# A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem

@article{Kuperberg2005ASQ, title={A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem}, author={Greg Kuperberg}, journal={SIAM J. Comput.}, year={2005}, volume={35}, pages={170-188} }

We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity $2^{O(\sqrt{\log\ N})}$. In this problem an oracle computes a function $f$ on the dihedral group $D_N$ which is invariant under a hidden reflection in $D_N$. By contrast, the classical query complexity of DHSP is $O(\sqrt{N})$. The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group.
The algorithm begins as usual with a quantum character…

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## References

SHOWING 1-10 OF 29 REFERENCES

### Groups with Representations of Bounded Degree

- MathematicsCanadian Journal of Mathematics
- 1949

Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that…

### Noise-tolerant learning, the parity problem, and the statistical query model

- Computer ScienceSTOC '00
- 2000

The paper describes a slightly sub-exponential time algorithm for learning parity functions in the presence of random classification noise and shows that any class of functions learnable (strongly or weakly) with t-wise queries for t = O(log n) is also weakly learnable with standard unary queries.

### Quantum computation of Fourier transforms over symmetric groups

- MathematicsSTOC '97
- 1997

A quantum polynomial time algorithm for the Fourier transform for the symmetric groups Sn is given, adapting results obtained by Clausen and Diaconis–Rockmore to the quantum setting.

### 20世紀の名著名論：Peter Shor : Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

- Environmental Science
- 2004

Shorの論文は,そのアルゴリズム開発でのひらめき, 理論的解析の深さと美しさだけでも名論文に値するもの だが,最も重要な点は 21世紀の直前に新しいコンピュー タモデルとしての量子力学原理を情報処理に用いた量子 コンピュータの研究を先導したということだ.色々な観 点はあるかもしれないが,この論文が 21世紀で量子計 算と呼ばれている分野を爆発的に広め,いま少し歴史の…

### Linear representations of finite groups

- MathematicsGraduate texts in mathematics
- 1977

Representations and characters: generalities on linear representations character theory subgroups, products, induced representation compact groups examples. Representations in characteristic zero:…

### Quantum measurements and the Abelian Stabilizer Problem

- Mathematics
- 1995

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on…

### Optimal On-Line Simulations of Tree Machines by Random Access Machines

- Computer ScienceSIAM J. Comput.
- 1992

This paper shows that every tree machine of time complexity t can be simulated on-line by a log-cost random access machine (RAM) of time complexity $O((t\log t)/\log \log t)$. Using…

### Quantum computation and quantum information

- PhysicsMathematical Structures in Computer Science
- 2007

This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal…

### On Time Versus Space

- Computer ScienceJACM
- 1977

The context-sensitive languages cannot be recognized in linear time by deterministic multitape Turing machines, and are strictly contained in the class of languages recognized by Turing machines of tape complexity.

### Generic quantum Fourier transforms

- MathematicsSODA '04
- 2004

This paper uses Bratteli diagrams, Gel'fand-Tsetlin bases, and strong generating sets of small adapted diameter to provide efficient quantum circuits for the QFT over a wide variety of finite Abelian and non-Abelian groups, including all group families for which efficient QFTs are currently known and many new group families.