X Gate Swap - Quantum States & Qubits Step by Step

The X gate is a two-state ledger operation that swaps the zero and one amplitudes before probabilities are read out. The swap is exact only for the stated input vector and ideal gate; real devices have calibration error, pulse distortion, decoherence, leakage, and readout uncertainty.

The X gate swaps the two amplitudes of a single qubit. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The X gate swaps amplitudes

The input amplitudes are 3/5 and 4/5. The checked X output swaps them.

X\begin{bmatrix}\frac{3}{5}\\\frac{4}{5}\end{bmatrix}=\begin{bmatrix}\frac{4}{5}\\\frac{3}{5}\end{bmatrix}

The probabilities swap too

The output probabilities are 16/25 for zero and 9/25 for one.

P_{\text{zero,out}} = \frac{16}{25},\quad P_{\text{one,out}} = \frac{9}{25}

Applying X twice returns the input

The swap is reversible. A second X swaps the amplitudes back to the original order.

X^2\begin{bmatrix}\frac{3}{5}\\\frac{4}{5}\end{bmatrix}=\begin{bmatrix}\frac{3}{5}\\\frac{4}{5}\end{bmatrix}