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arXiv:1711.09546

Quantum speedup to some types of polynomial equations

Changpeng Shao

Published Nov 27, 2017• 0 citations
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Abstract

In this paper, we consider three types of polynomial equations in quantum computer: linear divisibility equation, which belongs to a special type of binary-quadratic Diophantine equation; quadratic congruence equation with restriction in the solution and exponential congruence equation in finite field. Quantum algorithms based on Grover's algorithm and Shor's algorithm to these problems are given. As for the exponential congruence equation, which has been considered by Dam and Shparlinski \cite{dam} at 2008, a relatively simple quantum algorithm is given here. And some other results and generalizations are discovered.

DOI
10.48550/arxiv.1711.09546

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In this paper, we consider three types of polynomial equations in quantum computer: linear divisibility equation, which belongs to a special type of binary-quadratic Diophantine equation; quadratic congruence equation with restriction in the solution and exponential congruence equation in finite field. Quantum algorithms based on Grover's algorithm and Shor's algorithm to these problems are given. As for the exponential congruence equation, which has been considered by Dam and Shparlinski \cite{dam} at 2008, a relatively simple quantum algorithm is given here. And some other results and generalizations are discovered.
It is assumed that P is the product of two prime numbers q1and q2. If there is an integer 02= C (mod P), i.e., the congruence has a solution, then C is said to be a quadratic congruence (mod P). Quadratic congruence (mod P) is a NP-complete problem. If the value of C is equal to one, then four integer solutions for M2= 1 (mod P) are, respectively, b, P - b, 1 and P - 1, where 12= 1 (mod P) can be found by means of the proposed quantum algorithms with polynomial quantum gates, polynomial quantum bits and the successful probability that is the same as that of Shor's order-finding algorithm.
Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer values for the unknown variables and check for equality. Grover’s algorithm is a quantum search algorithm which can find marked indices in a list very efficiently. By treating the indices as the integer variables in the Diophantine equation, Grover’s algorithm can be used to find solutions in a brute force way more efficiently than classical methods. We present a hand-coded example for the simplest possible Diophantine equation, and results for a more complicated, but still simulable, equation encoded with a high-level quantum language.
In this work, we present a multi-layer quantum search method that generates an exponential speedup of the standard Grover's algorithm. As direct applications, any NP problems can be solved efficiently on a quantum circuit with only polynomial gate complexity. In particular, such multi-layer search can solve the factoring problem with an exponential speedup, providing an alternative to Shor's algorithm. Our results show that the exponential speedup of quantum circuits is ubiquitous, and Grover's search is much more powerful than that has been demonstrated. With no contradiction to the quadratic optimality of single-layer query complexity, the great potential of Grover's search is fully released by such multi-layer search design.
Quantum computing introduces an efficient way for unfolding complex problems on computing systems. It uses concept of superposition and entanglement, & hence supports intrinsic parallelism for obtaining the speedup in context of time over classical computing. Quantum algorithms can be efficiently implemented over quantum computing system. Most popular quantum algorithms like Grover's and Shor's provide a great foundation to specifically design the algorithms. These algorithms work on improving the lower runtime bounds of classical algorithms. Grover's algorithm exhibits a quadratic polynomial speedup while searching an element among `N' elements unstructured database. The algorithmic strategy can be applied over various computer science related applications and some important are being considered in this article i.e. pattern matching, data mining and machine learning. This article explores quantum fundamentals; history based evolution followed by Grover's search algorithm in detail with cascading advantages, limitations & applications and also finally concludes with the significance of quantum computing & quantum algorithmic domains.
The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.