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Finding conjugate stabilizer subgroups in PSL(2; q) and related groups

We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an almost 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as PSL(2; q).

Different subgroups yield different coset states, which must then be distinguished by some series of quantum measurements.
For abelian subgroups, sampling these states in the Fourier basis of the group G is sufficient to completely determine a hidden subgroup in an efficient manner. For nonabelian subgroups, the Fourier basis takes the form {|ρ, i, j } where ρ is the name of an irreducible representation and i and j index a row and column in a chosen basis. Although a number of interesting results have been obtained on the nonabelian HSP, the groups for which efficient solutions are known remain woefully few. Friedl, Ivanyos, Magniez, Santha, and Sen solve a problem they call the Hidden Translation Problem, and thus generalize this further to what they call "smoothly solvable" groups: these are solvable groups whose derived series is of constant length and whose abelian factors are each the direct product of an abelian group of bounded exponent and one of polynomial size [4]. Moore, Rockmore, Russell, and Schulman give an efficient algorithm for the affine groups AGL(1; p) = Z p ⋊ Z * p , and more generally Z p ⋊ Z q where q = (p − 1)/polylog(p). Bacon, Childs, and van Dam derive algorithms for the Heisenberg group and other "nearly abelian" groups of the form A ⋊ Z p , where A is abelian, by showing that the "Pretty Good Measurement" is the optimal measurement for distinguishing the corresponding coset states [1]. Recently, Ivanyos, Sanselme, and Santha [7,8] give an efficient algorithm for the HSP in nilpotent groups of class 2.
However, for groups of the greatest algorithmic interest, such as the symmetric group S n for which solving the HSP would solve Graph Isomorphism, the hidden subgroup problem appears to be quite hard. Moore, Russell, and Schulman showed that the standard approach of Fourier sampling individual coset states fails [14]. Hallgren et al. showed under very general assumptions that highly-entangled measurements over many coset states are necessary in any sufficiently nonabelian group [6]. For S n in particular, Moore, Russell andŚniady showed that the main proposal for an algorithm of this kind, a sieve approach due to Kuperberg [9], cannot succeed [15].
It is tempting to think that the difficulty of the HSP on the symmetric group is partly due to the appearance of the alternating group A n as a subgroup. For n ≥ 5, A n forms one of the families of nonabelian finite simple groups. All known algorithmic techniques for the HSP work by breaking the group down into abelian pieces, as a semidirect product or through its derived series. Since simple groups cannot be broken down this way, it seems that any positive results on the HSP for simple groups is potentially valuable.
We offer a small advance in this direction. We show how to efficiently solve a restricted case of HSP for the family of finite simple groups PSL(2; q), and for two related finite groups of Lie type. No new quantum techniques are introduced; instead, we point out a group-theoretic reduction to a mild extension of a previously solved case of the HSP. Unfortunately, this reduction only applies to one set of subgroups, and there is no obvious generalization that covers the other subgroups. On the other hand, we show that a similar reduction works in any group which acts on some set in a sufficiently transitive way, though this is unhelpful in many obvious cases.

Reduction
We start with a trivial observation: suppose we have a restricted case of the hidden subgroup problem where we need to distinguish among a family of subgroups H 1 , ..., H t ⊂ G. If there is a subgroup F whose intersections K i = H i ∩F are distinct, then we can reduce the original hidden subgroup problem to the corresponding one on F , consisting of distinguishing among the K i , by restricting the oracle to F , rather than the original domain G.
The subgroups in question will be the stabilizers of one or more elements under a suitably transitive group action. Recall the following definitions: Definition 1. A group action of a group G on a set Ω is a homomorphism φ from G to the group of permutations on Ω. In other words, When the group action is understood, we will often write just g 1 (x) for φ(g 1 )(x).

