An Efficient Quantum Algorithm for the Hidden Subgroup Problem
over some Non-Abelian Groups
†D.N. GONC¸ ALVES1*, T.D. FERNANDES2 and C.M.M. COSME3
Received on December 20, 2015 / Accepted on May 2, 2017
ABSTRACT.The hidden subgroup problem (HSP) plays an important role in quantum computing because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exists a new efficient quantum algorithm for the HSP on groups
ZN ⋊ Zqs whereN is an integer with a special prime factorization,qprime number andsany positive integer.
Keywords:Quantum Algorithms, Hidden Subgroup Problem, Quantum Computational Group Theory.
1 INTRODUCTION
The most important problem in group theory in terms of quantum algorithms is called hidden subgroup problem (HSP) [14]. The HSP can be described as follows: given a groupG and a function f :G→X on some setX such that f(x)= f(y)iffx·H =y·Hfor some subgroup
H, the problem consists in determining a generating set forH by querying the function f. We say that the function f hides the subgroupH inGor that f separates the cosets ofH inG. A quantum algorithm for the HSP is said to be efficient when the running time isO(poly(log|G|)). There are many examples of efficient quantum algorithms for the HSP in particular groups [17, 18]. It is known that for finite abelian groups, the HSP can be solved efficiently on a quantum computer [14]. On the other hand, an efficient solution for a generic non-abelian group is not known. Two important groups in this context are the symmetric and the dihedral groups. An efficient algorithm for solving the HSP for the former group would imply in an efficient solution for the graph isomorphism problem [1, 2, 12, 8] and for the latter one would solve instances of the problem of finding the shortest vector in a lattice, which has applications in cryptography [16].
One way to design new quantum algorithms for the HSP is to investigate the structures of all subgroups of a given group, and then to find a quantum algorithm applicable to each subgroup
†Short version in: III CMAC-SE, Vit´oria, 2015
*Corresponding author: Demerson Nunes Gonc¸alves – E-mail: demerson.goncalves@cefet-rj.br 1Coordenac¸˜ao de Licenciatura em F´ısica, CEFET-RJ, Petr´opolis, RJ, Brasil.
structure. Following this, Inui & Le Gall presented an efficient quantum algorithm for the HSP on groups of the form Zpr ⋊ Zp with prime p and positive integerr[13]. Later, Cosme [6] presented an efficient quantum algorithm for the HSP inZpr ⋊φZps where pis any odd prime number,randsare positives integers and the homomorphismφis given by the roott pr−s+l+1 such thatr ≥2s−l. Subsequently, in [10] the authors presented an efficient quantum algorithm for the HSP inZp⋊Zqs, withp/q=poly(logp), wherep,qare distinct odd prime numbers and
sis an arbitrary positive integer. The caseZpr ⋊ Zqs withp,qdistinct odd prime numbers and
r,s>0 such that pr/qt =poly(logpr)was discussed in [11]. The parametert ∈ {0,1, . . . ,s} characterizes the group. The case t = 0 reduces to the abelian groupZpr ×Zqs and the case
t = 1 was addressed by the authors, with unsuccessful results. This work established, for the first time, a complete description of the structure of the subgroups ofZpr⋊ Zqs. Recently, using the algebraic structure of the subgroups ofZpr ⋊ Zqs, van Dam & Dey [7] presented a new quantum algorithm for the HSP overZpr ⋊ Zqs for all possible values oft ∈ {0,1, . . . ,s}by imposing a restriction on the parameters pandq: the relative sizes of subgroups are bounded by
pr/qt−j ∈O(poly(logpr)), where j∈ {0, . . . ,t−1}.
In this article, we describe a new efficient quantum algorithm to solve the HSP in the specific class of non-abelian groups, i.e., the semi-direct product groups of the formG = ZN ⋊ Zqs, where, N is factorized as pr1
1 . . .p rn
n and there exists a 1 ≤ k ≤ n such thatqt (qodd prime)
divides pk−1 andq does not dividepi−1 for alli =k. The parametert is related to the type
of the homomorphism that describes the group, as can be checked in Section 3. Using a similar approach presented in [5], we define an isomorphism betweenZN⋊ Zqs and the direct product ofZpr ⋊ψZqs with cyclic groups, reducing the HSP inGto similar HSPs, solutions which are already known.
This work is organized as follows: In Section 2, we review some fundamental notations and definitions of finite groups. In Section 3, we give the relevant definitions and results concerning the semi-direct product groups and explain its homomorphisms and their properties. In Section 4, we present our main result and we show that there exist an efficient quantum algorithm for the HSP in the groups. In Section 5, we draw our conclusions.
