Generic Quantum Fourier Transforms

GENERIC QUANTUM FOURIER TRANSFORMS

Cristopher Moore Daniel Rockmore University of New Mexico Dartmouth College [email protected] [email protected] Alexander Russell University of Connecticut [email protected]

Abstract problems over Abelian groups are well-understood, for The quantum Fourier transform (QFT) is the principal non-Abelian groups our understanding of these problems ingredient of most efficient quantum algorithms. We remains embarrassingly sporadic. Aside from their natu- present a generic framework for the construction of ral appeal, these lines of research are motivated by their efficient quantum circuits for the QFT by “quantizing” direct relationship to the graph isomorphism problem: an the highly successful separation of variables technique efficient solution to the hidden subgroup problem over the for the construction of efficient classical Fourier trans- (non-Abelian) symmetric groups would yield an efficient forms. Specifically, we apply the existence of computable quantum algorithm for graph isomorphism. Bratteli diagrams, adapted factorizations, and Gel’fand- Over the cyclic group Zn, the quantum Fourier Tsetlin bases to provide efficient quantum circuits for the transform refers to the transformation taking the state ∑ f (z)|zi to the state ∑ω fˆ(ω)|ωi, where QFT over a wide variety of finite Abelian and non-Abelian z∈Zn ∈Zn ˆ ω groups, including all group families for which efficient f : Zn → C is a function with k f k2 = 1 and f ( ) = 2πiωz/n QFTs are currently known and many new group families. ∑z f (z)e denotes the familiar discrete Fourier trans- Moreover, our method provides the first subexponential- form at the frequency ω. Over an arbitrary finite group size quantum circuits for the QFT over the linear groups G, this analogously refers to the transformation taking the ∑ ∑ ˆ ρ ρ GLk(q), SLk(q), and the finite groups of Lie type, for any state z∈G f (z)|zi to the state ρ∈Gˆ f ( )i j | ,i, ji, where fixed prime power q. f : G → C, as before, is a function with k f k2 = 1 and fˆ(ρ)i j denotes the i, j entry of the Fourier transform at the 1 Introduction representation ρ. This is explained further in Section 2. Peter Shor’s seminal discovery of efficient quantum While there is no known explicit relationship between algorithms for factoring and discrete logarithm [25] re- the quantum Fourier transform and the hidden subgroup lies crucially on the fact that the Fourier transform over problem over a group G, all known efficient hidden subgroup algorithms rely on an efficient quantum Fourier the cyclic group Zn can be carried out efficiently on a quantum computer, even when n is exponentially large. transform. Indeed, it is fair to say that the quantum This has motivated broad interest in the problem of effi- Fourier transform—the so-called transform and measure cient quantum computation over arbitrary groups; see e.g., approach—is the only known non-trivial quantum algo- [3, 9, 11, 13, 14, 20, 21, 27]. While this research effort has rithmic paradigm for such problems. already become quite ramified, two related themes have In this article we focus on the construction of efficient emerged: quantum Fourier transforms. Our research is motivated by dramatic progress over the last decade in the theory of ef- (i.) development of efficient quantum Fourier trans- ficient classical Fourier transforms, e.g. [4, 5, 8, 18, 22]. forms, and These developments have provided a collection of techniques which, taken together, yield a uniform framework (ii.) development of efficient quantum algorithms for the for the efficient computation of Fourier transforms over a hidden subgroup problem. wide variety of important families of groups. These in- The complexity of these two problems appears to be include, for example, the finite groups of Lie type (properly timately related to the structure of the group in question: parametrized) and the symmetric groups. while quantum Fourier transforms and hidden subgroup Our main result is an adaptation to the quantum 2 Representation theory background setting of the most successful and general of these tech- Fourier analysis over a group G consists of expressing niques, namely the “separation of variables” approach. arbitrary functions f : G → C as linear combinations of While almost all efficient classical Fourier transforms basis functions which reflect the group’s structure and are divide-and-conquer algorithms, which recursively per- symmetries. If G is Abelian, these are the characters of form the Fourier transform for a series of subgroups and G, i.e., the homomorphisms of G into C; for a general combine the results according to their coset structure, the group, they are the irreducible matrix elements. Then the separation of variables approach uses the existence of Fourier transform is the change of basis from the basis of adapted bases to streamline this process considerably. delta functions to the basis of irreducible matrix elements. Specifically, we define a broad class of polynomially In order to be precise we need the language of (finite) uniform groups and show group representation theory (see, e.g., Serre [24] for an ρ THEOREM 1.1. If G is a polynomially uniform group excellent introduction). A representation of a finite ρ with a subgroup tower G = Gm > Gm−1 > ··· > {1} group G is a homomorphism : G → U(V), where U(V) with adapted diameter D, maximum multiplicity M, and denotes the group of unitary