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 sub- group 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 tech- niques 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 in- clude, 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) .