2 PRELIMINARIES
We begin by reviewing some fundamental notations and definitions of finite groups that will be used throughout the text. More details can be found in lots of textbooks of abstract algebra such as in [3, 9].
LetGbe a finite group. We use|G|to denote theorderofG. A nonempty subsetH of a group
G is called a subgroup ofG, denoted by H ≤ G, if H2 ⊆ H and H−1 ⊆ H, where H2 = {h1h2|h1,h2∈ H}andH−1= {h−1|h ∈ H}. For a subgroupHofGand every group element
g∈G, theleft cosetofH determined bygis the setg H = {gh,h ∈H}.
GorGis generated byM. A group generated by one element is called acyclic group. For an elementg∈ G, we call the order of the subgroupg theorderofg, denoted by ord(g), that is, ord(g)= | g |. One can show that ord(g)is the smallest positive integernsatisfyinggn=1.
Given two groupsGandH, a mapα:G→ His called ahomomorphismofGintoHif
α(ab)=α(a)α(b), ∀a,b∈G.
If the homomorphismαis a bijection, thenαis anisomorphismfromGontoH. In this case,
G andH are said to beisomorphic, denoted byG ∼= H. An isomorphism fromG to itself is called anautomorphismofG. We use Aut(G)to denote the set of automorphism ofG. For the composition of maps, Aut(G)is a group, called theautomorphism groupofG.
A subgroupN of a groupGis callednormal, denoted by NGif N g =g N,∀g ∈ G. The groupnZ, for any integer n, is a normal subgroup of the integer groupZ. The factor group Zn = Z/nZ = {0,1, . . . ,n−1}is called the (additive) group of integers modulo n. In this
group,(nZ)a.(nZ)b = (nZ)cif and only if a+b = c(modn). The unit group, denoted by Z∗
n, for any positive integern, is the group of invertible integers modn (i.e, thosea ∈ Zn with
gcd(a,n)=1).
Thedirect productof two groupsGandH, denoted byG×H, is the set{(g,h)|g ∈G,h ∈H}, where the multiplication operation is defined by(g,h)(g′,h′) = (gg′,hh′)for allg,g′ ∈ G
andh,h′ ∈ H. In the same way, one may define the direct product ofngroupsG1, . . . ,Gn as
G=G1×. . .×Gn.
3 SEMI-DIRECT PRODUCT GROUPS
The semi-direct product of two groupsG and H is defined by a homomorphism φ : H → Aut(G). The semi-direct productG⋊φ H is the set{(g,h) : g ∈ G,h ∈ H}with the group
operation defined as(g,h)(g′,h′)=(g+φ (h)(g′),h+h′). One can easily check that the group inversion operation satisfies(g,h)−1=(φ (−h)(−h),−h).
In this paper we consider the HSP on the semi-direct product groups G = ZN ⋊φ Zqs for positive integersN andsand odd prime numberq. We assume that the prime factorization of
N ispr1
1 . . .p rn
n and there exists a 1≤k ≤nsuch thatqt divides pk−1 andqdoes not divide
pi −1 for alli = k. The parametert ∈ {0,1, . . . ,s}characterizes the group as shown in the
following.
The elementsx = (1,0)and y = (0,1)generate the groupsZN ⋊φZqs. Since Aut(ZN)is isomorphic toZ∗
N, the homomorphismφis completely determined byα:=φ (1)(1)∈ Z
∗
N and
φ (b)(a) = aαb for alla ∈ ZN andb ∈ Zqs. Now, note thatφ (0) = φ (qs) : ZN → ZN is the identity element of the group Aut(ZN). Thenαq
s
= φ (qs)(1) = 1. If the elementα ∈
Z∗
N satisfies the congruence relationXq
s
= 1 mod N, then it defines the semi-direct product ZN⋊αZqs. In this case, we must have ord(α)=qt for some integer 0≤t ≤s. The caset =0 reduces to the direct productZN×Zqs, which is an abelian group. An efficient solution for the HSP is known for this case [14]. Sinceα∈ Z∗
phi-function[9]. Sinceϕ(N)= pr1−1
1 . . .p rn−1
n (p1−1) . . . (pn−1), we can choose the option
qt | pn−1 with no loss of generality.
For instance, let G = ZN⋊φZqs, withN =45125,q = 3 ands = 4. The homomorphism φcan be described by an elementα = 2626∈ Z∗N with order 32. Since 45125 = 192.53, the order ofαsatisfies 32|(19−1)and 3 ∤(5−1). ThenGis an example of group for which the Theorem 4.1 holds.
