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Physics & Astronomy International Journal

Standard quantum algorithms and the property of a certain function

Koji Nagata, Tadao Nakamura

Published Jan 1, 2018• 0 citations
Publisher

Abstract

\n\tWe discuss a new mathematical structure for standard quantum algorithms. It says a certain property in case of a special function f that the relation f(x)=f(-x) holds. That is, the particular mathematical structure of the special function is that f(x) should be even if we assume -x=-x.
DOI
10.15406/paij.2018.02.00076

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We present a new quantum algorithm. It determines the property of a function. It is f(x)=f(−x)f(x)=f(-x)f(x)=f(−x). How fast can we succeed? The quantum algorithm does not use the Hadamard transformation. All we need is of evaluating ∣0,0,...,1⏞N⟩|\overbrace{0,0,...,1}^N\rangle∣0,0,...,1​N​⟩. And we can know the global property, that is, we can realize f(x)=f(−x)f(x)=f(-x)f(x)=f(−x) for numbers. Our quantum algorithm overcomes a classical counterpart by a factor of O(2N)O(2^N)O(2N).
We discuss a character of quantum algorithms. In fact, all of them determine the property of a certain function. The function under study must have the property f(x)=f(−x)f(x) = f(-x)f(x)=f(−x) when f(x)≠0f(x)\neq 0f(x)=0.
We introduce and examine quantum states of a special kind, referred to as E-states, whose properties are both structurally and functionally analogous to the sieve of Eratosthenes. More broadly, the concept of an E-state is related to a certain noncommutative extension of the Dirichlet ring, also discussed here for the first time. Furthermore, we demonstrate that E-states can be implemented on a universal quantum computer and, as a particular application, we construct an algorithm which implements the Dirichlet multiplication of sequences on a quantum computer. We also discuss the potential applicability of E-states to the problem of integer factorization although, we haste to add, we are not aware at present of the possibility of using this approach to obtain algorithms of sub-exponential complexity.