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√X

Square-root-of-X gate tile from the Operations catalog

√X ("square root of X") is a gate that, applied twice, is exactly equal to one Pauli-X gate. It's a common native gate on real quantum hardware.
Qubits 1
Parameters None
Inverse √X† (not self-inverse)
Also called SX gate

What it does

√X creates an equal superposition from either basis state, similar to H, but with a different pattern of complex phases. Applying it twice in a row reproduces a full bit-flip: \( (\sqrt{X})^2 = X \).

Matrix

\[ \sqrt{X} = \frac{1}{2}\begin{pmatrix} 1+i & 1-i \\ 1-i & 1+i \end{pmatrix} \]

Action on basis states

  • \( |0\rangle \rightarrow \dfrac{1}{2}\big[(1+i)|0\rangle + (1-i)|1\rangle\big] \)
  • \( |1\rangle \rightarrow \dfrac{1}{2}\big[(1-i)|0\rangle + (1+i)|1\rangle\big] \)

Example

Two √X gates in a row on q[0] (starting at \( |0\rangle \)) land exactly on \( |1\rangle \) - the same result one X gate would give in a single step.

Other notations

Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.

Language Typical form
OpenQASM 2.0 sx q[0];
Qiskit circuit.sx[q[0]]
Cirq cirq.X(q[0])**0.5
Q# -