Definition 2.
A transitive group action on a set Ω is one such that for any α, β ∈ Ω there is at least one g ∈ G such that g(α) = β. A k-transitive group action is one such that any k-tuple of distinct elements (α 1 , . . . , α k ) can be mapped to any k-tuple of distinct elements (β 1 , . . . , β k ). That is, given that α i = α j and β i = β j only when i = j, there is at least one g such that g(α i ) = β i for all i = 1, . . . , k A group is called k-transitive if it has a k-transitive group action on some set. Definition 3. Given an element α ∈ Ω, the stabilizer of α with respect to a given action by a group G is the subgroup G α = {g ∈ G | g(α) = α}. Given a subset S ⊆ Ω, the pointwise stabilizer is When S is small we will abuse notation by writing, for instance, G α or G α,β .
Let's consider the case of the HSP where we wish to distinguish the one-point stabilizers G α from each other. If G is transitive, these are conjugates of each other, since G β = gG α g −1 for any g such that g(α) = β. Conversely, gG α g −1 = G g(α) , so any conjugate of a stabilizer is a stabilizer. Similarly, for each α, the two-point stabilizers G α,β labeled by β are conjugate subgroups in G α . Now suppose we restrict our queries to the oracle to G α . We then get a coset state corresponding to G α ∩ G β = G α,β : This reduces the problem of distinguishing the one-point stabilizers G β , as subgroups of G, to that of distinguishing the two-point stabilizers G α,β as subgroups of G α -a potentially easier problem. Note that we can test for the possibility that α = β with a polynomial number of classical queries, since we just need to check that f (1) = f (g) for a set of O(log |G|) generators of G α . Of course, this whole procedure is only useful if G α,β are distinct when G β are distinct, or if there are only a (polynomially) small number of one-point stabilizers corresponding to each two-point stabilizer. Below we give sufficient conditions for this to be true, and use this reduction to give an explicit algorithm for distinguishing conjugates of the Borel subgroups in some finite groups of Lie type, including the finite simple groups PSL(2; q). Using the transitivity of the group action we can bound the size of these stabilizers relative to each other and to the original group, and hence show that they are distinct.
In particular, .
Proof. The index of G S in G is the number of cosets. There is one coset for each j-tuple to which we can map S, and since G is j-transitive this includes all s!/(s − j)! ordered j-tuples.
For groups that are at least 3-transitive, the following then holds: the intersection of two subgroups that are single-point stabilizers of Ω has size 1/(s − 1) of both of the subgroups. The intersection with a third stabilizer subgroup is 1/(s − 2) this size again. In particular, this means that when subgroups G β and G γ are distinct, then their intersections G α ∩ G β = G α,β and G α ∩ G γ = G α,γ are distinct, because their intersection G α,β ∩ G α,γ = G α,β,γ is smaller than either.
In fact, we don't need full 3-transitivity for this argument to hold. The crucial fact we used was that the number of cosets of G α,β,γ was greater than G α,β or G α,γ . Consider the following definition:

Definition 4. A group is almost k-transitive if there is a constant b such that G has an action on a set
Ω which is (k − 1)-transitive, and such that we can map any k-tuple of distinct elements (α 1 , . . . , α k ) to at least a fraction b of all ordered k-tuples (β 1 , . . . , β k ) of distinct elements.
Strictly speaking, there is a different notion of "almost" for different values of b. Obviously, for any group there is some value of b low enough that this definition applies. However, by fixing b and considering a family of groups we still have a useful concept.
As an example, a group action is k-homogeneous if any set of points of size k can be mapped (setwise) to any other set of the same size. Since this means that any ordered k-tuple can be mapped to at least 1/k! of the ordered k-tuples, and since all k-homogeneous group actions are (k − 1)-transitive [3], a group with such an action is almost k-transitive with b = 1/k! (in fact, with b = 1/k).
Applying the above argument to almost 3-transitive groups shows that the stabilizer of 3 distinct elements is smaller than the stabilizer of 2 distinct elements by a factor of (s − 2)b. So long as b ≥ 1/(s − 2), two-point stabilizers of distinct elements will be distinct. In the group families we cover, b = 1/2, and s grows.

