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Most quantum algorithms that give an exponential speedup over classical algorithms exploit the Fourier transform in some way.
Elementary Hadamard difference sets
J. Dillon · 1975
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O. S. Rothaus · 1976
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The Theory of Error–Correcting Codes
F. J. MacWilliams and N. J. A. Sloane · 1977
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An exact polynomial–time algorithm for Simon’s problem
G. Brassard and P. Høyer · 1997
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E. Bernstein and U. Vazirani · 1997
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Quantum computations: algorithms and error correction
A. Yu. Kitaev · 1997
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Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
P. Shor · 1997
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The hidden subgroup problem and eigenvalue estimation on a quantum computer
M. Mosca and A. Ekert · 1998
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Quantum Computation and Quantum Information
M. Nielsen and I. Chuang · 2000
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Succinct quantum proofs for properties of finite groups
J. Watrous · 2000
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Quantum fingerprinting
Buhrman, H. and Cleve, R. and Watrous, J. and de Wolf, R · 2001
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W. van Dam, S. Hallgren, and L. Ip · 2003
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Hidden translation and orbit coset in quantum computing
K. Friedl, G. Ivanyos, F. Magniez, M. Santha, and P. Sen · 2003
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Limitations of quantum coset states for graph isomorphism
S. Hallgren, C. Moore, M. Rötteler, A. Russell, and P. Sen · 2006
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Gowers uniformity, influence of variables, and PCPs
A. Samorodnitsky and T. Trevisan · 2006
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Quantum algorithms for learning and testing juntas
A. Atici and R. Servedio · 2007
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Quantum algorithms for hidden nonlinear structures
A. Childs, L. J. Schulman, and U. Vazirani · 2007
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Low-degree tests at large distances
A. Samorodnitsky · 2007
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An inverse theorem for the Gowers U 3 ( G ) U^{3}(G) norm
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Classical and quantum function reconstruction via character evaluation
A. Russell and I. Shparlinski · 2004
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From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
D. Bacon, A. Childs, and W. van Dam · 2005
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A subexponential-time quantum algorithm for the dihedral hidden subgroup problem
G. Kuperberg · 2005
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Quantum algorithms for highly non-linear boolean functions
M. Rötteler · 2008
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Efficient quantum algorithm for identifying hidden polynomials
Th. Decker, J. Draisma, and P. Wocjan · 2009
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Quantum algorithms for shifted subset problems
A. Montanaro · 2009
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