Shor’s quantum factoring algorithm

Lomonaco, Samuel J.

doi:10.1090/psapm/058/1922897

Cited by 20 publications

(28 citation statements)

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“…. , N − 1} with hidden subgroup structure given by the commutative diagram From section 6, we know that Shor's algorithm [21], [25], [35], [36] solves the hidden subgroup problem ϕ : Z −→ Z N with hidden subgroup structure…”

Section: Does Grover's Algorithm Have Symmetries That We Can Exploit?mentioning

confidence: 99%

Quantum Hidden Subgroup Algorithms

Lomonaco¹,

Kauffman²

2007

Mathematics of Quantum Computation and Quantum Technology

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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 Deutsch-Jozsa, 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.

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Section: Does Grover's Algorithm Have Symmetries That We Can Exploit?mentioning

confidence: 99%

Quantum Hidden Subgroup Algorithms

Lomonaco¹,

Kauffman²

2007

Mathematics of Quantum Computation and Quantum Technology

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“…With high probability, the integers y and P are relatively prime, and thus, the unknown period P is found. , called a group probe 5 .…”

Section: Iia3 Shor's Factoring Algorithmmentioning

confidence: 99%

Search for New Quantum Algorithms

Lomonaco¹,

Kauffman²

2006

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching data sources, Gathering and maintaining the data needed, and completing and reviewing the collection of information. SPONSOR/MONITOR'S ACRONYM(S) 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)DARPA AFRL-IF-RS-TR-2006-180 DISTRIBUTION AVAILABILITY STATEMENTApproved for public release; distribution unlimited. PA# 06-356 SUPPLEMENTARY NOTES ABSTRACTThe first objective of this effort, searching for new quantum algorithms, created six new quantum hidden subgroup algorithms. The second objective, improving the theoretical understanding of existing quantum algorithms, produced three new systematic procedures. Also, application of combinatorial group theory led to substantial progress in the understanding and analysis of nonabelian quantum hidden subgroup algorithms. Additionally, methods and techniques of quantum topology have been used to obtain new results in quantum computing including discovery of a relationship between quantum entanglement and topological linking. The last objective, analyzing issues associated with algorithm implementation proposed distributed quantum computing (DQC) as a fast track to scalable quantum computing with technology available within the next five years. A universal set of DQC primitives has been created and used to transform the quantum Fourier transform and the Shor algorithm into DQC. The additional computational overhead needed for DQC algorithms is insignificant and DQC is found to simplify the decoherence problem. SUBJECT TERMSquantum algorithms, quantum hidden subgroup algorithms, hidden subgroups, distributed quantum computation, distributed algorithms, Shor's algorithm, Grover's algorithm, non-abelian quantum hidden subgroup algorithms, decoherence, quantum computer architectures, topological quantum computation, quantum gates, anyonic computation SECURITY

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“…Portanto, podemos rearranjar os termos de forma a fatorar o segundo registrador: assegura uma alta probabilidade em medir um valor de j carregando a informação desejada. Uma análise cuidadosa da expressão (4.4.15) é feita em [15], e um estudo meticuloso da forma do pico é feita em [9].…”

Section: Generalização Por Meio De Um Exemplounclassified

Uma Introdução à Computação Quântica

2004

Notas Em Matemática Aplicada

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Shor’s quantum factoring algorithm

Abstract: Abstract. This paper is a written version of a one hour lecture given on Peter Shor's quantum factoring algorithm.

Cited by 20 publications

References 2 publications

Quantum Hidden Subgroup Algorithms

Quantum Hidden Subgroup Algorithms

Search for New Quantum Algorithms

Uma Introdução à Computação Quântica

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