Families of transitive groups
Which families of groups and subgroups have the kind of transitivity that let us take advantage of this idea?
Unfortunately, not many do. We can categorize based on faithful group actions, i.e., those that do not map any group element other than the identity to the trivial action. Any non-faithful group action corresponds to a faithful action of a quotient of the group. Even the requirement of 2-transitivity in faithful group actions restricts the choices to a few sporadic groups, or one of eight infinite families [3]: The symmetric group S n , the alternating group A n , and six different families of groups of Lie type.
Obviously the symmetric group S n is n-transitive, and the alternating group A n is almost n-transitive. However, the size n of the set these groups act on is only polynomially large (i.e., polylogarithmic in the size of the groups) so we can distinguish the one-point stabilizers with a polynomial number of classical queries.
The other infinite families are finite groups of Lie type which are defined in terms of matrices over finite fields F q subject to some conditions. These groups have natural actions by matrix multiplication on column vectors, or on equivalence classes of column vectors. The actions of most of these groups are rather complicated to describe; for more details, see [3, §7.7]. Of these, two are 3-transitive: PSL(2; q), and AGL(d; 2).
There are also a number of sporadic finite groups that are up to 5-transitive, such as the Matthieu groups M 11 , M 12 , M 22 , M 23 , M 24 , built on finite geometries. However, an interesting fact is that if a group action has a threshold of transitivity, then it contains all permutations, or at least all even ones: for k > 5, all finite groups with a k-transitive action on a set of size n must contain A n [3].

PSL(2; q) and some relatives
The most interesting family of simple groups with a faithful almost 3-transitive group action is PSL(2; q). To discuss it, consider instead GL(2; q), the group of invertible 2 × 2 matrices with entries in the finite field F q , where q = p n is the power of some prime p. Its elements are of the form α β γ δ where α, β, γ, δ ∈ F q , and αδ − βγ = 0. We will assume that q is odd; some details change when it is a power of 2, but the basic results still hold.
A little thought reveals that |GL(2; q)| = (q 2 − 1)(q 2 − q) = (q + 1)q(q − 1) 2 . The subgroup SL(2; q) consists of the matrices with determinant 1, so |SL(2; q)| = (q + 1)q(q − 1). If we take the quotient of these groups by the normal subgroup consisting of the scalar matrices, we obtain PGL(2; q) and PSL(2; q) respectively. For SL(2; q) the only scalar matrices are ±1, so |PSL(2; q)| = (q + 1)q(q − 1)/2. GL(2; q) and SL(2; q) act naturally on nonzero 2-dimensional vectors. For PGL(2; q) and PSL(2; q), we must identify vectors which are scalar multiples. This identification turns F 2 q − {0, 0} into the projective line PF q . Each element of PF q corresponds to a "slope" of a vector: the vector x y has slope x/y, i.e., xy −1 if y = 0 and ∞ if y = 0. Thus we can think of PF q as F q ∪ {∞}, and it has q + 1 elements. The action of PGL(2; q) and PSL(2; q) on PF q is given by This fractional linear transformation is analogous to the Möbius transformation defined by PGL 2 (C): which can map any 3 points in the complex projective line PC (i.e., the complex plane augmented by the point at infinity, or the Riemann sphere) to any other 3 points. When we replace C with the finite field F q , the action of PGL(2; q) remains 3-transitive. The action of PSL(2; q) is 2-transitive, but cannot be 3-transitive, since there are half as many elements as there are 3-tuples. However, PSL(2; q) is almost 3-transitive in the sense defined above with b = 1/2, since 1/2 of all 3-tuples can be reached. SL(2; q) is also almost 3-transitive: from a given tuple, it reaches the same set of tuples as PSL(2; q), with each tuple being hit twice. As a result, this action is obviously not faithful, for the kernel is ±1.

