Phase Stage Keeps Direct Counts - Quantum Computing Physical Bases: Control, Coupling, and Error Signals Step by Step

A phase stage can change the state relation while leaving direct zero/one probabilities unchanged, so a first readout can miss the stored phase. The equality of counts is exact only for the declared balanced state, ideal phase stage, and chosen basis; real devices add phase noise, decoherence, gate error, drift, and finite-shot uncertainty.

A phase stage can preserve direct probabilities while changing what later stages do. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A phase stage can hide from direct counts

After the phase stage, the direct zero and one probabilities remain 1/2 and 1/2.

p_{\text{zero}}=\frac{1}{2},\quad p_{\text{one}}=\frac{1}{2}

Direct counts stay fixed across the phase stage

The phase row changes the state relation, but the directly measured zero and one budgets still match the balanced row.

\begin{array}{c|c|c}\text{row}&p_{\text{zero}}&p_{\text{one}}\\\text{balanced}&\frac{1}{2}&\frac{1}{2}\\\text{phased}&\frac{1}{2}&\frac{1}{2}\\\text{total}&1&1\\\end{array}

Hidden does not mean lost

The phase relation is still in the state. It becomes visible only after another checked stage asks the right question.

\text{phase affects later recombination}