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Table of Contents
Single-qubit gates
Single-qubit gates are unitary operations that act on one qubit. They are represented by $2 \times 2$ unitary matrices and form the Lie group $\mathrm{SU}(2)$. Every quantum computation can be decomposed into single-qubit rotations and two-qubit entangling gates (e.g., CNOT), making single-qubit gates the fundamental building blocks of quantum circuits.
A single-qubit gate is applied to a state $|\psi\rangle$ to produce a new state $U|\psi\rangle$. The gates compose: two gates in sequence form a new gate (matrix product). Some gates are self-inverse (like Pauli gates and Hadamard); others have finite order ($S^4 = I$, $T^8 = I$); some are parameterized (RX, RY, RZ with rotation angle $\theta$).
Bloch sphere
Universality
Any single-qubit unitary matrix belongs to $\mathrm{SU}(2)$ and can be decomposed into rotations around chosen axes. The universal gate $U(\theta, \phi, \lambda)$ parameterizes all single-qubit unitaries; any gate in the list below can be recovered by setting its three angles appropriately.
Decomposition gates
A common decomposition is the ZYZ decomposition: any single-qubit unitary can be written as $U = R_Z(\phi) R_Y(\theta) R_Z(\lambda)$ for angles $\phi, \theta, \lambda$. Other decompositions exist (e.g., XYX); the choice depends on available native gates in hardware.
Clifford gates
Clifford gates (X, Y, Z, H, S) map Pauli strings to Pauli strings and can be simulated classically in polynomial time. They form the stabilizer code formalism and are cheap to implement on most quantum hardware.
Non-Clifford gates (primarily the T gate) are required for universal quantum computation. T gates are expensive in fault-tolerant quantum error correction, making their count a key resource metric. The magic state distillation protocol converts many noisy non-Clifford states into fewer high-fidelity ones.
Parametrized rotation (RX, RY, RZ)
The rotation gates $R_X(\theta)$, $R_Y(\theta)$, $R_Z(\theta)$ are parameterized rotations around the x, y, z axes by angle $\theta$. The Lie group $\mathrm{SU}(2)$ is isomorphic to the rotation group $\mathrm{SO}(3)$, so these three families generate all single-qubit rotations; any two rotations around different axes compose to form a rotation around a third axis.
Quaternion rotation (U gate)
The universal gate $U(\theta, \phi, \lambda)$ represents a quaternion rotation on the Bloch sphere. The Lie group $\mathrm{SU}(2)$ is isomorphic to the group of unit quaternions $\mathrm{Sp}(1)$, which parameterize all 3D rotations; the three parameters of $U$ encode the axis and angle of rotation. Pauli and Hadamard gates are special cases of $U$ with specific angle choices.