Hidden shift problem

In quantum computing, the hidden shift problem is a type of oracle-based problem. Various versions of this problem have quantum algorithms which can run much more quickly than known non-quantum methods for the same problem. In its general form, it is equivalent to the hidden subgroup problem for the dihedral group. ^[1] It is a major open problem to understand how well quantum algorithms can perform for this task, as it can be applied to break lattice-based cryptography. ^{[2] [3]}

Problem statement

The hidden shift problem states: Given an oracle ${\displaystyle O}$ that encodes two functions ${\displaystyle f}$ and ${\displaystyle g}$, there is an ${\displaystyle n}$-bit string ${\displaystyle s}$ for which ${\displaystyle g(x)=f(x+s)}$ for all ${\displaystyle x}$. Find ${\displaystyle s}$. ^[4]

Functions such as the Legendre symbol and bent functions, satisfy these constraints. ^[5]

Algorithms

With a quantum algorithm that is defined as ${\displaystyle |s\rangle =H^{\otimes n}O_{f}H^{\otimes n}O_{\hat {g}}H^{\otimes n}|0^{n}\rangle }$, where ${\displaystyle H}$ is the Hadamard gate and ${\displaystyle {\hat {g}}}$ is the Fourier transform of ${\displaystyle g}$, certain instantiations of this problem can be solved in a polynomial number of queries to ${\displaystyle O}$ while taking exponential queries with a classical algorithm.

References

1. Childs, Andrew M.; van Dam, Wim (2007), "Quantum algorithm for a generalized hidden shift problem", in Bansal, Nikhil; Pruhs, Kirk; Stein, Clifford (eds.), Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7-9, 2007, SIAM, pp. 1225–1232, arXiv: quant-ph/0507190
2. Lomont, Chris (November 4, 2004), The Hidden Subgroup Problem - Review and Open Problems, arXiv: quant-ph/0411037
3. Regev, Oded (January 2004). "Quantum Computation and Lattice Problems" . SIAM Journal on Computing. 33 (3): 738–760. doi:10.1137/S0097539703440678. ISSN   0097-5397.
4. Dam, Wim van; Hallgren, Sean; Ip, Lawrence (2002). "Quantum Algorithms for some Hidden Shift Problems". SIAM Journal on Computing . 36 (3): 763–778. arXiv: quant-ph/0211140 . doi:10.1137/S009753970343141X. S2CID   11122780.
5. Rötteler, Martin (2008). "Quantum algorithms for highly non-linear Boolean functions". Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms. Vol. 402. Society for Industrial and Applied Mathematics. pp. 448–457. arXiv: 0811.3208 . doi:10.1137/1.9781611973075.37. ISBN   978-0-89871-701-3. S2CID   9615826.