Exact & formalGatesSingle-qubit gate2026-07-16

SX (√X) gate

A Clifford square root of the Pauli-X gate that is the native physical single-qubit gate (an X_{π/2} pulse) on most superconducting hardware.

Stabilizer

cliffordsingle qubithardware native

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SX matters less as an algorithmic primitive and more as a hardware fact: many transmon-qubit backends implement exactly one calibrated single-qubit pulse, and SX is that pulse.

Circuit & simulation

q[0]

SXSX

P(|0⟩) after SX50%

P(|1⟩) after SX50%

P(|1⟩) after SX·SX (=X)100%

What this takes and returns

Takes1 qubitWhat joins here

Takes a 1-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 1 qubits can feed it.

15 entries meet this end. Named below.

Returns1 qubitWhat joins here

Returns a 1-qubit register a next stage can take. This is the end that joins.

13 entries meet this end. Named below.

A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →

Fits before this 15

Fits after this 13

How it works

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.

Implementation

FrameworkNative

sx-sqrt-x.qasm

OPENQASM 3.0;
include "stdgates.inc";
qubit q;
sx q;

Quantum vs classical

Classical baseline

Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.

Quantum claim

The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.

How to compare

Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.

Declared gaps

Nobody has reviewed this record for gaps yet.

Literature & references

Efficient Z-Gates for Quantum Computing2016 · David C. McKay, Christopher J. Wood, Sarah Sheldon, Jerry M. Chow, Jay M. Gambetta

Shows how a fixed physical X_{π/2} pulse (the SX gate) combined with zero-duration virtual Z rotations synthesizes any single-qubit unitary on superconducting hardware.

arxiv.org/abs/1612.00858 ↗

Quantum Computation and Quantum Information: 10th Anniversary Edition2010 · Michael A. Nielsen and Isaac L. Chuang

Standard reference for single- and multi-qubit gate matrices, Euler-angle decomposition, and universal gate sets.

doi.org/10.1017/cbo9780511976667 ↗