An efficient algorithm for distinguishing the conjugates of the Borel subgroup
Now consider the case of the HSP on these groups where the hidden subgroup is one of B's conjugates, or equivalently, one of the one-point stabilizers G s . As discussed above, we solve this by restricting the oracle to B, and distinguishing the two-point stabilizer subgroups B ∩ G s = G s,∞ as subgroups of B. To do this, we need to describe the structure of B explicitly. For all three families of matrix groups we discuss, namely SL(2; q), PSL(2; q), and PGL(2; q), B is closely related to the affine group.
In PGL(2; q) a generic representative of B can be written α β 0 1 , α = 0, so |B| = q(q − 1). This is exactly the affine group AGL(1; q) ∼ = F q ⋊ F * q . To see this, recall that AGL(1; q) consists of the set of affine functions on F q of the form x → αx + β under composition. Now consider B's action on PF q − {∞}, which we (re)identify with F q . For PGL(2; q), we have α β 0 1 Obviously these elements compose as AGL(1; q), so B ∼ = AGL(1; q). The cases of SL(2; q) and PSL(2; q) are more complicated. The unit determinant requirement limits B to elements of the form α β 0 α −1 . Thus |B| = q(q − 1) again in SL(2; q). For PSL(2; q) we identify α with −α, so |B| = q(q − 1)/2. For SL(2; q), we can enumerate the elements as: Composing two such elements gives us: Here, we still have a semidirect product of the groups F q and F * q . Unlike the affine group, where the multiplicative group acts directly as an automorphism on the additive group by multiplication, it instead acts "doubly" by multiplying twice, analogous to the "q-hedral" groups in [13] (with q = p/2, in their notation). Finally, PSL(2; q) merely forgets the difference between ±α. This quotient group of SL(2; q) can also be seen as a subgroup of the affine group that can only multiply by the square elements.
In all three cases the HSP on B can be solved efficiently using small generalizations of the algorithms of [13]. We need to generalize slightly as [13] deals only with the case of Z n ⋉ F p with p prime -not a prime power q = p n , as here. The basic methods remain effective, though we construct and analyze a slightly different final measurement. The number and size of the representations remains the same (with q replacing p), and the methods for constructing Gelf'and-Tsetlin adapted bases are similar. As this has not been published in the literature, we describe the details more fully in the next section, though only what is necessary for our purposes.
6 Generalizing the affine group to the prime power case Although there can be more types of subgroups than the ones covered in [13], we are only concerned about one particular type whose analog was covered there: H = (a, 0) and its conjugates H b = (1, b)H(1, −b), stabilizing the finite field element b. The representation theory is analogous, with q − 1 one-dimensional representations (characters) depending only on a. As in the prime case, we have q conjugacy classes: the identity, all pure translations, and each multiplication by a different a, combined with all translations. This leaves us with one (q − 1) dimensional representation, ρ.
In the prime case we had: where ω p = exp(2πi/p). The roots of unity are the non-trivial additive characters of F p , indexed by j, evaluated at b. We can extend this to the prime power case simply by replacing bj, with b · j b · j = Tr bj = Tr F p n /Fp bj = n m=1 (bj) p m , which as b varies, exactly covers the full set of non-trivial linear operators from F p n to F p , and ω b·j p exactly covers the set of additive characters. Performing weak measurement on the coset state yields ρ with probability P (ρ) = 1 − 1/q. Conditioned on that outcome, we get the following projection operator: As in [13], we then perform a Fourier transform on the rows, and ignore the columns. There they performed the Fourier transform over Z p−1 , as there were p − 1 rows. However, the structure for general q is not Z * q ≡ Z q−1 , but F * q . The interaction we want to capture is the additive one, not the multiplicative one. We can still perform the abelian transform over the additive group F q ≡ Z n p -the zero component we lack is, of course, zero. The probability of observing a frequency ℓ ∈ Z n p is then: For the case of B in PSL(2; q), we can analyze the equivalent measurements via the embedding in the full affine group, just as in the prime case. Let a be a generator of the "even" multiplicative subgroup of F * q , consisting of elements that are squares. H b a is then elements of the form (a t , (1 − a t )b) stabilizing b. For these subgroups, the trivial representation, a "sign" representation, and the large representation occur with non-zero probability. The first two have vanishingly small probability, O(1/q).
In the following we use the notation G(m, a) = x∈F * q m(x)a(x) for the Gauss sum of a multiplicative and an additive character, where χ k (j) = ω k·j p is the additive character of F q with frequency k ∈ Z n p . We follow the common convention that non-trivial multiplicative characters vanish at 0. We use the quadratic character η of F * q , which is 1 for squares, and −1 for non-squares, to select rows and columns which differ by values in the "even" subgroup mentioned above.
Weak measurement gives us the representation ρ with overwhelming probability. Conditioning on this event, we get the mixed state Measuring the column k gives us, up to a phase, ρ(b) j = 2 q−1 ω b·j p (1 ± η(j))/2. We again include the zero component, with zero weight, and perform the abelian Fourier transform over the additive group F q ≡ Z n p . The probability of measuring frequency ℓ is where d is odd for odd n if p n ≡ 3 (mod 4), and d is even otherwise. For ℓ = b we have P (ℓ) = (q − 1) 2 /2q(q − 1) = (q − 1)/2q. For ℓ = b we have P (ℓ) = (q + 1)/4q(q − 1) if d is odd. If ℓ = b and d is even, we have P (ℓ) = (q ± 2q 1/2 + 1)/4q(q − 1). In any case, the probability of observing b is so repeating this measurement will allow us to identify ℓ = b with any desired probability. As SL(2; q) is a small extension of PSL(2; q), we can handle it similarly, by Theorem 8 in [13].

