.

• Task 2 (Depth-Induced Barren Plateau): Implements a deep circuit with a local cost function. This faithfully replicates the central result of McClean et al. (2018), demonstrating an exponential decay in variance.

• Task 3 (Cost-Induced Barren Plateau): Implements a shallow circuit with a global cost function. This replicates the findings of Cerezo et al. (2021), showing that a global observable induces a barren plateau even when the circuit is shallow.

• Task 4 Real Challenge: Algorithmic Mitigation with ADAPT-VQE: where need to replicate the highly effective, state-of-the-art mitigation strategy

An additional cell was added to tackle the advanced challenge of demonstrating NIBPs.

• Methodology: Takes the known "trainable" setup from Task 1 (shallow circuit, local cost) and injects a qml.DepolarizingChannel after each layer of the ansatz.

• Result: This successfully shows that noise alone can induce a barren plateau, causing the gradient variance to decay exponentially where it was previously stable.

Complete Summary and Analysis

The final markdown cells need to be filled out:

• The summary table with analysis of each mitigation strategy's performance and complexity.

• The written analysis section answers the guiding questions based on the results from the implemented experiments.