Let us consider the usual decompositionZN ∼=Zpr1 1
×. . .×Zprn
n , which can be found in quantum polynomial time [4]. Thus, the following isomorphism holds
ZN⋊φZqs ∼=(Z
pr11 ×. . .×Zprnn )⋊φZqs. (3.1)
The elements of(Zpr1
1
×. . .×Zprn
n )⋊φZqshave the form((a1, . . . ,an),b), where(a1, . . . ,an)∈
Zpr1
1
×. . .×Zprn
n andb ∈ Zqs. For eachb inZqs the elementφ (b)is an automorphism on
Zpr1
1
×. . .×Zprn
n such thatα=φ (1)(1)is an element inZ ∗
pr11×. . .×Z
∗
prnn of orderq
t. Note that
Zpri
i is isomorphic to the subgroup I1×Z
prii ×I2ofZp1r1 ×. . .×Zprnn , whereI1is the identity onZpr1
1
×. . .×Zpri−1
i−1
andI2is the identity on
Zpri+1
i+1
×. . .×Zprnn , for all i=1, . . . ,n.
Thus, we can identify an elementaiinZpri
i with the pointaiin I1×Z
prii ×I2such that it has an
integer valueaiin thei-th coordinate and 0′selsewhere.
Now we are ready to state the following results.
Lemma 3.1. Let Zpr1
1
×. . .×Zprn
n andZqs be finite abelian groups with distinct odd prime
numbers p1, . . . ,pn,q and positive integers r1, . . . ,rn and s. Define the semi-direct product
group(Zpr1
1
×. . .×Zprnn )⋊φZqs. Then, for each b∈Zqs and ai ∈Zprii there exists a ci ∈Zprii
such thatφ (b)(ai)=ci.
Proof. Letei be elements inZpr1
1 ×. . .×Zp
rn
n with all components equal zero except thei-th one which is 1. Because φ (b) ∈ Aut(ZN), it is enough to show thatφ (b)(ei) = di, for some
di ∈Zpri
i . In fact,φ (b)(ai)=φ (b)(aiei)=aiφ (b)(ei)=aidi =ci, for someci ∈Zprii . Now let us suppose thatφ (b)(ei)=(d1, . . . ,dn). Note that
(0, . . . ,0) = φ (b)(0, . . . ,0)=φ (b)(0, . . . ,0,pri
i ,0, . . . ,0)
= pri
i φ (b)(ei)=(pirid1, . . . ,priidn).
Then, for all j =1, . . . ,nwe have pri
i dj ≡ 0 modp rj
j and this implies thatdj ≡ 0 modp rj
j
for all j=i. Hence,φ (b)(ei)=(0. . . ,di,0, . . . ,0)=di as was to be shown.
The next lemma shows that there exists an isomorphism betweenZN⋊αZqsand the non-abelian groupZprn
Lemma 3.2.Let N be a positive integer with prime factorization pr1
1 . . .p rn
n and q an odd prime
such that q= piand s a positive integer. Define the semi-direct product group G=ZN⋊αZqs
for anα ∈ Z∗
N. Let t ∈ {1, . . . ,s}be the smallest positive integer such thatα qt
= 1. Let us assume that there exists a1 ≤ k ≤n such that qt | pk−1and q ∤ pi−1for all i = k. By
choosing k=n (WLOG) we have
ZN⋊φZqs ∼=(Z
pr1
1 ×. . .×Zprn−n−11
)×(Zpr ⋊ψZqs), (3.2)
for some homomorphismψfromZqs into the group of automorphisms ofZpr and p = pnand
r =rn.