linear operators on a finite- maximum index I = maxi[Gi : Gi−1], then there is a dimensional vector space V whose dimension we denote quantum circuit of size poly(I × D × M × log|G|) which dρ. Once we fix an orthonormal basis for V, each computes the quantum Fourier transform over G. ρ(g) is a dρ × dρ unitary matrix and is called a matrix 2 representation of G. Each of the dρ functions ρi j(g) = This quantifies the complexity of the quantum Fourier [ρ(g)] is called a matrix element of ρ; note that while ρ transform in exactly the same fashion as Corollary 3.1 i j is a homomorphism, in general ρi j is not. of [17] does for the classical case. In fact, for many of A matrix representation ρ of G on V is called irre- the group families we study, the quantum and classical ducible if the only subspaces it preserves are the trivial circuit complexities of the Fourier transform differ by a one, {0}, and V itself. This is equivalent to the statement | | factor of G . We extend this class further by showing that there is no change of basis that simultaneously gives a that it is closed under a certain type of Abelian extension block diagonalization (of a given shape) of ρ(g) for all g. which may have exponential index. Otherwise the representation is said to be reducible. The This framework allows us to give efficient QFTs— irreducible representations will play a role in the theory (| |) that is, circuits of polylog G size—for many new fami- analogous to that of the characters of an Abelian group. lies of groups, as well as to place existing QFT algorithms Two representations ρ and σ are equivalent if they dif- in a uniform framework. These include fer only by a change of basis, so that for some fixed uni- σ −1σ (i.) the Clifford groups CLn; tary matrix U, (g) = U (g)U for all g ∈ G. Up to equivalence, a finite group G has a finite number of irre- (ii.) symmetric groups, recovering Beals’ algorithm [3]; ducible representations equal to the number of its conju- ˆ (iii.) wreath products G o Sn where |G| = poly(n); gacy classes. For a group G, we let G denote a collection of representations of G containing exactly one from each (iv.) metabelian groups (semidirect products of two isomorphism class of irreducible representations. Abelian groups) including metacyclic groups such as Selecting explicit bases for the representations of Gˆ the dihedral and affine groups, recovering the algo- results in a set of (inequivalent irreducible) matrix rep- rithm of Høyer [13]; resentations, whose matrix elements then form an or- (v.) bounded extensions of Abelian groups such as the thonormal basis for the |G|-dimensional vector space of generalized quaternions, recovering the algorithm of complex-valued functions on G. Since there must be Puschel¨ et al. [21]. enough matrix elements to span this space, this implies the following important relationship between |G| and the Our methods also give the first subexponential size dimensions of the irreducible representations: quantum circuits for the linear groups GLk(q), SLk(q), 2 PGLk(q), and PSLk(q) for fixed prime power q, finite ∑ dρ = |G| . groups of Lie type, and the Chevalley and Weyl groups. ρ∈Gˆ The paper is structured as follows. Sections 2 and 3 briefly summarize the representation theory of finite We are now equipped to give the general definition of groups, the Bratteli diagram, and adapted bases. We give the Fourier transform over arbitrary groups. Marvelously, our algorithms in Section 4 along with a list of group fam- this definition possesses many of the properties of the ilies for which our techniques provide efficient circuits for Fourier transform over Zn that we know and love; for the QFT. We conclude with open problems in Section 5. instance, it transforms convolution into (matrix) product. ρ e0 • DEFINITION 1. Let f : G → C; let : G → U(V) be a NNN NNg matrix representation of G. The Fourier transform of f at e1 • NN MMM NN e0 ρ, denoted fˆ(ρ), is the matrix MM N•O e2 MMf s O •KK MMss OOOg KK y ssMMM OOO 1 s sKsKse • o • s KK qqe1 oo dρ e ss qKqK ow o ˆ ρ ρ 3 •s qx q KK ooo f ( ) = ∑ f (g) (g) . qqq •o |G| qq ppe2 g∈G e4 •q pw pp ppp e5 •pp We typically restrict our attention to fˆ(ρ) where ρ is (4) •U (3) irreducible. UUUU (2) ii•UUU (3,1) •Liii UU•L LLL rrr LLL (1) The Fourier transform is linear in f ; with the constants (2,2) • L•r(r2,1) L •• φ p r LL r dρ/|G| we use here, it is in fact unitary, taking the rr LL rr (2,1,1) •rr L•rr 2 UUUU iii |G| complex numbers h f (g)ig∈G to a total of ∑dρ = Ui•ii (1,1) (1,1,1,1) •iiii(1,1,1) |G| complex numbers organized into |Gˆ| matrices with varying dimensions dρ. Figure 1: The Bratteli diagrams for the subgroup towers For two complex-valued functions f1 and f2 on a 6 > 3 > 1 (top) and S4 > S3 > S2 > 1 (bottom). Cyclic group G, there is a natural inner product h f1, f2i given Z Z 1 ∗ groups of order n have representations indexed by the in- by ∑g f1(g) f2(g) . The orthonormality of the matrix |G| tegers mod n, and (assuming m|n) the representation cor- elements can then be expressed as follows: for any pair of responding to j restricts to the representation correspond- matrix representations ρ,σ ∈ Gˆ, ing to j mod m. The lower diagram uses the well-known ( correspondence between irreducible representations of S 0 if ρ =6∼ σ , n (2.1) ρ ,σ = and partitions of n. In this case restrictions from Sn to i j kl 1 δ δ if ρ = σ . dρ ik jl Sn−1 are determined by those partitions obtained via the decrement of a part