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Quantum computational algorithms exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms.
Feynman, R. P. Simulating Physics with computers. Int. J. Theor. Phy
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Shor, P. W. in Algorithms for Quantum Computation: Discrete Logarithms and Factoring (ed. Goldwasser, S.) 124-134 ( Proc. 35th Annu. Symp. Foundations of Computer Science
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Mosca, M. & Ekert, A. The hidden subgroup problem and eigenvalue estimation on a quantum computer 174–188 ( Lecture Notes in Computer Science
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While the motivation of Ref. Parker and Plenio 2000 was to show that efficient factoring remains possible in systems where the preparation of pure states is challenging, it has not been widely appreciated that a key step—recycling of the control qubit also described in Ref. Mosca and Ekert 1999 —dramatically reduces the resource requirement in all implementations: the reduction in resources comes with only a polynomial cost in time, while maintaing exponential speed-up of the algorithm
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Parker, S. & Plenio, M. B. Efficient Factorization with a Single Pure Qubit and logN Mixed Qubits. Phys. Rev. Lett
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Fidelity between two quantum states ρ \rho and σ \sigma is defined Nielsen and Chuang 2000 to be F ( ρ , σ ) ≡ Tr ρ 1 / 2 σ ρ 1 / 2 F(\rho,\sigma)\equiv\operatorname{Tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}
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Vandersypen, L. M. K. et al. Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature
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Hofmann, H. F. & Takeuchi, S. Quantum phase gate for photonic qubits using only beam splitters and postselection. Phys. Rev. A
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Lanyon, B. P. et al. Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement. Phys. Rev. Lett
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Veis, L. & Pittner, J. Quantum computing applied to calculations of molecular energies: CH2 benchmark. The Journal of Chemical Physics
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O’Brien, J. L., Pryde, G. J., White, A. G., Ralph, T. C. & Branning, D. Demonstration of an all-optical quantum controlled-NOT gate. Nature
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Lu, C.-Y., Browne, D. E., Yang, T. & Pan, J.-W. Demonstration of a Compiled Version of Shor’s Quantum Factoring Algorithm Using Photonic Qubits. Phys. Rev. Lett
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Cited alongside, same era.
In fact the same is generally true of a control register based on d d dimensional systems (qudits) and orders that are a power of d d , r = d p r=d^{p}
Cited in the paper.
Shor’s algorithm is designed to work for even orders only, as the classical subroutine must calculate gcd ( x r 2 ± 1 , N ) \gcd(x^{\frac{r}{2}}\pm 1,N) . However, for certain choices of square coprime x x and odd order, the algorithm works. For example, in the case of x = 2 2 j x=2^{2j} with j j an integer, a numerical simulation of Shor’s algorithm for the first (odd) 4851 4851 composite N N (product of two primes) finds approximately 58 % 58\% of this class of coprimes produces an odd order; of these approximately 36 % 36\% successfully lead to non trivial factors of N N in the classical gcd \gcd subroutine. Order finding for N = 21 N=21 with x = 4 x=4 , resulting in r = 3 r=3 is one such successful example
Cited in the paper.
Note that, while encoding in higher dimensions will reduce the number of required photons, encoding the entire work register in a single 2 m 2^{m} level system typically requires resources exponential in m m
Cited in the paper.
2010
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Whitfield, J. D., Biamonte, J. & Aspuru-Guzik, A. Simulation of Electronic Structure Hamiltonians Using Quantum Computers. Mol. Phys
2011
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Li, Z. et al. Solving Quantum Ground-State Problems with Nuclear Magnetic Resonance. Sci. Rep
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Zhou, X.-Q. et al. Adding control to arbitrary unknown quantum operations. Nature. Commun
2011
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