= Heralded Pauli X correction using controlled-Z and measurements
Two angle-zero <one-bit teleportations> with outcomes $p,q$ implement $X^qH X^pH=X^qZ^p$ on an arbitrary input, up to <global phase>. A <Controlled-Z gate> with a fixed $|1\rangle$ <quantum ancilla> implements $Z$, so $Z^p$ removes the known $Z$ factor. The result is $X^q$ up to phase. With probability one half the desired <Pauli X gate> is applied; otherwise the input is unchanged. Repeating until $q=1$ gives an exact heralded correction with two attempts on average and almost-sure termination. It has no finite worst-case measurement count. A <Pauli frame> avoids this repeat-until-success procedure when only logical action or classical output statistics are required.
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