Grover's algorithm

In quantum computing, Grover's algorithm, also known as the quantum search algorithm, is a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just ${\displaystyle O({\sqrt {N}})}$ evaluations of the function, where ${\displaystyle N}$ is the size of the function's domain. It was devised by Lov Grover in 1996.^[1]

The analogous problem in classical computation would have a query complexity ${\displaystyle O(N)}$ (i.e., the function would have to be evaluated ${\displaystyle O(N)}$ times: there is no better approach than trying out all input values one after the other, which, on average, takes ${\displaystyle N/2}$ steps).^[1]

Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function ${\displaystyle \Omega ({\sqrt {N}})}$ times, so Grover's algorithm is asymptotically optimal.^[2] Since classical algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for unstructured search, this suggests that Grover's algorithm by itself will not provide polynomial-time solutions for NP-complete problems (as the square root of an exponential function is still an exponential, not a polynomial function).^[3]

Unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts, Grover's algorithm provides only a quadratic speedup. However, even quadratic speedup is considerable when ${\displaystyle N}$ is large, and Grover's algorithm can be applied to speed up broad classes of algorithms.^[3] Grover's algorithm could brute-force a 128-bit symmetric cryptographic key in roughly 2⁶⁴ iterations, or a 256-bit key in roughly 2¹²⁸ iterations. It may not be the case that Grover's algorithm poses a significantly increased risk to encryption over existing classical algorithms, however.^[4]