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Quantum hidden subgroup algorithms on free groups, (in preparation
"... Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In thi ..."
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Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents
A CONTINUOUS VARIABLE SHOR ALGORITHM
, 2004
"... Abstract. In this paper, we use the methods found in [21] to create a continuous variable analogue of Shor’s quantum factoring algorithm. By this we mean a quantum hidden subgroup algorithm that finds the period P of a function Φ: R − → R from the reals R to the reals R, where Φ belongs to a very ge ..."
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Cited by 2 (2 self)
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Abstract. In this paper, we use the methods found in [21] to create a continuous variable analogue of Shor’s quantum factoring algorithm. By this we mean a quantum hidden subgroup algorithm that finds the period P of a function Φ: R − → R from the reals R to the reals R, where Φ belongs to a very general class of functions, called the class of admissible functions. One objective in creating this continuous variable quantum algorithm was to make the structure of Shor’s factoring algorithm more mathematically transparent, and thereby give some insight into the inner workings of Shor’s original algorithm. This continuous quantum algorithm also gives some insight into the inner workings of Hallgren’s Pell’s equation algorithm. Two key questions remain unanswered. Is this quantum algorithm more efficient than its classical continuous variable counterpart? Is this quantum
Continuous Quantum Hidden Subgroup Algorithms
, 2003
"... In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Pat ..."
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Cited by 2 (1 self)
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In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Paths of L 2 paths x: [0, 1] → R n in real nspace R n. Also included is a curious discrete QHS algorithm which is dual to Shor’s algorithm. Contents 1
POSITIVE SESQUILINEAR FORM MEASURES AND GENERALIZED EIGENVALUE EXPANSIONS
, 2008
"... Abstract. Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are forma ..."
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Cited by 1 (1 self)
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Abstract. Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are formal similarities with earlier approaches to special cases of the problem, the paper differs e.g. from standard rigged Hilbert space constructions and avoids the introduction of nuclear spaces. The techniques are predominantly measure theoretic and hence the Hilbert spaces involved are separable. The results range from a Naimark type dilation result to direct integral representations and a fairly concrete generalized eigenvalue expansion for unbounded normal operators. Mathematics Subject Classification. 47A70 (Primary); 47A07, 47B15 (Secondary).
Symmetry, Integrability and Geometry: Methods and Applications Time Asymmetric Quantum Mechanics ⋆
"... Abstract. The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone–von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary gro ..."
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Abstract. The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone–von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution. Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)). If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ = � Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS’s of Hardy functions and connected with it, to a semigroup time evolution t0 ≤ t < ∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system. The decay of quasistable particles is used to illustrate this quantum mechanical time asymmetry. From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.
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, 2002
"... Abstract. In this paper, we use the methods found in [12] to create a continuous variable analogue of Shor’s quantum factoring algorithm. By this we mean a quantum hidden subgroup algorithm that finds the period P of a function Φ: R − → C from the reals R to the complex numbers C, where Φ belongs to ..."
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Abstract. In this paper, we use the methods found in [12] to create a continuous variable analogue of Shor’s quantum factoring algorithm. By this we mean a quantum hidden subgroup algorithm that finds the period P of a function Φ: R − → C from the reals R to the complex numbers C, where Φ belongs to a very general class of functions, called the class of admissible functions. This algorithm gives some insight into the inner workings of Shor’s quantum factoring algorithm.