Proof. Note thatφ (qs)is the identity mapI onZ
pr11 ×. . .×Zprnn . For alli =1, . . . ,n−1, follows from Lemma 3.1 thatei =φ (qs)(ei) =(0. . . ,ciq
s
, . . . ,0). Thenciq
s
= 1 modpri
i ,
which implies thatci is an element inZ∗
prii with orderq t′
, for somet′ ∈ {1, . . . ,s}. Let us
supposeci =1. Sinceqt
′
divides the order ofZ∗
prii and gcd(pi,q)=1, we have thatq
t′divides
pi−1. But that leads to an absurd, hencecimust be 1 andφacts trivially onZpr1
1
×. . .×Zprn−1
n−1 . Thus, there exists a homomorphismψfromZqs into the group of automorphisms ofZpr (p= pn andr=rn), such that for allb∈Zqs and all(a1, . . . ,an)∈Z
pr1 1
×. . .×Zprn
n we have
φ (b)(a1, . . . ,an)=(a1, . . . ,an−1, ψ (b)(an)). (3.3)
Now for two elementsg=((a1. . . ,an),b)andg′=((a′1. . . ,an′),b′)in(Zpr1 1
×. . .×Zprn
n )⋊φ
Zqs, the group operation is defined by
gg′ = ((a1, . . . ,an)+φ (b)(a′1, . . . ,a
′
n),b+b
′)
= (a1+a1′, . . . ,an−1+an′−1,an+ψ (b)(an′),b+b′). (3.4)
Define the map
Ŵ:ZN⋊φ Zqs →(Z
pr1 1
×. . .×Zprn−1
n−1
)×(Zpr ⋊ψZqs), (3.5)
such thatŴ(a1, . . . ,an,b)) =((a1, . . . ,an−1), (an,b)). The group operation inZpr ⋊ψZqs is (a,b)(c,d)=(a+ψ (b)(c),b+d)for alla,c∈Zpr andb,d ∈Zs
q. Note that
Ŵ(gg′) = Ŵ(a1+a1′, . . . ,an−1+an′−1,an+ψ (b)(an′),b+b′)
= ((a1+a′1, . . . ,an−1+an′−1), (an+ψ (b)(a′n),b+b′)
(an,b).ψ(an′,b′) )
= ((a1, . . . ,an−1), (an,b))((a′1, . . . ,an′−1), (an′,b′))
= Ŵ(g)Ŵ(g′). (3.6)
Thus,Ŵis an group homomorphism. One can easily see that Ŵis injective and from the fact that|ZN ⋊φ Zqs| = |(Z
pr1 1
×. . .×Zprn−1
n−1
)×(Zpr ⋊ψ Zqs)| = N qs we have that Ŵis an
Lemma 3.3. Let G1and G2be finite groups with relatively prime orders. If H is a subgroup of
G then H=H1×H2for some subgroups H1of G1and H2of G2.
Proof. Letπi : G1×G2 → Gi such thatπi(g1,g2) = gi,i = 1,2. For any subgroup H
ofG1×G2 defineH1 = π1(H) ≤ G1 andH2 = π2(H) ≤ G2. ThenH ≤ H1×H2. We
claim that H = H1×H2. In fact, if(h1,h2)∈ H1×H2it follows from definition of H1and
H2that there existsh′1∈ G1andh′2∈ G2such that(h1,h′2), (h′1,h2)∈ H. From the fact that
gcd(|G1|,|G2|) = 1 and by the Chinese remainder theorem [9], there exist integersr1 andr2
such that
r1≡1 mod|G1|
r1≡0 mod|G2|
and
r2≡0 mod|G1|
r2≡1 mod|G2|.
(3.7)
It follows from (3.7) that there are integersk1,k2,k3,k4such that
r1=k1|G1| +1
r1=k2|G2|
and
r2=k3|G1|
r2=k4|G2| +1.
Thus,
(h1,h′2)
r1 =(hr1
1,h
′r1
2 )=(h k1|G1|+1
1 ,h
′k2|G2|
2 )=(h1,e2)∈H
(h′1,h2)r2 =(h′1r2,hr22)=(h1′k3|G1|,h2k4|G2|+1)=(e1,h2)∈H
where e1 and e2 are the identities elements in the groups G1 and G2, respectively. Hence,
(h1,h2)=(h1,e2)(e1,h2)∈ H.
4 QUANTUM ALGORITHM FOR HSP INZN⋊φZqs
In this section we present an efficient quantum algorithm that can solve the HSP inZN⋊φZqs, whereN is factorized asN = pr1
1 . . .p rn
n and given a 1≤t ≤s, there exists a 1≤k ≤nsuch
thatqt | pk−1 andq ∤ pi−1 for alli =k.
Before stating our main theorem, let us introduce the last two intermediate results.
Proposition 4.1.Let G be a finite group, H a subgroup of G and f :G→ X the oracle function that hides H in G. For any subgroupG of G we have f = f
G : G→ X hides H= H∩G
inG.
Proof. We must show that f(a)= f(b)if and only ifaH=bH, for alla,b∈ G. In fact, let
a,b∈ Gsuch that f(a)= f(b). Since f hidesH inG,a H =b Hwhich impliesa =bh, for someh ∈ H. Sincea,b∈G, we haveh =b−1a ∈Gwhich impliesh ∈ H, henceaH=bH. Conversely, ifaH=bH, sinceaH⊂a H andbH⊂b Hwe havea H∩b H= ∅which implies
Although the Proposition 4.1 establish a very simple result to be verified, it has important appli-cations in solving the HSP. In fact, if there exists a subgroupGofGwhere the HSP can solved efficiently by a quantum computer, the Proposition 4.1 shows that is possible to obtain informa-tion aboutHby using the restriction of the hiding function f to the subgroupH=H∩G. Note that ifH ⊂Gthen the problem is completely solved.
