The Hidden shift problem states: Given an

oracle An oracle is a person or agency considered to provide wise and insightful counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. As such, it is a form of divination. Description The wor ...

O

that encodes two functions

f

and

g

, there is an n-bit string

s

for which

g(x) = f(x + s)

for all

x

. Find

s

. Many functions, such as the

Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residue ...

and Bent functions, satisfy these constraints. With a

quantum algorithm In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite seq ...

that's defined as "

, s\rangle = H^ O_ H^ O_ H^, 0^\rangle

" where

H

is the

Hadamard gate The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...

and

\hat

is the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

g

, this problem can be solved in a polynomial number of queries to

O

while taking exponential queries with a classical algorithm. The difference between the

Hidden subgroup problem The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. The framework captures problems such as factoring, discrete logarithm, graph isomorphism, and the shortest vector problem. This makes it esp ...

and the Hidden shift problem is that the former focuses on the underlying

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

while the latter focuses on the underlying

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

or field.

References

{{reflist Quantum algorithms