= Bernstein-Vazirani decoding controlled by a quantum register
{c}
{title2=$|z\rangle|y\rangle\mapsto|z\oplus a_y\rangle|y\rangle$}
= Coherent Bernstein-Vazirani decoding
{synonym}
For a <Boolean quantum oracle> whose function is $g(x,y)=a_y\cdot x$, place its target in $|{-}\rangle$. Conjugating the oracle by a <Walsh-Hadamard transform> on the $x$ register translates that register by $a_y$. This is <Bernstein-Vazirani phase kickback> without measuring its output, so the index $y$ may remain in an arbitrary quantum superposition. A second application undoes the translation. A phase evaluation between the two calls therefore transfers a hidden-label-dependent phase to the index register while <uncomputation> returns the decoding register to zero.
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