An important consequence of the Proposition 4.1 follows below:
Corollary 4.1. Let G1and G2be finite groups with relatively prime orders. Then, an efficient
solution to the HSP over G1 and G2 implies in an efficient solution to the HSP over the direct
product G1×G2.
Proof. By Lemma 3.3, ifHis a subgroup ofG1×G2thenH= H1×H2for some subgroup
H1ofG1and subgroupH2ofG2. Let f be the oracle function that hidesH inG1×G2. By
Proposition 4.1, the restrictions of f toG1andG2hide, respectively,H1andH2. Since the HSP
can solved efficiently over the groupsG1 andG2 one can efficiently find generators toH1and
H2, or equivalently, generators toH1×H2.
Now we are able to state and prove our main result.
Theorem 4.1. Let N be a positive integer with prime factorization pr1
1 . . .p rn
n, q an odd prime
such that q= piand s a positive integer. Define the semi-direct product group G=ZN⋊αZqs
for anα ∈ Z∗
N. Let t ∈ {1, . . . ,s}be the smallest positive integer such thatαq
t
= 1. Let us assume that there exists a1≤ k ≤n such that qt | pk−1and q ∤ pi−1for all i = k. Then
there exists an efficient quantum algorithm that solves the HSP in the semi-direct product groups
ZN⋊αZqs.
Proof. DefineN′=N/prn
n thenZpr1 1
×. . .×Zprn−1
n−1
∼
=ZN′. By Lemma 3.2,
ZN⋊φZqs ∼=ZN′×(Zpr ⋊ψ Zqs).
The groupZN′ is an abelian and the HSP can be solved efficiently for abelian groups by quantum computers [14]. On the other hand, the groupZpr⋊ψZqs was addressed in [10, 11] and recently generalized by [7]. Since the order ofZN′ is relatively prime to order of the groupZpr ⋊ψZqs, by Corollary 4.1, the HSP overZN⋊ Zqs can be solved efficiently on a quantum computer. A series of efficient quantum algorithms for the non-abelian HSP over semi-direct product groups have been discovered. Among these, is the algorithm presented by Inui & Le Gall for groups of the formZpr ⋊ Zpwith prime pand positive integerr, which uses enumeration of subgroups and blackbox techniques. Chi, Kim & Lee [5] extended the algorithm to the caseZN ⋊ Zp,
whereN is factored asN = pr1
1 . . .p rn
n, and pprime does not divide each pj −1. The idea is
positives. Using a similar approach of [5], they extended their algorithm to the classZN⋊φZps. In [10], the authors presented an efficient quantum algorithm for the HSP over certain metacyclic groupsZp⋊ Zqs, withp/q =poly(logp), where p,q are distinct odd prime numbers andsis an arbitrary positive integer. This work was extended in [7], which developed an efficient HSP algorithm inZpr ⋊ Zqs, with p,qdistinct odd prime numbers andr,spositive integers.
All those class of groups are special cases of the semi-direct productsZM⋊ ZN, for any positive
integers M and N. In this sense, our result increases the number of groups in this family for which efficient solutions are known.
We hope that these ideas will be useful for the understanding of the complexity of the HSP over semi-direct product groups and lead to new algorithms for other non-Abelian HSP instances.
5 CONCLUSION
We have addressed the HSP on the semi-direct product groups G = ZN ⋊ Zqs where N is factorized as N = pr1
1 . . .p rn
n and given a 1 ≤ t ≤ s, there exists a 1 ≤ k ≤ n such thatqt
divides pk−1q ∤ pi−1 for alli =k. By employing an isomorphism betweenZN⋊ Zqs and the direct product ofZpr⋊ψZqs with cyclic groups we have shown that the HSP can be reduced to similar HSPs the solutions of which are already known. This provides a new efficient solution for the HSP onG.
ACKNOWLEDGEMENTS
We would like to tank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.
RESUMO.O problema do subgrupo oculto (PSO) tem um papel importante na computac¸˜ao
quˆantica pois muitos algoritmos quˆanticos que s˜ao exponencialmente mais r´apidos que seus equivalentes cl´assicos s˜ao casos especiais do PSO. Neste artigo n´os mostramos a existˆencia
de um novo algoritmo quˆantico eficiente para o PSO sobre grupos da formaZN⋊ Zqs, onde
N ´e um n´umero inteiro positivo com uma particular decomposic¸˜ao em fatores primos,qum n´umero primo esum inteiro positivo qualquer.
Palavras-chave: Algoritmos quˆanticos, problema do subgrupo oculto, teoria de grupos
computacional.
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