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22
On quantum algorithms for noncommutative hidden subgroups
, 2000
"... Quantum algorithms for factoring and finding discrete logarithms have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. We present a quantum ..."
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Cited by 76 (3 self)
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Quantum algorithms for factoring and finding discrete logarithms have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. We present a quantum algorithm for the special case of dihedral groups which determines the hidden subgroup in a linear number of calls to the input function. We also explore the difficulties of developing an algorithm to process the data to explicitly calculate a generating set for the subgroup. A general framework for the noncommutative hidden subgroup problem is discussed and we indicate future research directions.
Fast Quantum Fourier Transforms for a Class of nonabelian Groups
"... . An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvabl ..."
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Cited by 30 (0 self)
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. An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2 n rst (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of nonabelian 2groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2 n is O(n 2 ) in all cases. 1 Introduction Quantum algorithms are a recent subject and possibly of central importance in physics and computer science. It has been shown that there are problems on which a putative quantum computer could outper...
The efficient computation of Fourier transforms on the symmetric group
 Mathematics of Computation
, 1998
"... Abstract. This paper introduces new techniques for the efficient computation of Fourier transforms on symmetric groups and their homogeneous spaces. We replace the matrix multiplications in Clausen’s algorithm with sums indexed by combinatorial objects that generalize Young tableaux, and write the r ..."
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Cited by 23 (4 self)
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Abstract. This paper introduces new techniques for the efficient computation of Fourier transforms on symmetric groups and their homogeneous spaces. We replace the matrix multiplications in Clausen’s algorithm with sums indexed by combinatorial objects that generalize Young tableaux, and write the result in a form similar to Horner’s rule. The algorithm we obtain computes the Fourier transform of a function on Sn in no more than 3 n(n − 1) Sn  multiplications 4 and the same number of additions. Analysis of our algorithm leads to several combinatorial problems that generalize path counting. We prove corresponding results for inverse transforms and transforms on homogeneous spaces. 1.
Separation of Variables and the Computation of Fourier Transforms on Finite Groups, I
 I. J. OF THE AMER. MATH. SOC
, 1997
"... This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler ..."
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Cited by 17 (7 self)
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This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler sums of products. This is the separation of variables algorithm. The conquer aspect is the final computation of matrix products which we perform efficiently using a special form of the matrices. This form arises from the use of subgroupadapted representations and their structure when evaluated at elements which lie in the centralizers of subgroups in a subgroup chain. We present a detailed analysis of the matrix multiplications arising in the calculation and obtain easytouse upper bounds for the complexity of our algorithm in terms of representation theoretic data for the group of interest. Our algorithm encompasses many of the known examples of fast Fourier transforms. We recover the b...
Automatic Generation of Fast Discrete Signal Transforms
, 2001
"... This paper presents an algorithm that derives fast versions for a broad class of discrete signal transforms symbolically. The class includes but is not limited to the discrete Fourier and the discrete trigonometric transforms. This is achieved by finding fast sparse matrix factorizations for the mat ..."
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Cited by 15 (7 self)
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This paper presents an algorithm that derives fast versions for a broad class of discrete signal transforms symbolically. The class includes but is not limited to the discrete Fourier and the discrete trigonometric transforms. This is achieved by finding fast sparse matrix factorizations for the matrix representations of these transforms. Unlike previous methods, the algorithm is entirely automatic and uses the defining matrix as its sole input. The sparse matrix factorization algorithm consists of two steps: First, the "symmetry" of the matrix is computed in the form of a pair of group representations; second, the representations are stepwise decomposed, giving rise to a sparse factorization of the original transform matrix. We have successfully demonstrated the method by computing automatically efficient transforms in several important cases: For the DFT, we obtain the CooleyTukey FFT; for a class of transforms including the DCT, type II, the number of arithmetic operations for our fast transforms is the same as for the bestknown algorithms. Our approach provides new insights and interpretations for the structure of these signal transforms and the question of why fast algorithms exist. The sparse matrix factorization algorithm is implemented within the software package AREP.
Decomposing Monomial Representations of Solvable Groups
, 2002
"... We present an efficient algorithm which decomposes... ..."
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Cited by 13 (4 self)
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We present an efficient algorithm which decomposes...
The Graphlet Spectrum
, 2009
"... Current graph kernels suffer from two limitations: graph kernels based on counting particular types of subgraphs ignore the relative position of these subgraphs to each other, while graph kernels based on algebraic methods are limited to graphs without node labels. In this paper we present the graph ..."
