Figure 1

The schematic diagram of our scheme. Exploiting the operations \(U_{1}\) and \(U_{2}^{\dagger}\), the database and target data T are encoded into \(\vert D B ' \rangle \) and \(\vert T ' \rangle \), respectively. Then, the oracle marks the state that overlaps with \(\vert T ' \rangle \) by the multi-controlled-phase gate \(MCP\) and the NOT gate X. As for the diffusion operator, it rotates the phase of \(\vert 0 \rangle ^{\otimes ( n_{D} + n_{X} )}\). Notably, the phase rotations in the oracle and the diffusion operator are equal to satisfy the phase matching condition [33]. The oracle and diffusion operator together is referred to as G operator, which efficiently amplifies the amplitudes of the state similar to \(\vert T ' \rangle \). After applying G operator J times, the index \(i^{*}\) is acquired with high probability through measurement