Cybersecurity and Quantum Computing: friends or foes?

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Hard Mathematical Problems in Cryptography and Coding Theory

In this thesis, we are concerned with certain interesting computationally hard problems and the complexities of their associated algorithms. All of these problems share a common feature in that they all arise from, or have applications to, cryptography, or the theory of error correcting codes. Each chapter in the thesis is based on a stand-alone paper which attacks a particular hard problem. The problems and the techniques employed in attacking them are described in detail. The first problem concerns integer factorization: given a positive integer $N$. the problem is to find the unique prime factors of $N$. This problem, which was historically of only academic interest to number theorists, has in recent decades assumed a central importance in public-key cryptography. We propose a method for factorizing a given integer using a graph-theoretic algorithm employing Binary Decision Diagrams (BDD). The second problem that we consider is related to the classification of certain naturally arising classes of error correcting codes, called self-dual additive codes over the finite field of four elements, $GF(4)$. We address the problem of classifying self-dual additive codes, determining their weight enumerators, and computing their minimum distance. There is a natural relation between self-dual additive codes over $GF(4)$ and graphs via isotropic systems. Utilizing the properties of the corresponding graphs, and again employing Binary Decision Diagrams (BDD) to compute the weight enumerators, we can obtain a theoretical speed up of the previously developed algorithm for the classification of these codes. The third problem that we investigate deals with one of the central issues in cryptography, which has historical origins in the theory of geometry of numbers, namely the shortest vector problem in lattices. One method which is used both in theory and practice to solve the shortest vector problem is by enumeration algorithms. Lattice enumeration is an exhaustive search whose goal is to find the shortest vector given a lattice basis as input. In our work, we focus on speeding up the lattice enumeration algorithm, and we propose two new ideas to this end. The shortest vector in a lattice can be written as ${\bf s} = v_1{\bf b}_1+v_2{\bf b}_2+\ldots+v_n{\bf b}_n$. where $v_i \in \mathbb{Z}$ are integer coefficients and ${\bf b}_i$ are the lattice basis vectors. We propose an enumeration algorithm, called hybrid enumeration, which is a greedy approach for computing a short interval of possible integer values for the coefficients $v_i$ of a shortest lattice vector. Second, we provide an algorithm for estimating the signs $+$ or $-$ of the coefficients $v_1,v_2,\ldots,v_n$ of a shortest vector ${\bf s}=\sum_{i=1}^{n} v_i{\bf b}_i$. Both of these algorithms results in a reduction in the number of nodes in the search tree. Finally, the fourth problem that we deal with arises in the arithmetic of the class groups of imaginary quadratic fields. We follow the results of Soleng and Gillibert pertaining to the class numbers of some sequence of imaginary quadratic fields arising in the arithmetic of elliptic and hyperelliptic curves and compute a bound on the effective estimates for the orders of class groups of a family of imaginary quadratic number fields. That is, suppose $f(n)$ is a sequence of positive numbers tending to infinity. Given any positive real number $L$. an effective estimate is to find the smallest positive integer $N = N(L)$ depending on $L$ such that $f(n) > L$ for all $n > N$. In other words, given a constant $M > 0$. we find a value $N$ such that the order of the ideal class $I_n$ in the ring $R_n$ (provided by the homomorphism in Soleng's paper) is greater than $M$ for any $n>N$. In summary, in this thesis we attack some hard problems in computer science arising from arithmetic, geometry of numbers, and coding theory, which have applications in the mathematical foundations of cryptography and error correcting codes

