A Methodology for Quantum Gates Homogenizing using Lie Group Representations

The burgeoning field of quantum computing presents a rich variety of quantum gates, from el-ementary single-qubit rotations to complex multi-qubit entanglers. While this diversity underpinspowerful algorithms, it can also obscure the underlying mathematical unity of these operations.This article demonstrates how the theory of Lie groups and their representations provides a pro-found framework for classifying, simplifying, and homogenizing quantum gates. We first discuss themultifaceted problem of gate homogenization from the perspectives of quantum program analysis,synthesis, transpilation, and hardware construction, highlighting the state-of-the-art in addressingthese challenges. Subsequently, we detail how Special Unitary groups, SU(N), encapsulate gate op-erations and how their irreducible representations (multiplets) simplify multi-qubit analysis. Keytheorems, such as the Euler Angle Decomposition for SU(2) and the Solovay-Kitaev Theorem, areintroduced with conceptual proofs and illustrated with practical Qiskit examples, showcasing theinitial (common-sense) versus transformed (group theory-informed) perspective. Furthermore, weelaborate on how irreducible representations offer a powerful reinterpretation of superposition andentanglement. Finally, we propose a general methodology for transforming quantum programs intoa homogeneous gate set, complete with practical Qiskit examples for Deutsch’s Algorithm, QuantumTeleportation, and Quantum Phase Estimation.