SX (√X) gate

Definition

$$\sqrt{X} \equiv SX = \frac{1}{2}\begin{pmatrix} 1+i & 1-i \\ 1-i & 1+i \end{pmatrix}$$

Squaring the matrix directly confirms $SX \cdot SX = X$: it is a genuine matrix square root of the Pauli-X gate, and one of exactly four square roots of $X$ up to global phase (the others being $-SX$, $SX^\dagger$ scaled, etc.).

Action on basis states

$$SX|0\rangle = \tfrac12\big[(1+i)|0\rangle + (1-i)|1\rangle\big], \qquad SX|1\rangle = \tfrac12\big[(1-i)|0\rangle + (1+i)|1\rangle\big]$$

Both outcomes are measured with equal probability $1/2$; the gate's effect is entirely in the phase relationship between the two amplitudes, which a second $SX$ resolves back into a deterministic bit flip.

Key identities

• $SX = e^{i\pi/4} R_X(\pi/2)$: up to global phase, $SX$ is precisely the $R_X$ rotation halfway to a full $X$ flip.
• $SX$ is a Clifford gate: conjugating any Pauli operator by $SX$ returns a Pauli operator (up to sign), e.g. $SX \cdot Z \cdot SX^\dagger = -Y$. This is why $SX$, together with $R_Z$ and $\mathrm{CX}$, forms a common native gate set for Clifford-heavy compilation.
• $SX^\dagger \cdot SX = I$ and $SX \cdot SX = X$, so $SX^4 = X^2 = I$ — $SX$ has order 4.
• Combined with virtual $R_Z$ frame changes, two calibrated $SX$ pulses (an $R_X(\pi/2)$–$R_Z$–$R_X(\pi/2)$ sequence) synthesize any single-qubit unitary, which is why $SX$ rather than a general $R_X(\theta)$ is the physically native gate on many devices.