of the original partition. This is one form of Schur’s lemma [24]. We can use this orthonormality to invert the Fourier transform, giving the 3 Making divide-and-conquer feasible: Bratteli Fourier inversion formula: diagrams, Gel’fand-Tsetlin bases, and adapted diameters s dρ The classic Cooley-Tukey Fast Fourier Transform relies f (s) = ∑ tr ρ(s) fˆ(ρ)−1 . on the fact that the cyclic group k can be decomposed ρ ˆ |G| Z2 ∈G into a tower of subgroups: ρ A reducible matrix representation : G → U(V) can Z2k > Z2k−1 > ··· > Z4 > Z2 > Z1 = {1} always be decomposed into a direct product of irreducible representations. Specifically, there is a basis of V in which The Cooley-Tukey algorithm works recursively, by calcu- ρ is block diagonal, where the ith block of ρ is precisely lating the FFT for each subgroup in the tower, and then combining the results from that subgroup’s two cosets to σi for some irreducible matrix representation σi. In this L form the FFT at the next level up. case we write ρ = σi. The number of times a given i Almost all efficient classical algorithms for the σi ∈ Gˆ appears in this decomposition is the multiplicity Fourier transform work in this way. However, in the non- of σi in ρ; denoting this multiplicity wi, we will write w w Abelian case, making this divide-and-conquer approach ρ = ⊕ 1 σ1 ... ⊕ r σr. A representation ρ of a group G is also automatically concrete is far from trivial. Even if the group has a natu- a representation of any subgroup H. We refer to this ral subgroup tower, we need to choose a set of bases for ρ the representations which allows us to embed the cosets restricted representation on H as |H . Note that in general, representations that are irreducible over G may of each subgroup in the next one up in an efficient way. be reducible when restricted to H. Furthermore, we need to choose a set of generators into which we can factor group elements efficiently, and our Remark. The familiar Discrete Fourier Transform (DFT) choice of bases should make the matrix representations corresponds to the case G = Zn. In this case the repre- for these generators sparse and highly structured, so that sentations are all one-dimensional, and the Fourier trans- they can be multiplied together efficiently. (Finally, in the form is an n × n Vandermonde matrix whose entries are quantum setting, we will have to write the resulting trans- nth roots of unity. form as a product of elementary unitary operations.) 3 Luckily, there are principled ways to choose these be a tower of subgroups of length m for G. The corre- bases and these generating sets. These techniques allow sponding Bratteli diagram, denoted B, is a leveled di- us to construct an efficient classical Fourier transform rected multigraph whose nodes at level i = 0,...,m are from the following ingredients: in one-to-one correspondence with the (inequivalent) irreducible representations of Gi. For convenience, we re- (i.) a tower (or “chain”) of subgroups, by which the fer to vertices in the diagram by the representation with Fourier transform on G can be built recursively as an which they are associated. The number of edges from an accumulation of Fourier transforms on increasingly irreducible representation σ of Gi to ρ of Gi+1 is equal to larger subgroups; the multiplicity of σ in the restriction of ρ to Gi. Since there is a unique irreducible representation of the trivial (ii.) a natural indexing scheme for the representations group, a Bratteli diagram for a given tower is in fact a given by paths in the Bratteli diagram corresponding rooted tree. Bratteli diagrams for the cyclic group and to that subgroup tower, which in turn provides a Z6 the symmetric group S are shown in Figure 1. convenient basis for each representation; and finally 4 We now describe how paths in the Bratteli diagram (iii.) a factorization of group elements in terms of a basic index the rows and columns of each representation, and set of generators, which, when judiciously chosen, thus provide a natural set of bases. Each edge, from a node σ ˆ ρ ˆ provide a factorization of the Fourier transform as a : Gi → U(Vσ) of Gi to a node : Gi+1 → U(Vρ) of Gi+1, product of structured (direct sums of tensor products) represents an embedding of Vσ into Vρ. Thus the edges ρ and sparse matrices. into describe a decomposition of Vρ into a direct sum of orthogonal subspaces Vσ, each of which are invariant The complexity of the resulting algorithm can then be under the (restricted) action of Gi; and conversely, the derived in terms of the basic representation-theoretic and edges out from σ correspond to embeddings of Vσ into combinatorial data of the subgroup tower, the Bratteli orthogonal subspaces Vρ. Thus these edges describe diagram, and the generating set. We describe the recipe by how the subspaces acted on by the representations of which these ingredients are made into efficient classical Gi+1 are decomposed into smaller subspaces acted on by transforms in the next two sections. representations of Gi, and conversely how the subspaces of Gi are embedded in the subspaces of Gi+1. 