Grover's Algorithm

Theory

Amplitude Amplification

The uniform superposition: $|u\rangle =H^{\otimes n}|0\rangle ^{n}$, where $H$ is the Hadamard gate.

Apply the oracle reflection $U_{w}|u\rangle =|u'\rangle$.

Apply an other reflection about the state $|u\rangle$, also $U_{u}=2|u\rangle \langle u|-1$. Thus we are at state $U_{u}U_{w}|u>$. This amplifies by two the amplitude of state $|w\rangle$.

Repeat $t<a/math>times,where<math>t\approx {\sqrt {N}}$.

References

https://medium.com/swlh/grovers-algorithm-quantum-computing-1171e826bcfb

https://quantumcomputinguk.org/tutorials/grovers-algorithm-with-code

https://jonathan-hui.medium.com/qc-grovers-algorithm-cd81e61cf248

https://arxiv.org/pdf/1804.03719.pdf