This repository contains an Octave implementation of a Batch Gradient Descent (BGD) algorithm designed to approximate a reference unitary matrix Uref through numerical optimisation.
The method uses finite-difference gradients to minimise the Frobenius norm squared difference between Uref and a trial unitary matrix Utest = UC(P).

The code produces detailed convergence plots, percentile analyses, and saves both the numerical results and the best unitary matrix obtained during optimisation.

The optimisation follows these main steps:

• Choose the matrix dimension d (the algorithm constructs an n×n unitary matrix where n = d).
• Generate a random complex matrix A of size d×d.
• Decompose it using QR factorisation:
  [Q, R] = qr(A)
• Construct the reference unitary matrix:
  U_ref = Q * diag(diag(R) ./ abs(diag(R)))
  ensuring that U_ref is unitary.

For each simulation:

• Initialise a random real matrix P of size d×d.
• Map P to a unitary matrix U_test = UC(P) using the composite parameterisation.
• Evaluate the cost function:
  C = ||U_ref - U_test||_F²
• Compute the gradient of the cost function using finite differences:
  ∂C/∂P_ij ≈ (C(P_ij + ε) - C(P_ij - ε)) / (2ε)
• Update the matrix P via gradient descent:
  P = P - α * ∇_P C

The algorithm terminates when:

• C < 0.01, or
• the maximum iteration count (num_iter) is reached.

The program tracks:

• Cost per iteration
• Best cost achieved
• Best unitary matrix (U_best)
• Iteration number of the best result

4. Outputs
   • Average and percentile cost plots.
   • Iteration counts until convergence.
   • The best unitary matrix found (best_U_overall) and the corresponding cost.

• The default α = 0.35 works well for most cases (d = 2–5).
• Reducing α (e.g. to 0.25) can improve stability for higher-dimensional matrices.
• Increasing α may accelerate convergence but risks overshooting minima.
• The learning rate is easy to change:

  learning_rate = 0.35;  % modify as needed
  
  

This repository includes the UC.m function authored by Christoph Spengler (University of Vienna, 2011),
which implements the Composite Parameterisation of the Unitary Group U(d).

The function is provided under its original academic copyright and
is used here solely for research and educational purposes.

If you use or reference this repository in your work, please cite the following papers as requested by the author:

Spengler, C., Huber, M., & Hiesmayr, B. C. (2010).
A composite parameterization of unitary groups, density matrices and subspaces.
Journal of Physics A: Mathematical and Theoretical, 43 (38), 385306.
arXiv: 1004.5252

Spengler, C., Huber, M., & Hiesmayr, B. C. (2011).
Composite parameterization and Haar measure for all unitary and special unitary groups.
arXiv: 1103.3408

Note: The file UC.m remains © Christoph Spengler (2011) and is not distributed under the MIT License.
All other code and documentation in this repository are © Liam. Wakefield (2025) and released under the MIT License.