Recursive Cartan decompositions for unitary synthesis

Understand

Recursive Cartan decompositions (CDs) provide a way to exactly factorize quantum circuits into smaller components, making them a central tool for unitary synthesis.

• Here we present a detailed overview of recursive CDs, elucidating their mathematical structure, demonstrating their algorithmic utility, and implementing them numerically at large scales.
• We adapt, extend, and unify existing mathematical frameworks for recursive CDs, allowing us to gain new insights and streamline the construction of new circuit decompositions.
• Based on this, we show that several leading synthesis techniques from the literature-the Quantum Shannon, Block-ZXZ, and Khaneja-Glaser decompositions-implement the same recursive CD.