https://github.com/brian-hepler-phd/kak-decomposition-qml
Cartan decomposition (SU(4)) via Lie theory and QML – exact + variational approaches for two-qubit gates.
https://github.com/brian-hepler-phd/kak-decomposition-qml
lie-theory machine-learning numpy pennylane qml quantum-computing quantum-gate-decomposition scipy su4
Last synced: 6 months ago
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Cartan decomposition (SU(4)) via Lie theory and QML – exact + variational approaches for two-qubit gates.
- Host: GitHub
- URL: https://github.com/brian-hepler-phd/kak-decomposition-qml
- Owner: brian-hepler-phd
- Created: 2025-03-29T03:06:10.000Z (7 months ago)
- Default Branch: main
- Last Pushed: 2025-04-11T22:01:04.000Z (6 months ago)
- Last Synced: 2025-04-12T03:58:57.450Z (6 months ago)
- Topics: lie-theory, machine-learning, numpy, pennylane, qml, quantum-computing, quantum-gate-decomposition, scipy, su4
- Language: Jupyter Notebook
- Homepage: https://www.bhepler.com
- Size: 17.6 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# KAK Decomposition of Two-Qubit Gates via Lie Theory and Quantum Machine Learning
> Developed as supplemental material for Quantum Formalism's "[Lie Groups with Applications](https://quantumformalism.academy/lie-groups-with-applications)" course (Lecture 5).
> *See my full portfolio at: [https://bhepler.com/portfolio/](https://brainhelper.wordpress.com/data-science-portfolio/)*
## Goal
To decompose a general two-qubit gate (an $SU(4)$ matrix) into simpler components using Lie-theoretic methods (Cartan Decomposition) and, complementarily, to synthesize the same gate using a Quantum Machine Learning (QML) approach with a Variational Quantum Circuit (VQC). This project explores both exact numerical solutions rooted in Lie theory and approximate, trainable quantum circuits.
## Key Concepts & Relevance
* **Lie Theory:** Cartan KAK Decomposition ($G = K_0 A K_1^{-1}$) of SU(4), Lie Algebras ($\mathfrak{su}(4)$), matrix logarithm/exponential.
* **Quantum Computing:** Two-Qubit Gates (SU(4)), Gate Decomposition, Gate Synthesis, Local vs. Non-local (Entangling) Operations.
* **Quantum Machine Learning (QML):** Variational Quantum Circuits (VQC), Parameter Optimization, Automatic Differentiation, Gradient Descent.
* **Numerical Methods:** Root-finding algorithms for solving decomposition equations.
* **Relevance:** Essential for understanding quantum gate structures, optimizing quantum circuits, quantum optimal control, and applying ML techniques within quantum computation.
## Implementation & Activities
* **Implemented two distinct methods** for handling arbitrary $SU(4)$ two-qubit gates:
* **Exact Decomposition:** Developed a numerical solution using the Khaneja-Glaser algorithm (**NumPy, SciPy**) based on the Cartan KAK decomposition, isolating local ($K_0, K_1$) and non-local ($A$) factors. Utilized **SciPy's optimize.root** for solving intermediate steps.
* **Variational Synthesis (QML):** Designed and implemented a Variational Quantum Circuit (VQC) in **PennyLane**, with a structure inspired by the Cartan decomposition (parameterizing local and non-local parts).
* **Leveraged PennyLane's automatic differentiation** capabilities to compute gradients of a fidelity cost function (comparing VQC output to the target gate).
* **Implemented gradient descent** optimization to train the VQC parameters, learning an approximate synthesis of the target $SU(4)$ gate.
* **Verified** the accuracy of both methods by reconstructing the original gate from the decomposed/synthesized components and calculating the approximation error (e.g., using matrix norm).
* **Bridged** advanced Lie theory concepts with practical quantum computing frameworks (**PennyLane**) and numerical optimization techniques.
## Technologies & Skills
* **Languages:** Python
* **Libraries:** NumPy, SciPy (linalg, optimize), PennyLane, Matplotlib (for potential convergence plots)
* **Concepts:** Lie Groups (SU(4)), Lie Algebras ($\mathfrak{su}(4)$), Cartan Decomposition (KAK), Matrix Logarithm/Exponential, Quantum Machine Learning (QML), Variational Quantum Circuits (VQC), Automatic Differentiation, Gradient Descent, Quantum Computing (Two-Qubit Gates, Gate Decomposition/Synthesis), Numerical Optimization, Linear Algebra.
## Theoretical Background
This project leverages the Cartan KAK decomposition, a fundamental result in Lie theory stating that any element $G$ of a semisimple Lie group (like SU(4)) can be written as $G = K_0 A K_1^{-1}$, where $K_0, K_1$ are in a maximal compact subgroup (here $SU(2) \otimes SU(2)$, representing local operations) and $A$ is in a specific abelian subgroup related to the Cartan subalgebra (representing the non-local/entangling part).