3.1 Bratteli diagrams and Gel’fand-Tsetlin bases Since the only representation of the trivial group {1} Much of Abelian Fourier analysis is simplified by the fact is one-dimensional, composing these edges into paths ˆ χ that in this case the the set of characters G = { : G → C}, from the root to a given node ρ ∈ Gî gives a decomposition also called the dual, forms a group isomorphic to the orig- of Vρ into a direct sum of orthogonal one-dimensional inal group G. Furthermore, in this isomorphism lies a nat- subspaces; but this is tantamount to providing a basis for ural correspondence which provides an indexing of the ir- Vρ. Moreover, since paths from the root to ρ consist of reducible representations, and thus the matrix elements of paths from the root to various σ composed with paths from the transform. However, in the general case there is no σ to ρ, where σ ∈ Gˆ j for some G j < Gi, this basis has the immediate indexing scheme for the dual Gˆ and the land- following property: for any G j < Gi, there is a partition scape is further complicated by the absence of a canonical of the basis vectors into subsets, each of which spans an basis for the (now multidimensional) representations. In- irreducible G j-invariant subspace. Therefore, in this basis deed, where efficient Fourier analysis is concerned, not all the matrix representation ρ is block diagonal according bases are created alike! to this partition when restricted to G j and, moreover, the A fairly general methodology for the construction blocks corresponding to some σ which appears in ρ with of group FFTs, the ”separation of variables” approach multiplicity greater than 1 are actually equal. Such bases [17, 18] relies on the use of Gel’fand-Tsetlin or adapted are said to be G j-adapted or Gel’fand-Tsetlin. bases. These bases allow us to carry out the recursive Note that the number of paths to a node ρ is equal divide-and-conquer approach described above, building to dρ (so, for instance, the Bratteli diagram of an Abelian the transform efficiently at each level of the subgroup group is a directed tree). Furthermore, each ordered pair tower. To construct these bases, we need a natural of paths with common endpoint ρ indexes an irreducible indexing scheme for the representations, and for each of matrix element of ρ, since one path indexes a row and the their matrix elements, Happily, such an indexing scheme other indexes a column. is given by the Bratteli diagram formalism, which we now Following the divide and conquer approach, the present. Given a finite group G, let Fourier transform on G = Gm can be written a sum of Fourier transforms on Gm−1, each of which is translated G = Gm > Gm−1 > ··· > G1 > G0 = {1} from a different coset. Specifically, if T ⊂ G is a transver- can condition on the ηi to find out which subspace of ρ sal, i.e., a set of representatives for the left cosets of Gm−1 we are in, we can write ρ(γ) as a series of poly(M) ele- in Gm, we define fα : Gm−1 → C by fα(x) = f (αx). Then mentary quantum operations where M = maxi mi in (3.3). Therefore, the total number of elementary quantum oper- ˆ ρ ρ α ρ α f ( ) = ∑ ( ) ∑ (x) f ( x) ators we need to implement ρ(α) is D × poly(M). α∈T x∈G m−1 Moreover, if γ is itself in a subgroup H > K, and (3.2) = ∑ ρ(α) · fˆα(ρ| ). ρ ρ γ Gm−1 is adapted to both H and K, then ( ) also possesses α∈T ρ the block structure corresponding to |H . This places These matrices ρ(α) are called the “twiddle factors”. an upper bound on M, namely the maximum multiplicity Note that the number of terms in this sum is |T| = with which representations of K appear in restrictions [Gm : Gm−1] = |Gm|/|Gm−1|, the index of Gm−1 in Gm. of representations of H. Thus we can minimize M by As we will see below, the recursion of (3.2) will be choosing generators γ which (1) are inside subgroups as greatly simplified by the fact that in the adapted basis, the low on the tower as possible, and (2) centralize subgroups restricted representations ρ| become block diagonal, as high on the tower as possible. G j where the blocks are simply the matrices σ. For instance, for the symmetric group Sn we take the tower to be 3.2 Strong generating sets and adapted diameters Sn > Sn−1 > ··· > {1} , Adapted representations are only part of the story for the where S fixes all elements of {1,...,n} greater than i. Let construction of efficient Fourier transforms. In general, i S be the set of pairwise adjacent transpositions ( j, j + 1); the twiddle factors ρ(α) in Equation (3.2) could be an each of these is contained in S and centralizes S . arbitrary matrices of exponential size, so an algorithm j+1 j−1 The maximum multiplicity with which a representation of which simply performs the sum in (3.2) could be costly. S appears in a representation of S is 2, correspond- Luckily, under fairly mild assumptions, these twiddle j−1 j+1 ing to the two orders in which we can remove two cells factors can be factored into polylog(|G|) sparse, highly from a Young diagram. In this case the adapted basis de- structured matrices, and can therefore be implemented fined by the Bratteli diagram is exactly the Young orthogo- with polylog(|G|) elementary quantum operations. nal basis, in which each block of ρ(( j, j+1)) differs from We say that S is a strong generating set for the tower the identity only by a 2×2 minor. Since the adapted diam- of subgroups {G } if S ∩ G generates G . Say that we i i i eter is easily seen to be O(n2), this means that the twiddle have chosen a transversal T for each i indexing the cosets i factors ρ(α) can be carried out in O(n2) = polylog(|S |) of G in G . Now define n i−1 i elementary quantum operations [3]. We will see in the j Di = min{` > 0 : ∪ j≤`(S ∩ Gi) ⊇ Ti} , next section that a similar situation obtains for a large class of groups. i.e., the length of words over S ∩ Gi we