Controlled Quantum Gate

quantum-computing

Definition

Controlled Quantum Gate

A controlled quantum gate is a quantum gate with a control qubit and a target system. It leaves the target unchanged when the control is $|0⟩$, and applies a chosen operation $U$ to the target when the control is $|1⟩$:

$$\begin{array}{rl}|0⟩|\psi ⟩& \mapsto |0⟩|\psi ⟩,\\ |1⟩|\psi ⟩& \mapsto |1⟩U|\psi ⟩.\end{array}$$

The juxtaposition of kets denotes a tensor product:

$$|c⟩|\psi ⟩=|c⟩\otimes |\psi ⟩.$$

Here $|c⟩$ is the state of the control qubit, and $|\psi ⟩$ is the state of the target system.

Example

Controlled-NOT gate

The controlled-NOT gate, or CNOT, is the controlled version of the Pauli-X gate.

It has two qubits:

• the first qubit is the control;
• the second qubit is the target.

The control qubit is not changed. It only decides whether the target qubit is flipped.

If the control is $|0⟩$, CNOT does nothing to the target:

$$\begin{array}{rl}|00⟩& \mapsto |00⟩,\\ |01⟩& \mapsto |01⟩.\end{array}$$

If the control is $|1⟩$, CNOT applies $X$ to the target, so the target flips:

$$\begin{array}{rl}|10⟩& \mapsto |11⟩,\\ |11⟩& \mapsto |10⟩.\end{array}$$

Equivalently, for bits $x,y\in \{0,1\}$,

$$|x⟩|y⟩\mapsto |x⟩|y\oplus x⟩,$$

where $\oplus$ is addition modulo $2$.

If $x=0$, then $y\oplus x=y$, so the target does not change. If $x=1$, then $y\oplus x$ flips $y$.