Single-qubit gates act on one qubit, represented by $2 \times 2$ unitary matrices. They rotate the qubit state on the Bloch sphere or apply phase shifts. See gate details for individual gates.

The Identity gate is the trivial gate that leaves the quantum state unchanged. It is the quantum analog of “do nothing” and appears as a placeholder in circuit padding and theoretical proofs.

Matrix: $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

Property: $I^2 = I$, commutes with all gates.

The Pauli X gate flips the qubit: $|0\rangle \leftrightarrow |1\rangle$. It is the quantum analog of the classical NOT gate and is the most fundamental bit-flip operation.

Matrix: $X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Property: $X^2 = I$ (self-inverse).

The Pauli Y gate combines a bit flip and phase. It rotates around the y-axis of the Bloch sphere.

Matrix: $Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$

Property: $Y^2 = I$ (self-inverse). Less commonly used directly than X or Z.

The Pauli Z gate applies a phase: $|0\rangle$ unchanged, $|1\rangle \to -|1\rangle$. It leaves the computational basis unchanged but introduces a relative phase.

Matrix: $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Property: $Z^2 = I$ (self-inverse).

The Hadamard gate creates equal superposition from computational basis states. Essential for quantum algorithms; appears in nearly every quantum circuit.

Matrix: $H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

Property: $H^2 = I$ (self-inverse). Creates $|0\rangle \to (|0\rangle + |1\rangle)/\sqrt{2}$ and $|1\rangle \to (|0\rangle - |1\rangle)/\sqrt{2}$.

The S gate applies a 90° phase to the $|1\rangle$ state. It is a quarter-turn phase gate.

Matrix: $S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$

Property: $S^2 = Z$, $S^4 = I$.

The T gate applies a 45° phase to the $|1\rangle$ state. Critical for quantum algorithms and fault-tolerant quantum computing.

Matrix: $T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$

Property: $T^2 = S$, $T^8 = I$. Often called the “magic gate” in quantum error correction.

The RX gate is a parameterized rotation around the x-axis of the Bloch sphere by angle $\theta$.

Matrix: $R_X(\theta) = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$

Special cases: $R_X(\pi) = iX$, $R_X(\pi/2)$ is a half-rotation.

The RY gate is a parameterized rotation around the y-axis of the Bloch sphere by angle $\theta$.

Matrix: $R_Y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$

Special cases: $R_Y(\pi) = iY$, $R_Y(\pi/2)$ creates superposition.

The RZ gate is a parameterized rotation around the z-axis (phase rotation) by angle $\theta$.

Matrix: $R_Z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$

Special cases: $R_Z(\pi) = iZ$, $R_Z(\pi/2) = S$, $R_Z(\pi/4) = T$.

The Universal U gate is a general single-qubit gate parameterized by three angles. Any single-qubit unitary can be expressed as a U gate, making it universal for single-qubit operations.

Matrix: $U(\theta, \phi, \lambda) = \begin{pmatrix} \cos(\theta/2) & -e^{i\lambda}\sin(\theta/2) \\ e^{i\phi}\sin(\theta/2) & e^{i(\phi+\lambda)}\cos(\theta/2) \end{pmatrix}$

Property: reduces to H, X, Y, Z, S, T when parameters are set appropriately. General decomposition: any rotation gate sequence can be expressed as a single U gate.