A *quantum logic gate* is a device which performs a fixed unitary operation on selected qubits in a fixed period of time. The gates listed below are common enough to have their own names. The matrices describing *n* qubit gates are written in the computational basis {∣*x*⟩}, where *x* is a binary string of length *n*. The diagrams provide schematic representation of the gates.

### Hadamard gate

The Hadamard gate is a common single qubit gate *H* defined as

The matrix is written in the computational basis {∣0⟩, ∣1⟩} and the diagram on the right provides a schematic representation of the gate *H* acting on a qubit in state ∣*x*⟩, with *x* = 0, 1.

### Phase gate

The phase shift gate **\phi ** defined as ∣ 0⟩ ↦ ∣ 0⟩ and ∣ 1⟩ ↦ *e**i**ϕ*∣ 1⟩, or, in matrix notation,

### Controlled NOT gate

The controlled-NOT (C-NOT) gate, also known as the XOR or the measurement gate is one of the most popular two-qubit gate. It flips the second (target) qubit if the first (control) qubit is ∣ 1⟩ and does nothing if the control qubit is ∣ 0⟩. The gate is represented by the unitary matrix

where *x*, *y* = 0or 1 and ⊕ denotes XOR or addition modulo 2. If we apply the C-NOT to Boolean data in which the target qubit is ∣0⟩ and the control is either ∣0⟩ or ∣1⟩ then the effect is to leave the control unchanged while the target becomes a copy of the control, *i.e.*

∣*x*⟩∣0⟩ ↦ ∣*x*⟩∣*x*⟩ *x* = 0, 1.

Category:Evolutions and Operations Category:Quantum Computation Category:Handbook of Quantum Information