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52
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 74 (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 parallel circuits for the quantum Fourier transform
 PROCEEDINGS 41ST ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’00)
, 2000
"... We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our ..."
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Cited by 50 (2 self)
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We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomialsize, in combination with classical polynomialtime pre and postprocessing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with boundederror probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the
Combinatorial Landscapes
 SIAM REVIEW
, 2002
"... Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, ne ..."
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Cited by 34 (2 self)
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Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, nearness, distance or accessibility. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the "static" properties of landscapes as well as their influence on the dynamics of adaptation. In this review we focus on the connections of landscape theory with algebraic combinatorics and random graph theory, where exact results are available.
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...
Quantum factoring, discrete logarithms and the hidden subgroup problem
"... Amongst the most remarkable successes of quantum computation are Shor’s efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential ingredients of these algorithms and draw out the unifying gener ..."
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Cited by 24 (0 self)
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Amongst the most remarkable successes of quantum computation are Shor’s efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential ingredients of these algorithms and draw out the unifying generalization of the socalled abelian hidden subgroup problem. This involves an unexpectedly harmonious alignment of the formalism of quantum physics with the elegant mathematical theory of group representations and fourier transforms on finite groups. Finally we consider the nonabelian hidden subgroup problem mentioning some open questions where future quantum algorithms may be expected to have a substantial impact. 1
Quantum algorithms: Entanglement enhanced information processing
 Phil. Trans. R. Soc. Lond. A
, 1998
"... Abstract: We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform algori ..."
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Cited by 23 (1 self)
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Abstract: We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform algorithm. We describe the implementation of the FFT algorithm for the group of integers modulo 2 n in the quantum context, showing how the grouptheoretic formalism leads to the standard quantum network and identifying the property of entanglement that gives rise to the exponential speedup (compared to the classical FFT). Finally we outline the use of the Fourier transform in extracting periodicities, which underlies its utility in the known quantum algorithms.
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 22 (3 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.
Quantum fourier sampling simplified
 IN PROCEEDINGS OF THE THIRTYFIRST ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1999
"... We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over Z p can be efficiently approximated by transforming over Z q for any q in a ..."
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Cited by 19 (2 self)
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We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over Z p can be efficiently approximated by transforming over Z q for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing previous applications. In addition, our proof easily generalizes to multidimensional transforms for any constant number of dimensions.
Amplitude spectra of fitness landscapes
 J. COMPLEX SYSTEMS
, 1998
"... Fitness landscapes can be decomposed into elementary landscapes using a Fourier transform that is determined by the structure of the underlying con guration space. The amplitude spectrum obtained from the Fourier transform contains information about the ruggedness of the landscape. It can be used f ..."
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Cited by 18 (9 self)
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Fitness landscapes can be decomposed into elementary landscapes using a Fourier transform that is determined by the structure of the underlying con guration space. The amplitude spectrum obtained from the Fourier transform contains information about the ruggedness of the landscape. It can be used for classi cation and comparison purposes. We consider here three very di erent types of landscapes using both mutation and recombination to de ne the topological structure of the con guration spaces. A reliable procedure for estimating the amplitude spectra is presented. The method is based on certain correlation functions that are easily obtained from empirical studies of the landscapes.
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...