What are the purposes of circuit gates in addition to matrix presentation

Question

If quantum circuit gates can be presented as a matrix, are there any other qualities for the circuit gates except for a compact presentation of a matrix?

Answer 1

The classical analog to your question, as far as I can see, is "what reasons, besides its simplicity and smaller size, do we use NAND-gates and NOR-gates to specify boolean functions?" But we write such circuits when we actually want to implement the logic in silicon, e.g., with VLSI-design. The classical circuit corresponds to the etchings in silicon and the connections of metal for the transistors. We do not directly implement the truth-table itself.

Similarly, quantum circuits are how quantum algorithms are intended to be run - we translate the $T$ gate or the $CZ$ gate, etc., to a sequence of microwave pulses or laser pulses, etc. The quantum circuit corresponds to the sequences of such electromagnetic pulses. We do not directly implement the unitary matrix itself.