Quantum-Enhanced Signal Processing for Brain-Computer Interfaces

This project demonstrates the integration of quantum and classical signal processing techniques for brain-computer interface (BCI) applications. It includes modules for generating and simulating quantum Fourier transforms (QFT) and for implementing both classical and quantum-enhanced Kalman filters. These tools explore how quantum algorithms could eventually enhance the processing and tracking of neural signals.

File Structure

• project/
  • qfft_module.py
  • qkalman_module.py
  • QFT_Notebook.ipynb
  • qKalmanFilter_Notebook.ipynb
  • README.md

Module Descriptions

qfft_module.py

• Purpose:
  Implements functions to:

  • Generate a Quantum Fourier Transform (QFT) circuit using Qiskit.
  • Compute a classical Fast Fourier Transform (FFT) on simulated EEG-like signals.
  • Compare QFT simulation outcomes with classical FFT results.

• Key Functions:

  • generate_qft_circuit(n): Creates an n-qubit QFT circuit with measurements.
  • classical_fft(signal): Computes the FFT of a signal.
  • simulate_qft_circuit(circuit, shots): Simulates a given QFT circuit using Qiskit’s Aer simulator.

qkalman_module.py

• Purpose:
  Provides implementations for:

  • A classical Kalman filter for tracking neural signal changes.
  • A quantum-enhanced Kalman filter where the only difference is in the matrix inversion step.

• Key Functions:

  • classical_kalman_filter(zs, A, H, Q, R, x0, P0): Runs the standard Kalman filter.
  • quantum_matrix_inversion(M, method="classical"): By default, uses np.linalg.inv(M) but can be configured to use a prototype HHL-based inversion (quantum_matrix_inversion_hhl) in the future.
  • quantum_kalman_filter(zs, A, H, Q, R, x0, P0, inversion_method="classical"): Runs the Kalman filter using the selected inversion method.

Notebook Descriptions

QFT_Notebook.ipynb

• Overview:
  Demonstrates how to:

  • Generate and visualize a QFT circuit.
  • Simulate the QFT circuit using Qiskit's Aer simulator.
  • Compute and compare a classical FFT on simulated EEG-like data.

• Highlights:

  • Visualizations of the QFT circuit and FFT plots.
  • Side-by-side comparison of quantum and classical Fourier transform methods.

KalmanFilter_Notebook.ipynb

• Overview:
  Illustrates the tracking of complex, realistic EEG-like signals using:

  • A classical Kalman filter.
  • A quantum-enhanced Kalman filter (with a placeholder for quantum matrix inversion).

• Highlights:

  • Simulation of realistic EEG amplitude envelopes featuring baseline modulation and transient spikes.
  • Visual comparisons of the true signal, noisy measurements, and estimates from both filters.
  • Performance evaluations using metrics such as MSE, RMSE, MAE, and Pearson correlation.
  • Additional plots (time-series, scatter plots, and bar charts) to assess and compare filter performance.

Requirements & Installation

This project requires:

• Python 3.9+
• Qiskit and Qiskit-Aer (for quantum circuit simulation)
• NumPy, SciPy, Matplotlib, and Pandas
• pylatexenc (for LaTeX rendering in Qiskit circuits)

You can install the necessary dependencies using pip or conda:

pip install qiskit qiskit-aer numpy scipy matplotlib pandas pylatexenc

conda install -c conda-forge qiskit qiskit-aer numpy scipy matplotlib pandas pylatexenc

How to Run

Place all files in the same directory. Run the Notebooks: Open QFT_Notebook.ipynb in Jupyter Notebook or JupyterLab and execute the cells sequentially to view the QFT and FFT demonstrations. Open KalmanFilter_Notebook.ipynb to simulate EEG signals and apply both the classical and quantum-enhanced Kalman filters. Remember to reload the modules (qfft_module.py and qkalman_module.py) if you make changes.

Notes

Quantum-Enhanced Kalman Filter:

Currently, the quantum-enhanced Kalman filter uses a placeholder for the quantum matrix inversion step (i.e., it calls the classical inversion routine). A prototype version based on the HHL algorithm is provided as an example, but practical benefits will be seen only when running on a true quantum device with a suitable quantum subroutine.

QFT Simulations:

The QFT simulations use Qiskit's Aer simulator. Adjust your Qiskit installation if you encounter module import issues.