[Paper Review] Quantum algorithm for the hidden subgroup problem on a class of semidirect product groups

Carlos Magno Martins Cosme, Renato Portugal|ArXiv.org|Mar 23, 2007

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper presents efficient quantum algorithms for solving the hidden subgroup problem (HSP) on semidirect product groups $π_{p^{r}}\rtimes_{ϕ}\mathbb{Z}_{p^{2}}$ for odd primes $p$ and $r>4$, using reductions to the abelian HSP and direct quantum techniques. The algorithm achieves polynomial-time complexity in $\log|G|$, offering exponential speedup over classical methods.

ABSTRACT

We present efficient quantum algorithms for the hidden subgroup problem (HSP) on the semidirect product of cyclic groups $\Z_{p^r} times_ϕ\Z_{p^2}$, where $p$ is any odd prime number and $r$ is any integer such that $r>4$. We also address the HSP in the group $\Z_{N} times_ϕ\Z_{p^2}$, where $N$ is an integer with a special prime factorization. These quantum algorithms are exponentially faster than any classical algorithm for the same purpose.

Motivation & Objective

• To develop efficient quantum algorithms for the hidden subgroup problem (HSP) on non-abelian semidirect product groups $\mathbb{Z}_{p^r}\rtimes_{\phi}\mathbb{Z}_{p^2}$.
• To extend known HSP solutions beyond abelian and simple non-abelian cases to a broader class of solvable groups with structured automorphism actions.
• To address the HSP on $\mathbb{Z}_N\rtimes_{\phi}\mathbb{Z}_{p^2}$ where $N$ has a specific prime factorization, leveraging group isomorphisms.
• To achieve polynomial-time quantum complexity by reducing the problem to abelian HSP instances and using known techniques for normal subgroups.

Proposed method

• Classify all subgroups of $\mathbb{Z}_{p^r}\rtimes_{\phi}\mathbb{Z}_{p^2}$ using three parameters: $t$, $i$, and $j$, based on the homomorphism $\phi$.
• Use abelian HSP reductions by restricting the function $f$ to abelian subgroups $G$ to extract information about the hidden subgroup $H$.
• Apply direct quantum algorithms when $H$ is cyclic or has specific structure, particularly in the first class of groups where $\gcd(\tau, p^2) = 1$.
• Employ the method from Ivanyos et al. (2006) for normal subgroups when $H$ is not contained in any abelian subgroup, ensuring completeness.
• Reduce the HSP on $\mathbb{Z}_N\rtimes_{\phi}\mathbb{Z}_{p^2}$ to HSP on $\mathbb{Z}_{p_1^{r_1}}\rtimes_{\psi}\mathbb{Z}_{p^2}$ and cyclic factors via group isomorphism, exploiting coprime orders.
• Bound the overall complexity to $O(\text{poly}((r+2)\log p))$, ensuring polynomial runtime in the input size.

Experimental results

Research questions

• RQ1 Can the HSP be solved efficiently on $\mathbb{Z}_{p^r}\rtimes_{\phi}\mathbb{Z}_{p^2}$ for odd primes $p$ and $r>4$ using quantum algorithms?
• RQ2 How can the structure of subgroups in $\mathbb{Z}_{p^r}\rtimes_{\phi}\mathbb{Z}_{p^2}$ be leveraged to reduce the HSP to abelian HSP instances?
• RQ3 What conditions on $N$ allow the HSP on $\mathbb{Z}_N\rtimes_{\phi}\mathbb{Z}_{p^2}$ to be reduced to HSP on $\mathbb{Z}_{p^r}\rtimes_{\psi}\mathbb{Z}_{p^2}$?
• RQ4 Can the HSP on $\mathbb{Z}_{p^r}\rtimes_{\phi}\mathbb{Z}_{p^2}$ be solved with polynomial-time quantum complexity, and what is the success probability?

Key findings

• The HSP on $\mathbb{Z}_{p^r}\rtimes_{\phi}\mathbb{Z}_{p^2}$ is solvable in polynomial time, with complexity bounded by $O(\text{poly}((r+2)\log p))$.
• The algorithm uses reductions to the abelian HSP and direct quantum techniques, achieving exponential speedup over classical algorithms.
• For the group $\mathbb{Z}_N\rtimes_{\phi}\mathbb{Z}_{p^2}$, the HSP reduces to HSP on $\mathbb{Z}_{p_1^{r_1}}\rtimes_{\psi}\mathbb{Z}_{p^2}$ and cyclic groups when $p$ does not divide $p_i - 1$ for all $i$.
• The success probability of the algorithm exceeds $1/2$, and the method is probabilistic but efficient.
• The classification of subgroups into three parameter families enables systematic reduction strategies based on the structure of $H$.
• The results generalize prior work on $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_p$, extending it to $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_{p^2}$ with $r>4$.