DeepAI
Log In Sign Up

Simon's problem for linear functions

10/29/2018
by   Joran van Apeldoorn, et al.
0

Simon's problem asks the following: determine if a function f: {0,1}^n →{0,1}^n is one-to-one or if there exists a unique s ∈{0,1}^n such that f(x) = f(x ⊕ s) for all x ∈{0,1}^n, given the promise that exactly one of the two holds. A classical algorithm that can solve this problem for every f requires 2^Ω(n) queries to f. Simon showed that there is a quantum algorithm that can solve this promise problem for every f using only O(n) quantum queries to f. A matching lower bound on the number of quantum queries was given by Koiran et al., even for functions f: F_p^n→F_p^n. We give a short proof that O(n) quantum queries is optimal even when we are additionally promised that f is linear. This is somewhat surprising because for linear functions there even exists a classical n-query algorithm.

READ FULL TEXT

page 1

page 2

page 3

page 4

11/20/2019

Lower Bounds for Function Inversion with Quantum Advice

Function inversion is that given a random function f: [M] → [N], we want...
02/14/2021

Simple vertex coloring algorithms

Given a graph G with n vertices and maximum degree Δ, it is known that G...
07/17/2019

Almost tight bound on the query complexity of generalized Simon's problem

Simon's problem played an important role in the history of quantum algor...
11/30/2017

Element Distinctness Revisited

The element distinctness problem is the problem of determining whether t...
01/22/2019

Solving linear program with Chubanov queries and bisection moves

This short article focus on the link between linear feasibility and gene...
03/14/2021

Quantum and Randomised Algorithms for Non-linearity Estimation

Non-linearity of a Boolean function indicates how far it is from any lin...
03/30/2021

A note about claw function with a small range

In the claw detection problem we are given two functions f:D→ R and g:D→...