# Quantum algorithms for algebraic problems

@article{Childs2008QuantumAF, title={Quantum algorithms for algebraic problems}, author={Andrew M. Childs and Wim van Dam}, journal={Reviews of Modern Physics}, year={2008}, volume={82}, pages={1-52} }

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a…

## 146 Citations

### Quantum Computation and Isomorphism Testing

- Computer Science
- 2015

It is shown that when the process is allowed to be randomized, there are problems that have a constant quantum query complexity but cannot be solved classically no matter how many queries are made, and how to construct such an infinity-vs-one separation from any weaker separation between the classical and quantum query complexities.

### Quantum Computing, Quantum Games and Geometric Algebra

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Geometric algebra (GA) was found to provide a simple algebraic solution to the exact Grover search problem as well providing a simple visual picture describing the general solution to Meyer’s quantum penny flip game, which is a simple two-player quantum game based on the manipulation of a single qubit and hence closely analogous to theGrover search process.

### Quantum Algorithm Implementations for Beginners

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This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations and shows how these algorithms can be implemented on IBM’s quantum computer.

### A quantum algorithm to solve nonlinear differential equations

- Computer Science
- 2008

A quantum algorithm to solve sparse systems of nonlinear differential equations whose nonlinear terms are polynomials and its expected resource requirements are polylogarithmic in the number of variables and exponential in the integration time.

### Mathematics on a Quantum Computer

- Physics
- 2011

In this chapter we gave but a taste of these developments. The quantum counting algorithm is especially noteworthy since it combines ideas from both Grover’s algorithm and phase estimation. Moreover,…

### Hybrid Quantum-Classical Approach to Quantum Optimal Control.

- Physics, Computer SciencePhysical review letters
- 2017

It is shown that the most computationally demanding part of gradient-based algorithms, namely, computing the fitness function and its gradient for a control input, can be accomplished by the process of evolution and measurement on a quantum simulator.

### Quantum Chemistry in the Age of Quantum Computing.

- Physics, ChemistryChemical reviews
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This Review provides an overview of the algorithms and results that are relevant for quantum chemistry and aims to help quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.

### Quantum algorithms for highly non-linear Boolean functions

- Computer Science, MathematicsSODA '10
- 2010

New quantum algorithms are presented that solve the hidden shift problems for several well-known classes of bent functions in polynomial time and with a constant number of queries, while the classical query complexity is shown to be exponential.

### Quantum algorithm for solving hyperelliptic curve discrete logarithm problem

- Computer ScienceQuantum Inf. Process.
- 2020

A quantum algorithm is proposed for solving the HCDLP by applying the framework of quantum algorithm designed by Shor and the key of the algorithm is the realization of divisor addition, which can be efficiently realized on a quantum computer using the basic modular arithmetic operations.

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