need to generate every representative in T, and define the adapted diameter 4 Efficient quantum Fourier transforms D = ∑ D . Then clearly any group element can be i i We describe our algorithm in this section. As in the classi- factored as a series of coset representatives, which in turn cal case, we perform the Fourier transform inductively on can be factored as a total of at most D elements of S. the tower of subgroups, using the structure of the Bratteli Of course, to perform the QFT efficiently we would diagram to construct the transform at each level from the like ρ(γ) to have a simple form for each γ ∈ S. Given a transform at the previous level. subgroup K < G, recall that the centralizer of K is the Recall that for each level of our tower of subgroups subgroup Z(K) = {g ∈ G : gk = kg for all k ∈ K}. The G = G > G > ··· > G = {1} we have chosen a following is implicit in the oft-cited lemma of Schur: m m−1 0 transversal Ti for the left cosets of Gi−1 in Gi. At the LEMMA 3.1. (Schur, [17, Lemma 5.1]) Let K < G, let beginning of the computation, we represent each group γ ∈ Z(K), and let ρ be a K-adapted representation of G. element g as a product α = αm ···α1 where αi ∈ Ti. This ρ m1 η mr η ρ γ string becomes shorter as we work our way up the tower, Suppose that |K = ⊕ 1 ··· ⊕ r. Then ( ) has the form and after having performed the Fourier transform for Gi the remaining string α = αm ···αi+1 indexes the coset of ( ( ) ⊗ ) ⊕ ··· ⊕ ( ( ) ⊗ ) (3.3) GLm1 C Id1 GLmr C Idr Gi in G in which g lies. At the end of the computation, we have a pair of paths where Ik is the k × k identity matrix and di = dηi . in the Bratteli diagram, s = s1 ···sm and t = t1 ···tm, which Since any unitary operator in GLm(C) can be carried out index the rows and columns of the representations ρ of G. with poly(m) elementary quantum gates [2], and since we These paths begin empty and grow as we work our way up 5 ˆ ρ the tower; after having performed the Fourier transform Recall that the matrix fα( |H ) is a direct sum of sub- for Gi, the paths p = p1 ··· pi and q = q1 ···qi of length i matrices of the form fˆα(σ), summed over the σ appearing σ ρ ˆ ρ index the rows and columns of representations of Gi. in . We will construct f ( |H ) via an embedding opera- With a compact encoding, one could store α in the tion which reverses the restriction to H, same registers as s and t, at each step replacing a coset σ ρ representative αi with a pair of edges si,ti. (This is how (4.7) | i → ∑ Aσ,ρ | i ρ:σ appears in ρ| Coppersmith’s circuit for the QFT over Z2k works; see H below.) However, our algorithm is simpler to describe if we double the number of qubits and store α and s,t in where this “scale factor” is separate registers. Padding out α, s, and t to length m with s |H| dρ zeroes, our computational basis consists of unit vectors of Aσ,ρ = . the form |G| dσ

α α α i m−i m−i 2 | i|s,ti = m ··· i+1 0 ⊗ s1 ···si 0 ,t1 ···ti 0 . Note that ∑ρ |Aσ,ρ| = 1. Thus the algorithm consists of (i.) embedding the σ in Keep in mind that the basis {|s,ti}, where s and t have the appropriate ρ, (ii.) applying the “twiddle factor” ρ(α), length i and end in the same representation, is just a per- and (iii.) summing over the cosets. However, as discussed σ mutation of our adapted Gel’fand-Tsetlin basis {| , j,ki} above, doing these things efficiently is no simple matter. ˆ σ for Gi, where ranges over the representations of Gi and First, a given σ might appear in a given ρ with an arbitrary 1 ≤ j,k ≤ dσ index its rows and columns. Therefore, change of basis; the twiddle ρ(α) could be an arbitrary ˆ we will sometimes abuse notation by writing f (s,t) and unitary matrix of exponential size; and, if [G : H] is ˆ σ f ( ) j,k for the Fourier transform over Gi indexed in these exponentially large, summing over the cosets will take two different ways. exponential time unless parallelized in some way. Each stage of the algorithm consists of calculating The Bratteli diagram, and the adapted basis it pro- the Fourier transform over Gi+1 from that over Gi. By vides, allow us to accomplish (i) and (ii) above with a induction it suffices to consider the last stage, where we minimum of trouble. For (i), the embedding operation, go from H = G to G = G . Specifically, choose a m−1 m note that fˆα(s,t) is nonzero only when s and t end in the transversal T of H in G such that every g ∈ G can be same representation σ of G , i.e., in the same vertex of α α t written h where ∈ T and h ∈ H. As in (3.2), for each the diagram. Moreover, recall that the Bratteli diagram α α ∈ T we define a function fα on H as fα(h) = f ( h); indexes an adapted basis in which ρ| is block-diagonal this is the restriction of f to the coset αH, translated into H with the σ j as its blocks. This means that the σ appear in H. After having performed the Fourier transform on H, the ρ in an extremely simple way: namely, where s and t our state will be are extended by appending the same edge e to both. The only change of basis required is to literally pick the matrix ∑ |αi ⊗ ∑ fˆα(s,t)|s,ti σ α∈T s,t of length m−1 elements of up and place them in the appropriate place in ρ, and we discuss below how to do this unitarily. (4.4) = ∑ |αi ⊗ ∑ fˆα(σ) |σ, j,ki . j,k Similarly, when coupled with a strong generating set α∈T (σ, j,k)∈Hˆ of small adapted diameter as discussed in Section 3.2, Our goal is to transform this state into the Fourier basis of the adapted basis allows us to carry (ii) out efficiently