[Paper Review] Solution to the Hidden Subgroup Problem for a Class of Noncommutative Groups

Demerson Nunes Gonçalves, Renato Portugal|arXiv (Cornell University)|Apr 7, 2011

Quantum Computing Algorithms and Architecture13 references3 citations

TL;DR

This paper presents a polynomial-time quantum algorithm that solves the Hidden Subgroup Problem (HSP) for the non-abelian group $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_{q^s}$, under the condition $p^r/q = \text{poly}(\log p^r)$, where $p$ and $q$ are distinct odd primes and $r,s$ are positive integers. The algorithm reduces the HSP to finding cyclic subgroups using the abelian quantum Fourier transform and a structured measurement procedure, achieving success probability greater than $1/2$.

ABSTRACT

The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms can be casted in the HSP structure. In this paper, we present a new polynomial-time quantum algorithm that solves the HSP over the group $\Z_{p^r} times \Z_{q^s}$, when $p^r/q= \up{poly}(\log p^r)$, where $p$, $q$ are any odd prime numbers and $r, s$ are any positive integers. To find the hidden subgroup, our algorithm uses the abelian quantum Fourier transform and a reduction procedure that simplifies the problem to find cyclic subgroups.

Motivation & Objective

• To solve the Hidden Subgroup Problem (HSP) efficiently for a class of non-abelian groups $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_{q^s}$, extending known results for abelian and simpler non-abelian groups.
• To generalize previous quantum algorithms for HSP in $\mathbb{Z}_p\rtimes\mathbb{Z}_{q^s}$ and $\mathbb{Z}_N\rtimes\mathbb{Z}_q$ by allowing higher powers $p^r$ and $q^s$.
• To establish conditions under which the HSP in this group class can be reduced to the problem of finding cyclic subgroups via quantum Fourier transforms.
• To achieve a success probability greater than $1/2$ for identifying the hidden subgroup in polynomial time under the given parameter constraint.

Proposed method

• The algorithm uses the abelian quantum Fourier transform on the $\mathbb{Z}_{p^r}$ component to extract periodic structure from the function $f$ that hides the subgroup.
• It applies a reduction procedure that transforms the non-abelian HSP into a problem of identifying cyclic subgroups of order $q^t$, where $t \leq s$.
• A controlled unitary operation $U$ is applied after measurement to entangle the state with the hidden parameter $a$, leveraging injectivity of a function $S(n)$ over $\mathbb{Z}_{q^{t-j}}$.
• The inverse quantum Fourier transform is applied to a state encoding the parameter $a$, enabling its extraction with probability $q/p^r$ per run.
• The algorithm repeats $O(\text{poly}(\log p^r))$ times to amplify the success probability to greater than $1/2$.
• The construction relies on the group structure where $\alpha = u^{k}$ with $k = \frac{lp^{r-1}(p-1)}{q^t}$, ensuring $\alpha^{q^t} \equiv 1 \pmod{p^r}$, and $\text{ord}(\alpha) = q^t$.

Experimental results

Research questions

• RQ1 Can the HSP in $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_{q^s}$ be solved efficiently in quantum polynomial time under the condition $p^r/q = \text{poly}(\log p^r)$?
• RQ2 How can the non-abelian HSP be reduced to a problem of finding cyclic subgroups in this group class?
• RQ3 What role does the abelian quantum Fourier transform play in extracting the hidden subgroup structure when the group is non-abelian?
• RQ4 Why does the algorithm fail to generalize to $t > 1$ due to non-unitary behavior of the key operator $U$?
• RQ5 To what extent can this approach be extended to other non-abelian groups using non-abelian Fourier transforms?

Key findings

• The HSP in $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_{q^s}$ is solvable in polynomial time when $p^r/q = \text{poly}(\log p^r)$, generalizing prior results for $r=1$ or $s=1$.
• The algorithm reduces the HSP to finding cyclic subgroups of order $q^t$, with $t=1$ being the only case where a unitary operator $U$ can be constructed to extract the hidden parameter $a$.
• The success probability of measuring the correct parameter $a$ per run is $q/p^r$, and repeating the algorithm $O(\text{poly}(\log p^r))$ times increases the total success probability to greater than $1/2$.
• The algorithm achieves its goal using only the abelian quantum Fourier transform and a measurement-based reduction, avoiding the need for non-abelian Fourier transforms.
• The result generalizes previous work by Bacon et al. [10] (for $s=1$) and Gonçalves et al. [11] (for $r=1$), unifying both cases under a single framework.
• For $t > 1$, the lack of injectivity in the function $S(n)$ prevents the construction of a unitary operator $U$, limiting the algorithm to the case $t=1$.