CNOT and controlled operations
CX (CNOT) flips its target qubit inside exactly those components of the joint state where the control qubit is 1 — a conditional applied per component, with no measurement and no runtime branch. The operand order [control, target] is part of the program: swap it and you run a different circuit.
A controlled operation is a two-qubit gate that applies some action to one qubit — the target — only on those parts of the state where another qubit — the control — has the value 1. The workhorse is CX, also called CNOT (controlled-NOT): its action is an X flip of the target.
For plain 0/1 inputs, CX is fully described by a truth table — a list of every input and its output. With control q0 and target q1 (displayed as the string q1q0, rightmost bit q0):
00 → 00— control is 0, nothing happens01 → 11— control is 1, target flips from 0 to 110 → 10— control is 0, nothing happens11 → 01— control is 1, target flips from 1 to 0
The control is never changed; the target flips exactly when the control bit is 1.
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