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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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Cited by 6 (2 self)
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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
Shor’s Algorithm for Factoring Large Integers ∗
, 2008
"... This work is a tutorial on Shor’s factoring algorithm by means of a worked out example. Some basic concepts of Quantum Mechanics and quantum circuits are reviewed. It is intended for nonspecialists which have basic knowledge on undergraduate Linear Algebra. 1 ..."
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This work is a tutorial on Shor’s factoring algorithm by means of a worked out example. Some basic concepts of Quantum Mechanics and quantum circuits are reviewed. It is intended for nonspecialists which have basic knowledge on undergraduate Linear Algebra. 1
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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.