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Completing the Physical Representation of Quantum Algorithms Provides a Quantitative Explanation of Their Computational Speedup

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Abstract

The usual representation of quantum algorithms, limited to the process of solving the problem, is physically incomplete. We complete it in three steps: (i) extending the representation to the process of setting the problem, (ii) relativizing the extended representation to the problem solver to whom the problem setting must be concealed, and (iii) symmetrizing the relativized representation for time reversal to represent the reversibility of the underlying physical process. The third steps projects the input state of the representation, where the problem solver is completely ignorant of the setting and thus the solution of the problem, on one where she knows half solution (half of the information specifying it when the solution is an unstructured bit string). Completing the physical representation shows that the number of computation steps (oracle queries) required to solve any oracle problem in an optimal quantum way should be that of a classical algorithm endowed with the advanced knowledge of half solution.

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Notes

  1. Note that Alice knows the function \(s\left( b\right) \), the only thing she does not know is the problem setting selected by Bob.

  2. We are making reference to game theory, whose object is studying the mathematical models of conflict and cooperation between intelligent rational decision makers.

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Acknowledgements

Thanks for useful discussions and comments are due to Eli Cohen, Artur Ekert, Avshalom Elitzur, David Finkelstein, Daniel Shehan, and Ken Wharton.

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  1. Elsag Bailey Quantum Laboratory, Genoa, Italy

    Giuseppe Castagnoli

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  1. Giuseppe Castagnoli
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Correspondence to Giuseppe Castagnoli.

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Castagnoli, G. Completing the Physical Representation of Quantum Algorithms Provides a Quantitative Explanation of Their Computational Speedup. Found Phys 48, 333–354 (2018). https://doi.org/10.1007/s10701-018-0146-3

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