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In this new video, Anette Messinger and Katharina Ludwig introduce Parity Flow, the core method behind ParityQC’s algorithmic innovations. Parity Flow is a powerful method for tracking the flow of quantum information in circuits, leading to quantum circuits with record-low gate count and circuit depth. It offers intuitive insights into quantum operations, particularly those involving Clifford gates and Pauli rotations, leading to efficient synthesis of quantum algorithms. Parity Flow also supports debugging, circuit verification, and the derivation of circuit identities, making it a versatile tool for understanding and optimizing quantum protocols. We're excited to share this comprehensive overview of the method, and looking forward to seeing how the scientific community will further build on this. You can find out more about Parity Flow in the papers below: "Parity flow formalism: Tracking quantum information throughout computation". Berend Klaver, Katharina Ludwig, Anette Messinger, Stefan M. A. Rombouts, Michael Fellner, Kilian Ender, and Wolfgang Lechner Phys. Rev. Research 8, 013095 (2026). https://doi.org/10.1103/6xlb-l92j "SWAP-less implementation of quantum algorithms". Berend Klaver, Stefan M. A. Rombouts, Michael Fellner, Anette Messinger, Kilian Ender, Katharina Ludwig, and Wolfgang Lechner. Phys. Rev. A 113, 012443 (2026). https://doi.org/10.1103/2wzk-fnhx