Quantum Speed-ups for Boolean Satisfiability and Derivative-Free Optimization

In this thesis, we have considered two important problems, Boolean satisfiability (SAT) and derivative free optimization in the context of large scale quantum computers. In the first part, we survey well known classical techniques for solving satisfiability. We compute the approximate time it would take to solve SAT instances using quantum techniques and compare it with state-of-the heart classical heuristics employed annually in SAT competitions. In the second part of the thesis, we consider a few classically well known algorithms for derivative free optimization which are ubiquitously employed in engineering problems. We propose a quantum speedup to this classical algorithm by using techniques of the quantum minimum finding algorithm. In the third part of the thesis, we consider practical applications in the fields of bio-informatics, petroleum refineries and civil engineering which involve solving either satisfiability or derivative free optimization. We investigate if using known quantum techniques to speedup these algorithms directly translate to the benefit of industries which invest in technology to solve these problems. In the last section, we propose a few open problems which we feel are immediate hurdles, either from an algorithmic or architecture perspective to getting a convincing speedup for the practical problems considered

A Brief Review on Mathematical Tools Applicable to Quantum Computing for Modelling and Optimization Problems in Engineering

Since its emergence, quantum computing has enabled a wide spectrum of new possibilities and advantages, including its efficiency in accelerating computational processes exponentially. This has directed much research towards completely novel ways of solving a wide variety of engineering problems, especially through describing quantum versions of many mathematical tools such as Fourier and Laplace transforms, differential equations, systems of linear equations, and optimization techniques, among others. Exploration and development in this direction will revolutionize the world of engineering. In this manuscript, we review the state of the art of these emerging techniques from the perspective of quantum computer development and performance optimization, with a focus on the most common mathematical tools that support engineering applications. This review focuses on the application of these mathematical tools to quantum computer development and performance improvement/optimization. It also identifies the challenges and limitations related to the exploitation of quantum computing and outlines the main opportunities for future contributions. This review aims at offering a valuable reference for researchers in fields of engineering that are likely to turn to quantum computing for solutions. Doi: 10.28991/ESJ-2023-07-01-020 Full Text: PD

Quantum Control Machine: The Limits of Quantum Programs as Data

Quantum algorithms for factorization, search, and simulation obtain computational advantage by performing control flow such as branching and iteration based on the value of quantum data in superposition. Complicating realization of these algorithms is the fact that in predominant quantum machine models, all control flow as embodied by the program counter is classical, and cannot exist in superposition. In this work, we identify that an alternative model to enable a program counter in superposition faces an obstacle -- no such machine can correctly support control flow constructs with non-injective semantics, including the conventional conditional jump. In fact, prior attempts to support this instruction cause programs to inappropriately collapse the superposition of data, meaning that quantum advantage is lost. We present a quantum machine model that supports both quantum effects on data and data-dependent control flow, using variants of conditional jump with injective semantics. We identify the necessary condition for programs for such a machine to preserve superposition of data, and show that expressible programs coincide with the unitary quantum circuits.Comment: 12 pages, 8 figure

Quantum Computing for Fusion Energy Science Applications

This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum computation, and quantum algorithms. Then, we explore the topic of using quantum computers to simulate both linear and nonlinear dynamics in greater detail. Because quantum computers can only efficiently perform linear operations on the quantum state, it is challenging to perform nonlinear operations that are generically required to describe the nonlinear differential equations of interest. In this work, we extend previous results on embedding nonlinear systems within linear systems by explicitly deriving the connection between the Koopman evolution operator, the Perron-Frobenius evolution operator, and the Koopman-von Neumann evolution (KvN) operator. We also explicitly derive the connection between the Koopman and Carleman approaches to embedding. Extension of the KvN framework to the complex-analytic setting relevant to Carleman embedding, and the proof that different choices of complex analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert space metric are covered in the appendices. Finally, we conclude with a review of recent quantum hardware implementations of algorithms on present-day quantum hardware platforms that may one day be accelerated through Hamiltonian simulation. We discuss the simulation of toy models of wave-particle interactions through the simulation of quantum maps and of wave-wave interactions important in nonlinear plasma dynamics.Comment: 42 pages; 12 figures; invited paper at the 2021-2022 International Sherwood Fusion Theory Conferenc