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 $\theta$).

• Identity (I)
• Pauli X (NOT)
• Pauli Y
• Pauli Z
• Hadamard (H)
• S (Phase) gate
• T gate
• RX (rotation around X)
• RY (rotation around Y)
• RZ (rotation around Z)
• Universal single-qubit (U)

Single-qubit gates act as rotations on the Bloch sphere. The three rotation gates rotate by angle $\theta$ around the x, y, z axes:

Since $\mathrm{SU}(2) \cong \mathrm{SO}(3)$, these three families generate all single-qubit gates. Composing rotations around different axes yields a rotation around a third axis. The universal gate $U(\theta, \phi, \lambda)$ parametrizes any rotation via axis and angle; all other standard gates are special cases.

Clifford gates conjugate Paulis to Paulis: $C P C^\dagger$ is Pauli if $C$ is Clifford and $P$ is Pauli. This closure means Clifford circuits simulate classically in polynomial time (stabilizer formalism) — no quantum advantage. Includes:

Non-Clifford gates (T gate) break Pauli closure and enable universal computation and exponential speedup. T gates require magic state distillation in fault-tolerant schemes, making T-count a key resource metric.