Quantum gates and algorithms in qg units: Hamming-weight conservation and readout classes

We write fifteen standard quantum gates and fourteen standard quantum algorithms in qg units [1]. An -qubit state is described by its qg values, for every Pauli string , and a gate acts as . With this one rule each gate becomes a table of qg transformations and each algorithm a readout identity: which qg values hold the answer. The formulation is the Pauli (Heisenberg) representation and adds no new physics. It gives three practical classifications in the units used by the rest of the qang project: which gates conserve Hamming weight, and therefore admit the qg symmetry witness and filter; which algorithms are read out from a few qg values and which need the full outcome histogram; and where reading qg values is strictly worse than the standard readout. Three readout identities are, as far as we know, not stated in this form elsewhere: the per-qubit polar bias after Grover iterations, Simon’s secret as the unique non-trivial Z-parity equal to one, and quantum counting as a single . Every identity is checked numerically and pinned by tests in the open-source library qang (version 0.5.1, pip install qang), module qang.formulation.