Algorithm 4: Repeat the following steps $\mathcal{O}\left(\frac{2^n}{n}\right)$ times

Prepare an n-QSL-bit query register in the $(0,r)$ state, and an answer register containing one QSL-bit in the $(1,r_y)$ state. Apply a Walsh-Hadamard transform on the query register Apply the oracle Apply a Walsh-Hadamard transform on the query register Apply ${\mathcal{X}}^{\times n}$ to the query register. Apply an n-controlled $\mathcal{Z}$ to the query register. Apply ${\mathcal{X}}^{\times n}$ to the query register. Apply a Walsh-Hadamard transform on the query register Measure the query register to obtain a candidate y for $x^{*}$.   Check the output candidates for $x^{*}$ using the oracle, and return the candidate that outputs 1, if any.