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--- quantum-gate-single-qubit [August 22, 2026 at 18:46] – Ivan Janevski
+++ quantum-gate-single-qubit [August 22, 2026 at 18:55] (current) – external edit 127.0.0.1
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 **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 [[quantum-gate-single-qubit|gate details]] for individual gates.
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+## Identity gate
-The **[[quantum-gate-i|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.
+**[[quantum-gate-i|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. Self-inverse ($I^2 = I$), it commutes with all gates.
-Matrix: $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
+$$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$
-Property: $I^2 = I$, commutes with all gates.
+## Pauli X gate
-## Pauli X (NOT)
+**[[quantum-gate-x|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. Self-inverse ($X^2 = I$).
-The **[[quantum-gate-x|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.
+$$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$
-Matrix: $X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
+## Pauli Y gate
-Property: $X^2 = I$ (self-inverse).
+**[[quantum-gate-y|Pauli Y]]** gate combines a bit flip and phase, rotating around the y-axis of the Bloch sphere. Self-inverse ($Y^2 = I$), though less commonly used directly than X or Z.
-## Pauli Y
+$$Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$$
-The **[[quantum-gate-y|Pauli Y]]** gate combines a bit flip and phase. It rotates around the y-axis of the Bloch sphere.
+## Pauli Z gate
-Matrix: $Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$
+**[[quantum-gate-z|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. Self-inverse ($Z^2 = I$).
-Property: $Y^2 = I$ (self-inverse). Less commonly used directly than X or Z.
+$$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$
-## Pauli Z
+## Hadamard (H) gate
-The **[[quantum-gate-z|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.
+**[[quantum-gate-h|Hadamard]]** gate creates equal superposition from computational basis states. Essential for quantum algorithms; appears in nearly every quantum circuit. Self-inverse ($H^2 = I$), it maps $|0\rangle \to (|0\rangle + |1\rangle)/\sqrt{2}$ and $|1\rangle \to (|0\rangle - |1\rangle)/\sqrt{2}$.
-Matrix: $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
+$$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$
-Property: $Z^2 = I$ (self-inverse).
+## S (Phase) gate
-## Hadamard (H)
+**[[quantum-gate-s|S gate]]** applies a 90° phase to the $|1\rangle$ state—a quarter-turn phase gate with $S^2 = Z$ and $S^4 = I$.
-The **[[quantum-gate-h|Hadamard]]** gate creates equal superposition from computational basis states. Essential for quantum algorithms; appears in nearly every quantum circuit.
+$$S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$$
-Matrix: $H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$
+## T gate
-Property: $H^2 = I$ (self-inverse). Creates $|0\rangle \to (|0\rangle + |1\rangle)/\sqrt{2}$ and $|1\rangle \to (|0\rangle - |1\rangle)/\sqrt{2}$.
+**[[quantum-gate-t|T gate]]** applies a 45° phase to the $|1\rangle$ state. Critical for quantum algorithms and fault-tolerant quantum computing, it is often called the "magic gate" in quantum error correction ($T^2 = S$, $T^8 = I$).
-## S (Phase) Gate
+$$T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$$
-The **[[quantum-gate-s|S gate]]** applies a 90° phase to the $|1\rangle$ state. It is a quarter-turn phase gate.
+## RX (Rotation around X) gate
-Matrix: $S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$
+**[[quantum-gate-rx|RX gate]]** is a parameterized rotation around the x-axis of the Bloch sphere by angle $\theta$; $R_X(\pi/2)$ is a half-rotation.
-Property: $S^2 = Z$, $S^4 = I$.
+$$R_X(\theta) = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$
-## T Gate
+## RY (Rotation around Y) gate
-The **[[quantum-gate-t|T gate]]** applies a 45° phase to the $|1\rangle$ state. Critical for quantum algorithms and fault-tolerant quantum computing.
+**[[quantum-gate-ry|RY gate]]** is a parameterized rotation around the y-axis of the Bloch sphere by angle $\theta$; $R_Y(\pi/2)$ creates superposition.
-Matrix: $T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$
+$$R_Y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$
-Property: $T^2 = S$, $T^8 = I$. Often called the "magic gate" in quantum error correction.
+## RZ (Rotation around Z) gate
-## RX (Rotation around X)
+**[[quantum-gate-rz|RZ gate]]** is a parameterized rotation around the z-axis (phase rotation) by angle $\theta$; note $R_Z(\pi/2) = S$ and $R_Z(\pi/4) = T$.
-The **[[quantum-gate-rx|RX gate]]** is a parameterized rotation around the x-axis of the Bloch sphere by angle $\theta$.
+$$R_Z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$$
-Matrix: $R_X(\theta) = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$
+## Universal Single-Qubit (U) gate
-Special cases: $R_X(\pi) = iX$, $R_X(\pi/2)$ is a half-rotation.
+**[[quantum-gate-u|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; it reduces to H, X, Y, Z, S, T when parameters are set appropriately.
-## RY (Rotation around Y)
+$$U(\theta, \phi, \lambda) = \begin{pmatrix} \cos(\theta/2) & -e^{i\lambda}\sin(\theta/2) \\ e^{i\phi}\sin(\theta/2) & e^{i(\phi+\lambda)}\cos(\theta/2) \end{pmatrix}$$
-The **[[quantum-gate-ry|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.
-## RZ (Rotation around Z)
-The **[[quantum-gate-rz|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: $R_Z(\pi) = iZ$, $R_Z(\pi/2) = S$, $R_Z(\pi/4) = T$.
-## Universal Single-Qubit (U)
-The **[[quantum-gate-u|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.