Single-Qubit Gates

Single-qubit gates are 2x2 unitaries that rotate the Bloch sphere; they cover phase, bit-flip, and superposition operations.

Rotations of one qubit

A single-qubit gate is a 2x2 unitary matrix. Geometrically every one is a rotation of the Bloch sphere about some axis by some angle, so the set of single-qubit gates is exactly the set of sphere rotations (up to global phase).

The standard set

• X, Y, Z: the Pauli gates, 180-degree rotations about x, y, z
• H: Hadamard, 180-degree rotation about the x+z diagonal
• S: 90-degree phase gate (Z^{1/2})
• T: 45-degree phase gate (Z^{1/4})
• Rx, Ry, Rz: continuous rotations by an arbitrary angle

Rotation gates

The parameterised gates Rx(theta), Ry(theta), Rz(theta) rotate by angle theta about the named axis, defined as exp(-i theta P/2) for the corresponding Pauli P. Any single-qubit unitary can be written as a product of three such rotations — the Euler-angle decomposition — which is how compilers translate an arbitrary requested gate into hardware primitives.

Phase gates

S and T change only the relative phase of |1>, leaving computational-basis probabilities untouched but reshaping interference. The T gate is special: {H, T} generate a dense set of single-qubit unitaries, so with CNOT they form a universal discrete set.

Limits

Single-qubit gates alone can never produce entanglement, because they act independently on each qubit and preserve product structure. Universal computation therefore requires at least one two-qubit entangling gate such as CNOT in addition to a rich single-qubit set.