# Quantum Algorithms to Solve the Hidden Shift Problem for Quadratics and for Functions of Large Gowers Norm

@article{Rtteler2009QuantumAT, title={Quantum Algorithms to Solve the Hidden Shift Problem for Quadratics and for Functions of Large Gowers Norm}, author={Martin R{\"o}tteler}, journal={ArXiv}, year={2009}, volume={abs/0911.4724} }

Most quantum algorithms that give an exponential speedup over classical algorithms exploit the Fourier transform in some way. In Shor's algorithm, sampling from the quantum Fourier spectrum is used to discover periodicity of the modular exponentiation function. In a generalization of this idea, quantum Fourier sampling can be used to discover hidden subgroup structures of some functions much more efficiently than it is possible classically. Another problem for which the Fourier transform has…

## 16 Citations

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This thesis shows how certain types of random walk search algorithms can be transformed into quantum algorithms that search quadratically faster and builds a quantum algorithm that generalizes the rejection sampling method first formalized by von Neumann in 1951.

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It is demonstrated that the easiest instances of the Boolean hidden shift problem correspond to bent functions, in the sense that an exact one-query algorithm exists if and only if the function is bent.

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Hidden Symmetry Subgroup Problems

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A general method for reducing the HSSP to the HSP is presented, which works efficiently in several cases related to symmetries of polynomials, and connects in a rather surprising way certain hidden polynomial problems with the H SP.

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- Computer ScienceMathematical Structures in Computer Science
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This work discusses quantum algorithms based on the Bernstein–Vazirani algorithm for finding which input variables a Boolean function depends on, and describes quantum algorithms for learning the exact form of certain quadratic and cubic Boolean functions.

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By employing the newly defined concepts, it is shown that there exists a quantum computational algorithm which solves the continuous hidden shift problem on Rn with a continuous oracle function in time polynomial in n.

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- Computer Science, PhysicsScience
- 2018

It is shown that parallel quantum algorithms running in a constant time period are strictly more powerful than their classical counterparts; they are provably better at solving certain linear algebra problems associated with binary quadratic forms.

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- Computer ScienceDCM
- 2010

We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for finding which variables a Boolean function depends on. There are 2^n possible linear Boolean functions of n variables;…

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The Hidden Subgroup Problem

- Mathematics, Computer Science
- 2010

An overview of the Hidden Subgroup Problem (HSP) as of July 2010, including new results discovered since the survey of arXiv:quant-ph/0411037v1, and analyse Regev's algorithm for the poly(n)-uniqueSVP, proving how the degree of the polynomial is related to the oracle complexity used and suggesting several variants.

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