The 50% advanced information rule of the quantum algorithms

TL;DR

This paper demonstrates that quantum algorithms effectively require only half the classical evaluations needed with prior knowledge, explaining their exponential or quadratic speedups in structured and unstructured problems.

Contribution

It confirms the 50% advanced information rule applies to key quantum algorithms, linking quantum speedups to classical algorithms with partial prior knowledge.

Findings

In structured problems, classical algorithms with 50% information need only one evaluation.

In unstructured search, classical algorithms with 50% information require order 2^{n/2} evaluations.

Quantum algorithms achieve exponential or quadratic speedups by effectively utilizing this 50% information rule.

Abstract

The oracle chooses a function out of a known set of functions and gives to the player a black box that, given an argument, evaluates the function. The player should find out a certain character of the function through function evaluation. This is the typical problem addressed by the quantum algorithms. In former theoretical work, we showed that a quantum algorithm requires the number of function evaluations of a classical algorithm that knows in advance 50% of the information that specifies the solution of the problem. Here we check that this 50% rule holds for the main quantum algorithms. In the structured problems, a classical algorithm with the advanced information, to identify the missing information should perform one function evaluation. The speed up is exponential since a classical algorithm without advanced information should perform an exponential number of function evaluations. In unstructured database search, a classical algorithm that knows in advance 50% of the n bits of the database location, to identify the n/2 missing bits should perform Order(2 power n/2) function evaluations. The speed up is quadratic since a classical algorithm without advanced information should perform Order(2 power n) function evaluations. The 50% rule identifies the problems solvable with a quantum sped up in an entirely classical way, in fact by comparing two classical algorithms, with and without the advanced information.