Analysis and Implementation of Quantum Computing Algorithms

Analysis and Implementation of Quantum Computing Algorithms poster

Audio Presentation:

Audio Transcript

Research Authorship:

Caroline Fedele, Dr. Asai Asaithambi

Faculty Mentor:

Dr. Asai Asaithambi | College of Computing, Engineering and Construction | School of Computing

Abstract:

In this research, we investigate the relationships between classical and quantum computing, and the superior time complexity and memory allocation proposed theoretically for quantum algorithms. This is accomplished by building quantum circuits to represent algorithms and test in a quantum computer simulation. Classical circuit components have been continually reducing in size to the point where they are now being impacted by quantum properties, resulting in the need to investigate quantum computing. The inherent parallelism of quantum computing also allows us to solve problems for which classical computers are inept. The class of intractable problems in computing where the solution can only be found through exhaustive search is where we observe quantum computing supremacy. It is important to explore and advance our knowledge of quantum computing so we are prepared for when it becomes a reality, and so that in the future our understanding of encryption is deepened and new quantum-proof methods can be developed. Although Google and IBM have developed physical quantum computers, there is still a large deficit in knowledge of how they may be utilized. There is also a gap between proposed superior quantum solutions and problems demonstrably solved using quantum algorithms. Among the problems classically considered intractable is one extremely relevant to cybersecurity, known as Shor’s algorithm, that being an efficient algorithm for integer factorization, breaks down our well-established methods of cryptography. The aim of this research is to show the differences and advantages of quantum computing, and to specifically demonstrate how we can use Shor’s algorithm in a quantum system.

Hello, my name is Caroline Fedele and I would like to tell you about the research I have begun with Professor Asaithambi and the plan for this project going forward.

            First I want to introduce you to quantum computing. What is this radical computing paradigm that has become a buzzword in scientific communities? Fundamentally, quantum computing is a completely different approach to computation from what we have now – from our laptops and smartphones to industrial supercomputers – they all fall under the realm of classical computing. Quantum computing takes advantage of certain principles in quantum mechanics, particularly superposition, and allows for a drastically more powerful approach to solving certain types of problems. This was first proposed in the 1970s but it wasn’t until recently with certain advances in technology that quantum computing became feasible. IBM and Google both now have small scale, but fully-functioning quantum computers. There remains a gap however between what quantum computers can potentially and theoretically do, and what has been put into practice, particularly in the area of algorithm development. There is still lacking a reliable methodology of designing quantum algorithms, which are entirely different from classical algorithms.

The concept and implementation of quantum computing is so revolutionary because any field that deals with very large data sets and large numbers of variables will be impacted. We will be able to simulate or model systems that currently can only be approximated. This includes the medical field and pharmaceuticals, modeling a how a disease progresses through the body; materials science and chemistry, seeing exactly how electrons flow through lattice structures; economics and finance, greatly advancing prediction capabilities and statistical analyses. Perhaps one of the most important impacts quantum computing will have however, is in cybersecurity and cryptography, which is the crux of this research project. One particular quantum algorithm, Shor’s algorithm, has been shown mathematically and theoretically to break down one of the strongest existing encryption systems, RSA cryptography.

It is a key aim of this research to show this by developing a reliable methodology for building quantum circuits and then testing Shor’s algorithm. In order to get there however, we need to develop a reliable methodology of designing quantum algorithms, which are entirely different from classical algorithms. The key information in determining how efficient an algorithm is the time complexity, how the algorithm’s runtime scales with the amount of data tested, and how much memory it uses. We therefore must test both quantum and classical algorithms, and obtain this information to really understand different algorithms.

<Figure 1> Here is an artistic rendering of a qubit, or quantum bit, that quantum computers utilize, as opposed to classical bits.

The importance of quantum computing when it comes to cybersecurity cannot be stressed enough. The most prized utility in the world today is data, which can be accessed and transmitted anywhere in the world almost instantaneously. The security of data is vitally important at a national and international level, governments depending on it, all the way down to the individual level. Almost all current methods for secure data transmission use RSA cryptography, a system that relies on one very difficult math problem, known as the “factoring problem,” or the finding to two prime factors of some very large integer. Until Shor’s algorithm was proposed in 1994, this problem was deemed intractable, or nearly impossible for a classical computer to solve short of millions of years. Shor’s algorithm reduces this to a tractable, polynomial-time problem that can be solved in minutes! Inevitably, physical quantum computers will become sufficiently advanced to break down this system, which is where quantum algorithm development comes in.

This plan for this particular project requires several steps before working on Shor’s algorithm. So far this semester, I have been conducting an extensive background review, reviewing the necessary mathematical skills, including linear algebra, complex numbers, and Fourier transforms to name a few; reviewing quantum mechanics to understand the physics of why this works, namely superposition, qubits & vector space; and a study into the existing literature – what has been put into practice already and where is there a gap in our knowledge?

The next step, I will work on for the remainder of this semester,  is to solve several problems using classical algorithms, programming them with java and obtaining time complexity and memory information, and then try to solve those problems using quantum algorithms, using IBM’s online quantum circuit development tool, QISkit. After this, we will specifically simulate Shor’s algorithm.

Our key goal in this research is to develop an accurate and reliable approach to building quantum algorithms and attempt to help close the gap between theory and practice.

Just a brief review of the physical and mathematical principles we are utilizing:

Superposition is the reason quantum computing is so efficient, the quantum bits inherently work in parallel, as opposed to classical bits which inherently work sequentially. Classical bits must be in a state of 0 or 1, but quantum bits can have any of the infinite values between 0 and 1, hence a superposition of both 0 and 1. This <figure 2> Bloch sphere is a visualization of how qubit state space works. Two key algorithms we will utilize in Shor’s algorithm are the quantum Fourier transform and modular exponentiation, two processes that contribute to the algorithm’s efficiency.

The expected outcome is fully functioning quantum circuits. <figure 3> Here is what a quantum circuit looks like, this one in particular a section, or subroutine, of Shor’s algorithm. <figure 4> IBM’s free, open-source software, QISkit, is how we will develop Shor’s and other algorithms.

            In conclusion, this project aims to demonstrate the superiority of quantum algorithms when approaching certain problems and to contribute to closing that cap between what has been proposed theoretically and what has actually been tried. Right now, there is a global race going on in academia and government, not unlike the space race of the 1960s, where we all want to be the first to develop useful, large-scale quantum computers. I hope this research can contribute to the advancement of quantum computing, particularly by developing quantum-proof encryption methods so we are prepared for the future advancements in this technology.

I’d like to thank my mentor, Professor Asaithambi, as well as UNF’s departments of physics and computing.

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