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" course (Lecture 5).

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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).