G, namely by writing ρ(α) as a product of a small number of ρ(γ), each of which has the block-diagonal structure given by |0i ⊗ ∑ fˆ(s,t)|s,ti Lemma 3.1. s,t of length m For (iii), summing over the cosets, for now we simply (4.5) = |0i ⊗ ∑ fˆ(ρ) j,k |ρ, j,ki . take the time to sum over all the cosets serially, paying a (ρ, j,k)∈Gˆ cost of [Gi : Gi−1] per level as reflected in Theorem 1.1. This makes sense for subgroup towers where the index where |0i occupies the register that held the coset repre- of each subgroup in the one above it is polynomial, such α sentative before. as the tower for S above, and we focus on that case ˆ n As described in Equation (3.2) above, f can be in Section 4.1. However, in Section 4.2 we will see written as a sum over contributions from f ’s values on that even when the index of some level of the tower α each coset H, giving is exponentially large, in some cases we can use the ˆ ρ ρ α ˆ ρ parallelism of quantum mechanics to sum over all the (4.6) f ( ) = ∑ ( ) · fα( |H ) . α∈T cosets simultaneously, and still achieve an efficient QFT. We adopt the following notation. Given a path s in in at most [G : H] many ρ. We then carry out a series the Bratteli diagram of length m − 1 or m, denote the of [G : H] conditional rotations, each of which rotates the representation in which it ends by σ[s] or ρ[s] respectively, appropriate amplitude from |0i|s,ti to |0i|se,tei. Thus −1 and if s = s1 ···sm−1, denote s1 ···sm−1e as se. We will U, and therefore U , can be carried out in O([G : H]) index the edges of each vertex {1,...,k} where k is its out- quantum operations. degree. It will be convenient to carry out this embedding To apply the twiddle factor and sum over the cosets as only if the register containing the coset representative is in (4.6), we use a technique of Beals [3] and carry out the zero, and leave other basis vectors in (T ∪{0})⊗Hˆ fixed. following for-loop. For each α ∈ T, we do the following Then (4.7) becomes three things: left multiply fˆ(ρ) by ρ(α)−1; add fˆα(ρ) to fˆ(ρ); and left multiply fˆ(ρ) by ρ(α). This loop clearly ∑ |0i|s,ti → |0i e Aσ[s],ρ[se] |se,tei produces ∑α ρ(α) · fˆ(ρ), so we just need to show that (4.8) U : α α α ∈T | i|s,ti → | i|s,ti for all ∈ T each of these three steps can be carried out efficiently. Recall that fˆ(ρ) is given in the |s,ti basis, where where the sum is over all outgoing edges e of σ[s] = σ[t]. s and t index the row and column of ρ respectively. Note that we have not defined U on the entire space; To left multiply fˆ(ρ) by ρ(α), we apply ρ(α) to the s in particular, since we are moving probability from Hˆ to register and leave the t register unchanged. Since G is Gˆ, basis vectors |0i|se,tei ∈ (T ∪ {0}) ⊗ Gˆ cannot stay polynomially uniform, a classical algorithm can factor fixed. As we will see below, it does not matter precisely α as the product of D generators γ ∈ S, and provide how U behaves on the rest of the state space, as long i a factorization of each ρ(γ ) as the product of poly(M) as its behavior on Hˆ is as described in (4.8). This can i many elementary quantum operations, in polylog(|G|) be accomplished simply by putting the mth registers of s time. This implements ρ(α) and ρ(α)−1 in D×poly(M)+ and t in the superposition ∑ Aσ ρ |ei ⊗ |ei, and for a e [s], [se] polylog(|G|) operations. large class of extensions we can prepare this superposition The step “add fˆα(ρ) to fˆ(ρ)” is slightly more mys- efficiently. terious, and indeed it does not even sound unitary at first. We shall focus on group towers for which the Bratteli However, as Beals points out, at each point in the loop diagram data can be effectively computed: we are adding fˆα(ρ), which is the Fourier transform of a ∑ ρ α−1β ˆ ρ DEFINITION 2. For a group G and a tower of subgroups function with support only on H, to β<α ( ) fβ( ), which is the Fourier transform of a function with support Gi, let B be the corresponding Bratteli diagram, let Ti be a set of coset representatives at each level, and let S be only outside H. Thus these two states are orthogonal, and a strong set of generators for G. Then we say that G is adding two orthogonal vectors can be done unitarily by rotating one vector into the other while fixing the sub- polynomially uniform (with respect to {Gi}, B, {Ti}, and S) if the following functions are computable by a classical space perpendicular to both. Let Vα be the operation that α β algorithm in polylog(|G|) time: (i.) Given two paths s,t exchanges | i|s,ti with |0i|s,ti and leaves | i|s,ti fixed β α in B, whether ρ[s] = ρ[t]; (ii.) Given a path s in B, the for all ≤ ,0; then Beals showed that this step can be −1 dimension and the out-degree of ρ[s]; (iii.) Given a coset written U VαU where U is the embedding operator defined in (4.8). We showed earlier that U can be carried representative αi ∈ Ti, a factorization of α as a word of ∗ out in O([G : H]) quantum operations, and V is a simply a polylog(|G|) length in (S ∩ Gi) . Boolean operation on the α register. Finally, the for-loop 4.1 Extensions of small index We begin by focusing runs |T| = [G : H] times, and we are done. on groups and towers which are fairly refined, i.e., with Proof of Theorem 1.1. This follows by induction as the polynomial indexes at each level. depth of the Bratteli diagram is at most log|G|. LEMMA 4.1. If G is polynomially uniform with respect For many families of groups, the maximum index to a tower of subgroups where G = Gm and H = Gm−1 I = maxi[Gi : Gi−1], the adapted diameter D, and the max- and a strong generating set S with adapted diameter D imum multiplicity M are all polylog(|G|). In this case, and maximum multiplicity M, then the Fourier transform Theorem 1.1 gives circuits for the QFT of polylog(|G|) of G can obtained from the state (4.4) using poly([G : size. This includes the following three families of groups: H] × D × M × log|G|) elementary quantum operations. The symmetric groups Sn. As stated above, we take the Proof. First, to carry out the embedding transformation tower Sn > Sn−1 > ··· > {1}, so the I = n = o(log|Sn|). U, we use the classical algorithm to compute the list of The generators are the adjacent transpositions, so D = 2 edges e and dρ[se] conditional on s, and thus compute the O(n ) and M = 2. The adapted basis is precisely the Aσ,ρ (say, to n digits in poly(n) time). Note that σ appears Young orthogonal basis. 7 Wreath products G = H oSn for H of size poly(n). These 4.2 Extensions of large index; Coppersmith-type cir- groups arise naturally as automorphism groups of graphs cuits The reader familiar with Coppersmith’s circuit [7] obtained by composition [12]. As in [23] the tower is for the QFT over G = Z2n , where H = Z2n−1 , will recall that the Hadamard gate embeds a character σ ∈ Hˆ in two H o Sn > H × (H o Sn−1) > H o Sn−1 > ··· > {1} . characters ρ ∈ Gˆ, applies part of the twiddle factor, and sums over both cosets of H, all in one operation. This is Then I = max(n,|H|), the generators are the adjacent in contrast to the technique of the previous section, which transpositions and an arbitrary set of log|H| generators sums over the cosets serially—and which takes exponen- for each factor of H, D = O(n2 log|H|), and M = O(|H|). tial time if, for instance, G is an extension of H by Zp Then note that |H| = polylog(|G|). See [17] for details where p is exponentially large. and [15] for discussion on wreath products. For a certain type of extension, we can construct The Clifford groups. The Clifford groups CLn are circuits analogous to Coppersmith’s, which use quantum 2 σ ρ generated by x1,...,xn where xi = 1 and xix j = −xix j for parallelism to embed in the , sum over all the cosets all i 6= j [26]. We take the tower simultaneously, and apply the twiddle factor as well. Recall that G is a split extension or semidirect product of CLn > CLn−1 > ··· > {1} H by T, written T n H, if H C G and there is a transverse subgroup T < G so that T =∼ G/H. for which I = 2, and the generators {x1,x1x2,...,xn−1xn}. Then D = O(n), and since each xixi+1 centralizes CLi−1 DEFINITION 3. Suppose G is a split extension of H by T, we have M = 4. let S be a set of at most log2 |T| generators for T, and In addition to giving polylog(|G|)-size circuits for suppose that G is polynomially uniform with respect to a these groups, our techniques give the first subexponential- tower of subgroups where G = Gm and H = Gm−1 and a size circuits for the following classical groups: Bratteli diagram B. Then G is a homothetic extension of H by T if (i.) Given σ ∈ Hˆ and γ ∈ S, define σγ(h) = The linear groups GL (q), SL (q), PGL (q), and γ n n n σ(γ−1hγ); then for every σ ∈ H,ˆ either σ = σ, or the orbit PSL (q); the finite groups of Lie type; the Chevalley j n of q distinct representations σγ , for 0 ≤ j < q where q and Weyl groups. The case of GL (q) is emblematic of n divides the order of γ, appears among the representations all these families. We have a natural tower: of H given by B; (ii.) For each γ ∈ S, there is a classical (| |) GLn(q) > Pn(q) > GLn−1(q)×GL1(q) > GLn−1(q) > ··· algorithm which runs in polylog G time which, given a path s in B indexing a row of σ[s] and an integer j, returns Here Pk(q) is the so-called maximal parabolic subgroup, the size q of σ’s orbit under conjugation by γ, and returns j j j consisting of elements of the form a path sγ that indexes the same row of σ[sγ ] = σγ . A ~v THEOREM 4.1. If G is a homothetic extension of H by an 0···0 c Abelian group, then the Fourier transform of G can be obtained from the state (4.4) using polylog(|G|) elementary k−1 × n−1 where A ∈ GLk−1(q),~v ∈ Fq , and c ∈ Fq , so I = q . quantum operations . Our generators are the block-diagonal matrices with an arbitrary element of GL2(q) in the i,i − 1 block and all Proof. It is easy to show that a homothetic extension of other diagonal elements equal to 1. Then D = O(n2) and H by A × B consists of a homothetic extension of H by M = qO(n). Analogous factorizations arise for the finite A, followed by a homothetic extension by B. Therefore groups of Lie type as well as the finite unitary groups [18]. it suffices to prove the lemma for homothetic extensions Theorem 1.1 then implies a quantum circuit of size by cyclic groups of prime power order, so without loss of 2 qO(n) for the QFT over these groups. Since |G| = O(qn ) generality we let T be generated by γ of order pz. we can write this as |G|O(1/n), which is expO(plog|G|) We recall some representation theory from [6, 22]. if q is fixed. On the other hand, he best-known classical Given σ ∈ Hˆ , the stabilizer of σ is K = {x ∈ T : σx =∼ σ}, FFT for these groups [17] has complexity |G|qΘ(n) = and for a homothetic extension we can replace σx =∼ σ |G|1+Θ(1/n). Note for the group families above for which with σx = σ. Then K is the subgroup of T of order we obtain circuits of size polylog(|G|), there are classical p` generated by γq where q = pz−`, and σ’s orbit under algorithms of complexity |G|polylog(|G|). In both cases, conjugation by γ is of size q. it seems that the natural quantum speedup is a factor of The representations ρ in which σ appears can be |G|, modulo polylogarithmic terms; of course, we would obtained in two steps. First, we extend σ to K n H by like to know if this is the best possible. multiplying σ by one of the p` characters of K. This yields q j q j τb ∈ K\n H where τb(γ h) = χb( j)σ(h) and χb(γ ) = Closure under homothetic extensions and metacyclic b j p ω . Since dτ = dσ, we have Aσ τ = 1/p` and σ groups. Theorem 4.1 shows that the set of groups for p` b , b which circuits of polylog(|G|) size exist is closed under embeds in a uniform superposition over the τb, so we append a uniform superposition of edges 1 ≤ e ≤ p` where homothetic extensions by Abelian groups. It also general- izes the efficient quantum Fourier transform of Høyer [13] b = e−1. Combining this with the twiddle factor χb gives the unitary transformation for the metacyclic groups Zq n Zp, since these are homothetic extensions of Zp by Zq. Note that the metacyclic ` groups include the dihedral groups (where q = 2) and the E E 1 p (4.9) γq j+k |s,ti → γk ⊗ ∑ ω(e−1) j |se,tei . affine groups (where q = p − 1) as special cases. p ` p` p e=1 The quaternionic groups. The generalized quaternion 2 γ ≤ < ` group is an extension of H = Z2n by Z2 where γ is the Here we write the power of in two registers 0 j p p and 0 ≤ k < q. Then this operation Fourier transforms element of order 2 in H. Then C(γ) = σ(γ2) = 1 or i. Puschel,¨ Rotteler¨ and Beth [21] gave an efficient quantum the first register over Zp` and transfers the result to the mth register of s and t. This transform can be carried out Fourier transform for these groups in the case where n with O(log p` loglog p`) = O(log|G|loglog|G|) elemen- is a power of 2. Of course, these groups are extensions tary operations [10, 16]. Note that p` takes at most log|G| of Abelian groups with bounded index, so Lemma 4.1 different values, and can be obtained from the classical already provides an efficient QFT for them. algorithm which computes q. Metabelian groups. Even if an extension is neither ho- ρ ˆ σ If K = T, then the ∈ G containing are simply the mothetic nor of polynomial index, we can still construct τ extensions b and we’re done. If K < T, i.e., if q > 1, an efficient QFT if we can apply arbitrary powers of τ we carry out a second step as follows. Each b appears C(γ) in polynomial time—for instance, if C(γ) is of poly- ρ in a single induced representation b whose restriction to nomial size, which is true whenever all the representa- σ K nH is the direct product of all the representations in ’s tions of H are of polynomial size. This includes the χ ρ χ q−1 σγi orbit, times b: that is, b|H = b ⊕i=0 . The twiddle metabelian groups, i.e., split extensions of Abelian groups ρ γk factor b( ) is then a permutation matrix which cycles by Abelian groups, since all the representations of H are these p blocks k times, with an additional phase change one-dimensional. We discuss this further in the full paper. ωbk pz . This gives the unitary transformation The general case. In general, Abelian extensions can E k E γk ω(e−1)k γ be slightly more complicated; consider extensions by (4.10) |se,tei → pz |0i s e,te . γ Zp. If σ is isomorphic to σ, rather than equal to it, γ induces an additional twiddle factor C(γ) which changes γk Since s can be calculated by the classical algorithm in σ’s basis [22]. This occurs, for instance, if γp is an ωbk polylog(|G|) time, and since it is easy to implement pz element of H other than the identity, in which case the ω2yb γ with phase shifts pz for 0 < y < log2 k conditioned on cyclic group generated by is not transverse to H and the the binary digit sequence of bk, we can perform this op- extension is not split; then C(γ) is a pth root of σ(γp). eration in polylog(|G|) quantum steps. Composing (4.9) and (4.10) transforms the state (4.4) to the Fourier trans- 5 Conclusion and open problems form (4.5) over G. We have shown that a general technique for constructing efficient classical fast Fourier transforms on groups— separation of variables using an adapted basis—can be Relation to Coppersmith’s circuit. Let γ be a generator carried over to the quantum context, producing circuits of G = Z2n . Then G is an extension of H = Z2n−1 with of polylog(|G|) size for a wide variety of groups, and of transversal {1,γ}. Since γ2 6= 1, γ induces an additional subexponential size for classical linear groups mod q. γ pχ γ2 ωb phase shift C( ) = b( ) = 2n . (Similarly, the addi- While separation of variables is one of the most tional phase shift in (4.10) is due to the fact that Zpz is not general techniques for classical FFTs, it is not the only γ a split extension of Zp` .) In Coppersmith’s circuit, C( ) one. It is possible to use the Bratteli diagram in a appears as a set of phase shift gates conditional on the more precise fashion, looking for redundancy and sparsity low-order bit of j. Finally, the Hadamard gate in Copper- on the level of individual matrix elements. This finer smith’s circuit is precisely the operation (4.9) in the case analysis is responsible for the fastest known classical p = 2, ` = 1 and q = 1, and where we use the same qubit FFTs for the groups SL2(q), as well as Sn and its wreath register for e (the high-order bit of the frequency) as for α products [19]. 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