AGL(d; 2) and its stabilizer subgroups
An interesting question is whether it is useful to apply this approach to the other family of 3-transitive groups. This is the d-dimensional affine group AGL(d; 2), consisting of functions on F d 2 of the form Av + B, where A ∈ GL d (F 2 ) and B ∈ F d 2 . It can be expressed as a block matrix of the form A B 0 1 . It is the semidirect product GL(d; 2) ⋉ F d 2 , and hence obviously not simple. That it is triply transitive can be seen by realizing that the affine geometry it acts on has no three points that are collinear.
The stabilizer subgroups are 2 d conjugate subgroups of the original GL(d; 2). Obviously this stabilizes the point 0, and is the largest subgroup that will, as GL(d; 2) has two orbits: the zero vector, and all others. A general point P is stabilized by translating it to 0 with the element (A, B) = (1, P ), applying any element of GL(d; 2), and then translating back. To apply our method we need to look at the intersections.
As a result, distinguishing the stabilizers of points reduces to distinguishing conjugates of a smaller transposed copy of the affine group in the general linear group. This last reduction does not immediately yield an efficient new quantum algorithm.

Conclusion
It is interesting to note that although we can Fourier sample over AGL(d; 2) efficiently [12], we don't know how to do so in the projective groups. The fastest known classical Fourier transform for SL(2; q) or PSL(2; q) takes Θ(q 4 log q) time [10], and the natural quantum adaptation of this takes Θ(q log q) time [12]. If q is exponentially large, this is polynomial, rather than polylogarithmic, in the size of the group. In the absence of new techniques for the FFT or QFT, this suggests that we need to somehow reduce the HSP in PSL(2; q) to that in some smaller, simpler group-which was the original motivation for our work.
We conclude by asking whether our analysis of AGL(d; 2) can be extended to give an efficient algorithm distinguishing its stabilizer subgroups, or whether any of the other 2-transitive groups have usable "almost" 3-transitive actions.