Parity flow formalism: Tracking quantum information throughout computation

We propose the parity flow formalism, a method for tracking the information flow in quantum circuits. This method adds labels to quantum circuit diagrams such that the action of Clifford gates can be understood as a recoding of quantum information. The action of non-Clifford gates in the encoded space can be directly deduced from those labels without backtracking. An application of flow tracking is to design resource-efficient quantum circuits by changing any present encoding via a simple set of rules. Finally, the parity flow formalism can be used in combination with stabilizer codes to further reduce quantum circuit depth and to reveal additional operations that can be implemented in parallel.