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A New Quantum Algorithm in Case of a Special Function (New Version)

Koji Nagata, Tadao Nakamura, Han J.F. Geurdes, Ahmed Farouk, Josep Amengual i Batle, S. Abdalla, Germano Resconi

Published May 1, 2017• 0 citations
OpenAlex

Abstract

We present a new quantum algorithm. It determines the property of a function. It is $f(x)=f(-x)$. How fast can we succeed? The quantum algorithm does not use the Hadamard transformation. All we need is of evaluating $|\overbrace{0,0,...,1}^N\rangle$. And we can know the global property, that is, we can realize $f(x)=f(-x)$ for numbers. Our quantum algorithm overcomes a classical counterpart by a factor of $O(2^N)$.

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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).
Here, we propose a new type of quantum algorithm for determining the 2N2^N2N values of a function. By measuring the single output state, we determine all the values of f(x)f(x)f(x) for all xxx simultaneously. This is very interesting indeed: the quantum circuit gives us the ability to determine a perfect property of f(x)f(x)f(x), namely, f(x)f(x)f(x). This is faster than a classical apparatus by a factor of 2N2^N2N.
We propose quantum cryptography based on an algorithm of determining a function. The security of our cryptography is based on the Ekert 1991 protocol, that is, we use an entangled state. Eve must destroy the entangled state. Consider a function. Alice knows all the mappings concerning the function. Bob knows none of them. His aim is of obtaining all of them without Eve's attack. In classical case, Bob needs some queries. In quantum case, Bob needs just a query. By measuring the single entangled state, which is sent by Alice, Bob can obtain all the mappings concerning the function, simultaneously. This is faster than classical cryptography.
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 present the role that spectral methods play in the development of the most impressive quantum algorithms, such as the polynomial time number factoring algorithm by Shor. While the fast transform algorithms reduce the number of operations needed in obtaining the transforms from O(2/sup 2n/) to O(n/sup 2n/), quantum transforms are in comparison super-fast. The quantum Fourier transform can be performed in O(n/sup 2/) time, while the specific cases of Walsh-Hadamard and Chrestenson transforms require only O(n) operations. We derive quantum Fourier transform over non-binary quantum digits using the Chrestenson gate, which can serve as a basic block for quantum transforms.
Modern programming relies on our ability to treat preprogrammed functions as black boxes - we can invoke them as subroutines without knowing their physical implementation. Here we show it is generally impossible to execute an unknown quantum subroutine. This, as a special case, forbids applying black-box subroutines conditioned on an ancillary qubit. We explore how this limits many quantum algorithms - forcing their circuit implementation to be individually tailored to specific inputs and inducing failure if these inputs are not known in advance. We present a method to avoid this situation for certain computational problems. We apply this method to enhance existing quantum factoring algorithms; reducing their complexity, and the extent to which they need to be tailored to factor specific numbers. Thus, we highlight a natural property of classical information that fails in the advent of quantum logic; and simultaneously demonstrate how to mitigate its effects in practical situations.