A checked pair of stages can return a state to a basis outcome. 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 second stage receives a balanced state

A balanced input to a recombination gate can be checked directly by the gate box.

H+=0H\lvert +\rangle=\lvert 0\rangle
Recombination self checkThe H box recomputes its output.state1/sqrt(2) zero1/sqrt(2) onestate1 zero0 oneHgate

The paired-stage rows return to a basis state

The first row is balanced, the middle row is the checked gate action, and the final row is a basis outcome.

rowpzeroponeinput1212gate10checked10\begin{array}{c|c|c}\text{row}&p_{\text{zero}}&p_{\text{one}}\\\text{input}&\frac{1}{2}&\frac{1}{2}\\\text{gate}&1&0\\\text{checked}&1&0\\\end{array}
Paired-stage scanThe H stage computes the return row.state1/sqrt(2) zero1/sqrt(2) onestate1 zero0 oneHgate

Paired stages can undo each other

The checked output is zero with probability 1. This is a stage self-check, not a device calibration curve.

pzero=1p_{\text{zero}}=1
Paired-stage checkThe output ket is computed from the input.state1/sqrt(2) zero1/sqrt(2) onestate1 zero0 oneHgate