Abstract
The foundations of computer science are built upon the modified Church-Turing thesis. This thesis states that any reasonable model of computation can be simulated by a probabilistic Turing Machine with at most polynomial factor simulation overhead (see [10] for a discussion). Early interest in quantum computation from a computer science perspective was sparked by results indicating that quantum computers violate the modified Church-Turing thesis [3],[8]. The seminal work by Shor giving polynomial time algorithms for factorization and discrete logarithms [9] shook the foundations of modern cryptography, and gave a practical urgency to the area of quantum computation. All these quantum algorithms rely crucially upon properties of the Quantum Fourier transforms over finite Abelian groups. Indeed these properties are exactly what is required to solve a general problem known as the hidden subgroup problem, and it is easiest to approach the the algorithms for factoring and discrete logarithms as instances of this general approach. This survey paper focusses on presenting the essential ideas in a simple way, rather than getting the best results.
This research was supported by NSF Grant CCR-9800024, Darpa Grant F30602-00- 2-0601.
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Vazirani, U. (2002). Fourier Transforms and Quantum Computation. In: Khosrovshahi, G.B., Shokoufandeh, A., Shokrollahi, A. (eds) Theoretical Aspects of Computer Science. TACSci 2000. Lecture Notes in Computer Science, vol 2292. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45878-6_8
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