Grover's algorithm, along with variants like amplitude amplification, can be used to speed up a broad range of algorithms. In particular, algorithms for NP … 展开

As input for Grover's algorithm, suppose we have a function $${\displaystyle f\colon \{0,1,\ldots ,N-1\}\to \{0,1\}}$$. In the "unstructured database" … 展开

There is a geometric interpretation of Grover's algorithm, following from the observation that the quantum state of Grover's algorithm stays in a two-dimensional subspace after each step. Consider the plane spanned by
Grover's algorithm … 展开

To complete the algebraic analysis, we need to find out what happens when we repeatedly apply $${\displaystyle U_{s}U_{\omega }}$$. A natural way to do this is by eigenvalue analysis of a matrix. Notice that during the entire computation, the state of the algorithm … 展开

Multiple matching entries
If, instead of 1 matching entry, there are k matching entries, the same algorithm works, but the number of iterations must be $${\textstyle {\frac {\pi }{4}}{\left({\frac {N}{k}}\right)^{1/2}}}$$instead of 展开

Grover's algorithm is optimal up to sub-constant factors. That is, any algorithm that accesses the database only by using the operator Uω must apply Uω at least a $${\displaystyle 1-o(1)}$$ fraction as many times as Grover's algorithm. The extension of Grover's algorithm to k … 展开

• Amplitude amplification
• Brassard–Høyer–Tapp algorithm (for solving the collision problem)
• Shor's algorithm (for factorization)
• Quantum walk search 展开