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Hidden translation and orbit coset in quantum computing
 IN PROC. 35TH ACM STOC
, 2003
"... We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently ..."
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Cited by 48 (8 self)
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We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Z n p, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful selfreducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G. Our selfreducibility framework combined with Kuperberg’s subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.
Quantum approximation I. Embeddings of finite dimensional Lp spaces
 J. COMPLEXITY
, 2003
"... We study approximation of embeddings between finite dimensional Lp spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain ..."
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Cited by 17 (6 self)
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We study approximation of embeddings between finite dimensional Lp spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain quantum computation can essentially improve the rate of convergence of classical deterministic or randomized approximation, while there are other regions where the best possible rates coincide for all three settings. These results serve as a crucial building block for analyzing approximation in function spaces in a subsequent paper [11].
Quantum integration in Sobolev classes
, 2001
"... We study high dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes W r p ([0, 1]d) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optim ..."
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Cited by 13 (10 self)
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We study high dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes W r p ([0, 1]d) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Novak on integration of functions from Hölder classes.
From Monte Carlo to Quantum Computation
 PROCEEDINGS OF THE 3RD IMACS SEMINAR ON MONTE CARLO METHODS MCM2001, SALZBURG, SPECIAL ISSUE OF MATHEMATICS AND COMPUTERS IN SIMULATION
, 2003
"... Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring over classical deterministic or randomized methods for this ty ..."
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Cited by 11 (3 self)
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Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring over classical deterministic or randomized methods for this type of problem. In this
Quantum testers for hidden group properties
 FUNDAMENTA INFORMATICAE 1–16
, 2008
"... We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and for their testing problem in the classical model of computing. These are periodicity in groups and the c ..."
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Cited by 11 (1 self)
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We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and for their testing problem in the classical model of computing. These are periodicity in groups and the common coset range property of two functions having identical ranges within each coset of some normal subgroup. Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers. This is achieved by introducing a technique refining the majority correction process widely used for proving robustness of algebraic properties. The periodicity tester in nonAbelian groups and the common coset range tester are query efficient.
Why philosophers should care about computational complexity
 In Computability: Gödel, Turing, Church, and beyond (eds
, 2012
"... One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that ..."
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Cited by 9 (0 self)
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One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction, Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing
Data Structures in Natural Computing: Databases as Weak or Strong Anticipatory Systems
"... Abstract. Information systems anticipate the real world. Classical databases store, organise and search collections of data of that real world but only as weak anticipatory information systems. This is because of the reductionism and normalisation needed to map the structuralism of natural data on t ..."
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Abstract. Information systems anticipate the real world. Classical databases store, organise and search collections of data of that real world but only as weak anticipatory information systems. This is because of the reductionism and normalisation needed to map the structuralism of natural data on to idealised machines with von Neumann architectures consisting of fixed instructions. Category theory developed as a formalism to explore the theoretical concept of naturality shows that methods like sketches arising from graph theory as only nonnatural models of naturality cannot capture realworld structures for strong anticipatory information systems. Databases need a schema of the natural world. Natural computing databases need the schema itself to be also natural. Natural computing methods including neural computers, evolutionary automata, molecular and nanocomputing and quantum computation have the potential to be strong. At present they are mainly at the stage of weak anticipatory systems.
Locality, Weak or Strong Anticipation and Quantum Computing. I. Nonlocality in Quantum Theory
"... Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Categ ..."
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Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Category theory provides the necessary coordinatefree mathematical language which is both constructive and nonlocal to subsume the various interpretations of quantum theory in one pullback/pushout Dolittle diagram. This diagram can be used to test and classify physical devices and proposed algorithms for weak or strong anticipation. Quantum Information Science is more than a merger of ChurchTuring and quantum theories. It has constructively to bridge the nonlocal chasm between the weak anticipation of mathematics and the strong anticipation of physics.
Numerical Analysis on a Quantum Computer
"... We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bound ..."
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We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bounds for the quantum setting, we are in a position to compare them with those for the classical deterministic and randomized setting, previously obtained in informationbased complexity theory. This enables us to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.