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Scaling of quantum error correction
Technically, the right parameter is the number of "locations" in the circuit, since errors can occur on qubits even when they are sitting around doing nothing. To a first approximation, the number of locations is number of qubits times the depth of the circuit. (The depth is the number of time steps needed when the circuit is parallelized.) If you ignore storage errors, then the number of gates is the right parameter. Note that the number of gates will typically be much larger than the number of qubits, except for the shortest computations.
Proof of Pauli group preservation by Clifford group conjugation?
I don't understand what you are asking. How does one prove a definition? There are some standard generalizations to qudit Clifford groups, which share many of the properties of the qubit version. The general group theory construction of preserving a subgroup under conjugation is called the "normalizer".