HomeQuantum PhysicsLessonsPauli-X Gate (NOT)

Pauli-X Gate (NOT)

BasicSingle-qubitbit flip

Flips a qubit: $∣0 ⟩$ becomes $∣1 ⟩$ , and $∣1 ⟩$ becomes $∣0 ⟩$ .

Intuition

The X gate swaps $∣0 ⟩$ and $∣1 ⟩$ . It is the quantum version of a classical NOT gate. On the Bloch sphere, this is a 180° rotation around the X axis, which swaps the north and south poles. Unlike classical NOT, which flips a definite bit, the X gate acts on superpositions too. If your qubit is in the state $α$ $∣0 ⟩ + β$ $∣1 ⟩$ , X swaps the amplitudes to give $β$ $∣0 ⟩ + α$ $∣1 ⟩$ . All quantum information is preserved.

Matrix representation

(0110)

Action on states

∣0 ⟩ \to ∣1 ⟩, ∣1 ⟩ \to ∣0 ⟩

Bloch sphere

|0⟩

→

|1⟩

Circuit

State comparison

Before X Gate

|0⟩

|0\u27E9

100%

|1\u27E9

Bloch: (0.00, 0.00, 1.00)

→

After X Gate

|1⟩

|0\u27E9

|1\u27E9

100%

Bloch: (0.00, 0.00, -1.00)

Full technical statement

Pauli-X is one of three Pauli operators. The Pauli operators (X, Y, Z) are a set of three 2×2 matrices that form a basis for all single-qubit operations and correspond to spin measurements along the three spatial axes. X is unitary (it preserves probability), Hermitian (it equals its own conjugate transpose), and an involution (X² = I, meaning it is its own inverse). Its eigenvalues are +1 and −1. An eigenvalue is the factor by which an eigenstate is scaled when the operator acts on it. The corresponding eigenstates (states unchanged up to that scaling factor) are $∣ + ⟩ = (∣0 ⟩ + ∣1 ⟩) / 2$ and |−⟩ = ( $∣0 ⟩$ − $∣1 ⟩ / 2$ . In the computational basis, X acts as a bit flip: X( $α$ $∣0 ⟩ + β$ $∣1 ⟩ = α$ $∣1 ⟩ + β$ $∣0 ⟩$ . Combined with controlled operations, it becomes the building block for CNOT and Toffoli gates.

What changes in measurements

Every measurement outcome flips: 0 becomes 1, and 1 becomes 0.

Bloch sphere

Rotates the Bloch vector 180° around the X axis. The north pole $(∣0 ⟩)$ and south pole $(∣1 ⟩)$ swap. States on the X axis ( $∣ + ⟩$ , |−⟩) stay fixed because they are eigenstates of X.

Step-by-step example

Preparing |1⟩ from the default state

1Start with $∣0 ⟩$ . Every qubit begins in this state by default.
2Apply X. The state becomes $∣1 ⟩$ .
3Measure. The result is 1 with certainty.
4This is how you initialize helper qubits (ancillas) to $∣1 ⟩$ for algorithms like Deutsch’s.

Where this gate appears

Initializing qubits to |1⟩, since hardware starts all qubits at |0⟩
Building the CNOT gate, where X is the operation applied conditionally to the target
Preparing |−⟩ = H|1⟩ by applying X then H
Embedding classical reversible logic inside quantum circuits

Connected gates

X² = I — two bit flips cancel out and return the qubit to its original state
HXH = Z — when you switch to the diagonal basis using H, a bit flip becomes a phase flip
RX(π) = −iX — a full π-rotation around X equals X up to a global phase factor that does not affect measurements
CNOT = controlled-X, and Toffoli = doubly-controlled-X, both built from X

Tempting but wrong

It is tempting to think X is just classical NOT with a different name. That sounds right because the truth table matches. What actually happens is that X swaps amplitudes in superpositions, not just definite bits. When a qubit is in a superposition, X rearranges the amplitudes, and that rearrangement affects interference in ways that classical NOT cannot.

Why this gate matters for what comes next

X is the simplest non-trivial quantum gate and the foundation of all controlled operations. Understanding how it acts on superpositions, not just basis states, is the first step toward understanding multi-qubit gates like CNOT.

Test your understanding

What is X|+⟩?

Quick reference →↗ Nielsen and Chuang, Quantum Computation and Quantum Information ↗ MIT OCW 8.06: quantum computing notes

← Back to all lessons

HomeQuantum PhysicsLessonsPauli-X Gate (NOT)

Pauli-X Gate (NOT)

BasicSingle-qubitbit flip

Flips a qubit: $∣0 ⟩$ becomes $∣1 ⟩$ , and $∣1 ⟩$ becomes $∣0 ⟩$ .

Intuition

Matrix representation

(0110)

Action on states

∣0 ⟩ \to ∣1 ⟩, ∣1 ⟩ \to ∣0 ⟩

Bloch sphere

|0⟩

→

|1⟩

Circuit

State comparison

Before X Gate

|0⟩

|0\u27E9

100%

|1\u27E9

Bloch: (0.00, 0.00, 1.00)

→

After X Gate

|1⟩

|0\u27E9

|1\u27E9

100%

Bloch: (0.00, 0.00, -1.00)

Full technical statement

What changes in measurements

Every measurement outcome flips: 0 becomes 1, and 1 becomes 0.

Bloch sphere

Step-by-step example

Preparing |1⟩ from the default state

1Start with $∣0 ⟩$ . Every qubit begins in this state by default.
2Apply X. The state becomes $∣1 ⟩$ .
3Measure. The result is 1 with certainty.
4This is how you initialize helper qubits (ancillas) to $∣1 ⟩$ for algorithms like Deutsch’s.

Where this gate appears

Initializing qubits to |1⟩, since hardware starts all qubits at |0⟩
Building the CNOT gate, where X is the operation applied conditionally to the target
Preparing |−⟩ = H|1⟩ by applying X then H
Embedding classical reversible logic inside quantum circuits

Connected gates

X² = I — two bit flips cancel out and return the qubit to its original state
HXH = Z — when you switch to the diagonal basis using H, a bit flip becomes a phase flip
RX(π) = −iX — a full π-rotation around X equals X up to a global phase factor that does not affect measurements
CNOT = controlled-X, and Toffoli = doubly-controlled-X, both built from X

Tempting but wrong

Why this gate matters for what comes next

Test your understanding

What is X|+⟩?

Quick reference →↗ Nielsen and Chuang, Quantum Computation and Quantum Information ↗ MIT OCW 8.06: quantum computing notes

← Back to all lessons