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Cited by 11 (4 self)
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Current graph kernels suffer from two limitations: graph kernels based on counting particular types of subgraphs ignore the relative position of these subgraphs to each other, while graph kernels based on algebraic methods are limited to graphs without node labels. In this paper we present the graphlet spectrum, a system of graph invariants derived by means of group representation theory that capture information about the number as well as the position of labeled subgraphs in a given graph. In our experimental evaluation the graphlet spectrum outperforms stateoftheart graph kernels.
Algebraic signal processing theory: Foundation and 1D time
 IEEE TRANS. SIGNAL PROCESS
, 2008
"... This paper introduces a general and axiomatic approach to linear signal processing (SP) that we refer to as the algebraic signal processing theory (ASP). Basic to ASP is the linear signal model defined as a triple ( 8) where familiar concepts like the filter space and the signal space are cast as an ..."
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Cited by 11 (6 self)
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This paper introduces a general and axiomatic approach to linear signal processing (SP) that we refer to as the algebraic signal processing theory (ASP). Basic to ASP is the linear signal model defined as a triple ( 8) where familiar concepts like the filter space and the signal space are cast as an algebra and a module, respectively. The mapping 8 generalizes the concept of atransform to bijective linear mappings from a vector space of signal samples into the module. Common concepts like filtering, spectrum, or Fourier transform have their equivalent counterparts in ASP. Once these concepts and their properties are defined and understood in the context of ASP, they remain true and apply to specific instantiations of the ASP signal model. For example, to develop signal processing theories for infinite and finite discrete time signals, for infinite or finite discrete space signals, or for multidimensional signals, we need only to instantiate the signal model to one that makes sense for that specific class of signals. Filtering, spectrum, Fourier transform, and other notions follow then from the corresponding ASP concepts. Similarly, common assumptions in SP translate into requirements on the ASP signal model. For example, shiftinvariance is equivalent to being commutative. For finite (duration) signals shift invariance then restricts to polynomial algebras. We explain how to design signal models from the specification of a special filter, the shift. The paper illustrates the general ASP theory with the standard time shift, presenting a unique signal model for infinite time and several signal models for finite time. The latter models illustrate the role played by boundary conditions and recover the discrete Fourier transform (DFT) and its variants as associated Fourier transforms. Finally, ASP provides a systematic methodology to derive fast algorithms for linear transforms. This topic and the application of ASP to space dependent signals and to multidimensional signals are pursued in companion papers.
Fast Fourier Transform for Fitness Landscapes
, 2001
"... We cast some classes of fitness landscapes as problems in spectral analysis on various Cayley graphs. In particular, landscapes derived from RNA folding are realized on Hamming graphs and analyzed in terms of Walsh transforms; assignment problems are interpreted as functions on the symmetric group a ..."
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Cited by 10 (2 self)
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We cast some classes of fitness landscapes as problems in spectral analysis on various Cayley graphs. In particular, landscapes derived from RNA folding are realized on Hamming graphs and analyzed in terms of Walsh transforms; assignment problems are interpreted as functions on the symmetric group and analyzed in terms of the representation theory of Sn . We show that explicit computations of the Walsh/Fourier transforms are feasible for landscapes with up to 10 8 configurations using Fast Fourier Transform techniques. We find that the cost function of a linear sum assignment problem involves only the defining representation of the symmetric group, while quadratic assignment problems are superpositions of the representations indexed by the partitions (n), (n  1, 1), (n  2, 2), and (n  2, 1, 1). These correspond to the four smallest eigenvalues of the Laplacian of the Cayley graph obtained from using transpositions as the generating set on Sn .
Applications of the Generalized Fourier Transform in Numerical Linear Algebra
 Department of Information Technology, Uppsala University
"... Matrices equivariant under a group of permutation matrices are considered. Such matrices typically arise in numerical applications where the computational domain exhibits geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix v ..."
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Cited by 5 (1 self)
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Matrices equivariant under a group of permutation matrices are considered. Such matrices typically arise in numerical applications where the computational domain exhibits geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform. This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions. The theory for applying the Generalized Fourier Transform is explained, building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform. The classical convolution theorem and diagonalization results are generalized to the noncommutative case of block diagonalizing equivariant matrices. Our presentation stresses the connection between multiplication with an equivariant matrices and the application of a convolution. This approach highlights the role of the underlying mathematical structures such as the group algebra, and it also simplifies the application of fast Generalized Fourier Transforms. The theory is illustrated with a selection of numerical examples. Key words: Non commutative Fourier analysis, equivariant operators, block diagonalization. 1 Introduction.