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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |u\rangle} , also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_u = 2 |u\rangle \langle u| - 1} . Thus we are at state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_u U_w |u>} . This amplifies by two the amplitude of state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |w\rangle} .

Repeat Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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