H Gate Superposition - Quantum States & Qubits Step by Step

Start with a qubit prepared as zero, then pass it through one ideal H gate. The page shows the ledger in two visible moves: equal one-over-square-root-of-2 amplitudes first, then two one-half probability rows that still add to the full budget; real hardware adds calibration error, decoherence, readout noise, and finite-shot uncertainty.

The H gate creates equal amplitudes using exact one over square root of two. 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 H gate makes an equal superposition

Starting from zero, the H gate produces amplitudes one over square root of 2 and one over square root of 2.

H\lvert 0\rangle = \frac{1}{\sqrt{2}}\lvert 0\rangle + \frac{1}{\sqrt{2}}\lvert 1\rangle

Each equal branch has probability one half

The exact square of one over square root of 2 is 1/2.

\left(\frac{1}{\sqrt{2}}\right)^{2} = \frac{1}{2}

The equal branches still close the budget

The H output has two equal probability rows. Together they use the whole probability budget.

\begin{array}{c|c|c}P_{\text{zero}}&P_{\text{one}}&\text{sum}\\\frac{1}{2}&\frac{1}{2}&1\\\frac{1}{2}&\frac{1}{2}&1\\1&0&1\end{array}