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Civil War Book Review

Extension of Shor's period-finding algorithm to infinite dimensional Hilbert spaces

Jeremy J. Becnel

Published Mar 20, 2006• 0 citations
Publisher

Abstract

Over the last decade quantum computing has become a very popular field in various disciplines, such as physics, engineering, and mathematics. Most of the attraction stemmed from the famous Shor period--finding algorithm, which leads to an efficient algorithm for factoring positive integers. Many adaptations and generalizations of this algorithm have been developed through the years, some of which have not been ripened with full mathematical rigor. In this dissertation we use concepts from white noise analysis to rigorously develop a Shor algorithm adapted to find a hidden subspace of a function with domain a real Hilbert space. After reviewing the framework of quantum mechanics, we demonstrate how these principles can be used to develop algorithms which operate on a quantum computing device. We present a self-contained account of white noise analysis, including the main relevant results. Inspired by a generalized function in the algorithm, we develop a new distribution, the delta function for a subspace of an infinite dimensional Hilbert space. We then use this distribution to rigorously prove one of the main identities needed for the algorithm. Finally we provide a rigorous formulation of the hidden subspace algorithm in infinite dimensions.

DOI
10.31390/gradschool_dissertations.20

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Over the last decade quantum computing has become a very popular field in various disciplines, such as physics, engineering, and mathematics. Most of the attraction stemmed from the famous Shor period--finding algorithm, which leads to an efficient algorithm for factoring positive integers. Many adaptations and generalizations of this algorithm have been developed through the years, some of which have not been ripened with full mathematical rigor. In this dissertation we use concepts from white noise analysis to rigorously develop a Shor algorithm adapted to find a hidden subspace of a function with domain a real Hilbert space. After reviewing the framework of quantum mechanics, we demonstrate how these principles can be used to develop algorithms which operate on a quantum computing device. We present a self-contained account of white noise analysis, including the main relevant results. Inspired by a generalized function in the algorithm, we develop a new distribution, the delta function for a subspace of an infinite dimensional Hilbert space. We then use this distribution to rigorously prove one of the main identities needed for the algorithm. Finally we provide a rigorous formulation of the hidden subspace algorithm in infinite dimensions.
It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the information-processing in the three algorithms are presented from a different perspective, in terms of the way in which the algorithms exploit the non-Boolean quantum logic represented by the projective geometry of Hilbert space. From this quantum logical perspective, the XOR algorithm appears directly as a special case of Simon's algorithm, and all three algorithms can be seen as exploiting the non-Boolean logic represented by the subspace structure of Hilbert space in a similar way. Essentially, a global property of a function (such as a period, or a disjunctive property) is encoded as a subspace in Hilbert space representing a quantum proposition, which can then be efficiently distinguished from alternative propositions, corresponding to alternative global properties, by a measurement (or sequence of measurements) that identifies the target proposition as the proposition represented by the subspace containing the final state produced by the algorithm.
It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the information-processing in the three algorithms are presented from a different perspective, in terms of the way in which the algorithms exploit the non-Boolean quantum logic represented by the projective geometry of Hilbert space. From this quantum logical perspective, the XOR algorithm appears directly as a special case of Simon's algorithm, and all three algorithms can be seen as exploiting the non-Boolean logic represented by the subspace structure of Hilbert space in a similar way. Essentially, a global property of a function (such as a period, or a disjunctive property) is encoded as a subspace in Hilbert space representing a quantum proposition, which can then be efficiently distinguished from alternative propositions, corresponding to alternative global properties, by a measurement (or sequence of measurements) that identifies the target proposition as the proposition represented by the subspace containing the final state produced by the algorithm.
Quantum Information and Quantum Computation are rapidly evolving domains in their theoretical foundations, implementations and applications. The main scope of the present book is to give the basic ideas of this field to undergraduate students in Physics. A rigorous approach is used to provide the key concepts and methods. Many examples are discussed. To be self-contained, the principles of Quantum Mechanics are reviewed. The discussion is aimed to the specific, but paradigmatic, factorization problem of integers, dealt by the Quantum Shor’s Algorithm. Thus, notions of the Number Theory are given in the first Chapter, while the exposition of the algorithm and its implementation is reserved to the last Section.
This work presents a generalized period decomposition approach, significantly improving the practical reliability of Shor's quantum factoring algorithm. Although Shor's algorithm theoretically enables polynomial-time integer factorization, its real-world performance heavily depends on stringent conditions related to the period obtained via quantum phase estimation. Our generalized decomposition method relaxes these conditions by systematically exploiting arbitrary divisors of the obtained period, effectively broadening the applicability of each quantum execution. Extensive classical simulations were performed to empirically validate our approach, involving over one million test cases across integers ranging from 2 to 8 digits. The proposed method achieved near-perfect success rates, exceeding 99.998% for 7-digit numbers and 99.999% for 8-digit numbers, significantly surpassing traditional and recently improved variants of Shor's algorithm. Crucially, this improvement is achieved without compromising the algorithm's polynomial-time complexity and integrates seamlessly with existing quantum computational frameworks. Moreover, our method enhances the efficiency of quantum resource usage by minimizing unnecessary repetitions, making it particularly relevant for quantum cryptanalysis with noisy intermediate-scale quantum (NISQ) devices. This study thus provides both theoretical advancements and substantial practical benefits, contributing meaningfully to the field of quantum algorithm research and the broader field of quantum information processing.
The unique principles of quantum physics introduce an element of uncertainty into our previously deterministic world. Quantum computing, derived from quantum physics, enables the representation of a qubit in both |0> and |1> states, thus encoding more information and unleashing a realm of new possibilities. This paper conducts an analysis of the classical RSA encryption protocol and its vulnerability to the quantum algorithm—Shor's algorithm. Furthermore, this paper rigorously establishes the mathematical theories required for the analysis. Finally, the algorithm is thoroughly explored, and the potential efficiencies of quantum computing are discussed. As the study of quantum computing deepens, its application is poised to surpass that of traditional computers in various fields, leaving a profound impact.
In this paper, we use the methods found in quant-ph/0201095 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 F:R-->R from the reals R to the reals R, where F 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 algorithm or some approximation of it implementable?