Gate Output Three-Input Scan - Quantum States & Qubits Step by Step

A gate output three-input scan runs the same checked X gate on three different normalized starting states, so the swap rule is verified against more than one worked example. The three rows swap (3/5, 4/5) to (4/5, 3/5), swap (4/5, 3/5) back to (3/5, 4/5), and swap (5/13, 12/13) to (12/13, 5/13), each closing its own probability budget to exactly one for the stated toy qubit ledger; real single-qubit gates have finite fidelity, control-pulse error, and decoherence during the operation.

Three exact Pythagorean input pairs pass through the same X gate, each swapping to its own checked output. 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 the first checked input

Chapter three checked one X input. Here three-fifths and four-fifths swap to four-fifths and three-fifths, the same single case already reviewed.

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

Three normalized inputs swap through the same gate

The second row swaps the same two numbers in the other order. The third row uses a different exact Pythagorean pair, five-thirteenths and twelve-thirteenths.

\begin{array}{c|c|c|c}a_{\text{in}}&b_{\text{in}}&a_{\text{out}}&b_{\text{out}}\\\frac{3}{5}&\frac{4}{5}&\frac{4}{5}&\frac{3}{5}\\\frac{4}{5}&\frac{3}{5}&\frac{3}{5}&\frac{4}{5}\\\frac{5}{13}&\frac{12}{13}&\frac{12}{13}&\frac{5}{13}\end{array}

Every row still closes its own probability budget

Swapping amplitudes swaps probabilities too, but each row's zero and one probabilities still sum to one, independent of the other rows.

P_{\text{zero}}+P_{\text{one}}=1