Geometric Monte Carlo Post-Processing

This repository accompanies the article:

"Geometric Quantum Amplitude Estimation for Monte Carlo Post-Processing"

It contains a single, self-contained Python script that reproduces all numerical results and figures reported in the paper.

The purpose of this repository is reproducibility: running the provided script generates the data and plots used to demonstrate a reduction in measurement cost for a concrete Monte Carlo post-processing task.

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Problem description

We consider a Monte Carlo scenario model implemented as a quantum circuit.

• The circuit prepares a probability distribution over binary variables representing independent Bernoulli risk factors.
• A rare event is defined as a specific predicate on the sampled bitstrings (in this benchmark: all variables simultaneously equal to one).
• The goal is to estimate the probability p of this rare event.

This task is representative of Monte Carlo post-processing problems such as:

• tail-risk estimation,
• rare-event probability estimation,
• constraint-violation probability estimation.

The benchmark compares two estimation strategies applied to the same scenario generator:

1. Naive Monte Carlo sampling:

   • Sample the circuit output repeatedly.
   • Estimate p as the empirical frequency of the event.

2. Geometric amplitude-estimation-based post-processing:

   • Use amplitude amplification with several circuit depths.
   • Infer p from the geometric structure of the amplified probabilities using maximum-likelihood estimation.

The comparison is performed at fixed accuracy and confidence requirements.

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Benchmark criterion

For both methods we measure:

• Absolute estimation error |p_hat - p_true|
• Success probability: fraction of trials satisfying |p_hat - p_true| <= eps

The benchmark reports the minimal total number of shots required to achieve:

• Absolute error <= eps
• With success probability >= conf

In the results reported in the paper:

• eps = 1e-4
• conf = 0.95
• Each data point is averaged over 200 independent trials

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Repository contents

geometric_mc_postprocessing.py The main script. It:

• constructs the quantum circuits,
• runs both estimation methods,
• evaluates accuracy and success probability,
• generates publication-quality PNG figures.

LICENSE MIT license.

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Requirements

Python 3.9+ is recommended.

Install dependencies with:

pip install qiskit qiskit-aer numpy matplotlib

No additional packages are required.

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How to run

Simply execute:

python geometric_mc_postprocessing.py

The script will:

1. Run the Monte Carlo post-processing benchmark.
2. Generate two PNG files in the current directory:
   • mc_error_vs_shots.png
   • mc_success_vs_shots.png
3. Print the true probability p_true and the benchmark parameters.

The runtime depends on the shot budget and number of trials; on a typical workstation it completes within a few minutes.

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Output figures

mc_error_vs_shots.png Mean absolute error versus total number of shots for naive Monte Carlo sampling and the geometric estimator.

mc_success_vs_shots.png Probability of achieving the target absolute error as a function of the total number of shots. The 95% confidence threshold is indicated.

These figures are the ones used in the accompanying article.

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Scope and limitations

This repository demonstrates a post-processing advantage for probability estimation under the following conditions:

• The quantity of interest is a probability (or rare-event probability).
• The event can be implemented as a clean phase oracle.
• The amplitude amplification geometry is preserved.

The code does not claim a universal speedup for all quantum algorithms, nor does it replace general-purpose solvers. It demonstrates a concrete, reproducible advantage for a specific and practically relevant estimation task.

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Reproducibility

All numerical results and figures in the article were generated using this script without manual intervention.