Simplified Quantum Computing with Applications

Even today, quantum computing is still at a developing stage. The feature of this book is from introduction to deeply investigated research results. Therefore, the contents are fully satisfied with the authors'earnest will to transfer their concepts to the readers. Concretely speaking, the book is a simplified version of the classical quantum basic gate theory like Deutsch-Jozsa algorithm, Deutsch algorithm, Bernstein-Vazirani algorithm, Grover search algorithm, Simon algorithm, etc. with applications in the cryptography and coding theory. It reflects the actual status of research in these topics and at the same time it started from the beginning. The beginners can start to study these topics and the experts could find some new results. This book will compliment any standard textbook in quantum computing. With a simplified presentation, this book is suitable for both experts and beginners, helping researchers take their studies to the next level. Key Features •Focuses on the fundamental concepts of quantum computing from the beginning. •Examines the extraordinary connection between the Boolean algebra and quantum-gated computing. •Presents the new applications in quantum computing. •Compliments any standard textbook in quantum computing. •Provides a simplified presentation suitable for both experts and beginners PRELIMS.pdf Preface Acknowledgements Author biographies Koji Nagata Do Ngoc Diep Ahmed Farouk Tadao Nakamura Abstract CH001.pdf Chapter 1 Introduction 1.1 Introduction References CH002.pdf Chapter 2 Overview figures for a method of understanding quantum computing 2.1 What quantum-gated computing needs in its algorithms 2.2 Every reversibility in quantum circuits is by virtue of exclusive OR 2.3 Equivalence of the circuits by virtue of superposition of qubits to be applied by Hadamard gates 2.4 Bases of quantum computing 2.5 Preparation toward Deutsch’s algorithm using intuitive model of the quantum oracle Uf 2.6 Preparation with phase kickback toward Deutsch’s algorithm using an intuitive model of the quantum oracle Uf 2.7 Deutsch’s algorithm 2.8 Bernstein–Vazirani algorithm—general expression by eigenstate concept 2.9 Implementation of the phase oracle based on CNOT for the Bernstein–Vazirani algorithm 2.10 Implementation of the phase oracle based on CNOT for the Bernstein–Vazirani algorithm—secret string s = 101 case Reference CH003.pdf Chapter 3 Quantum key distribution based on a special Deutsch–Jozsa algorithm 3.1 Review of Deutsch’s algorithm 3.2 Deutsch’s algorithm with another input state 3.3 Deutsch’s algorithm using the Bell state 3.4 Quantum key distribution based on Deutsch’s algorithm 3.5 Review of the Deutsch–Jozsa algorithm 3.6 Special Deutsch–Jozsa algorithm 3.7 Special Deutsch–Jozsa algorithm with another input state 3.8 Special Deutsch–Jozsa algorithm using the GHZ state 3.9 Quantum key distribution based on the special Deutsch–Jozsa algorithm CH004.pdf Chapter 4 Quantum communication based on the Bernstein–Vazirani algorithm in a noisy environment 4.1 Review of the Bernstein–Vazirani algorithm 4.2 Quantum communication based on the Bernstein–Vazirani algorithm 4.3 Error correction based on the Bernstein–Vazirani algorithm 4.4 Evaluating simultaneously many functions using many parallel quantum systems 4.5 Method for evaluating a multiplication operation using the generalized Bernstein–Vazirani algorithm 4.6 Bernstein–Vazirani algorithm in a noisy environment CH005.pdf Chapter 5 Quantum communication based on Simon’s algorithm 5.1 Review of Simon’s algorithm 5.2 Quantum communication based on Simon’s algorithm CH006.pdf Chapter 6 Expansion of Deutsch’s algorithm 6.1 Expansion of Deutsch’s algorithm for determining all the mappings of a function 6.2 Deutsch’s algorithm 6.3 Expansion of Deutsch’s algorithm CH007.pdf Chapter 7 Some theoretically organized algorithm for quantum computers 7.1 New type of quantum algorithm for determining the 21 mappings of a function 7.2 New type of quantum algorithm for determining the 22 mappings of a function 7.3 Example using a logical function 7.4 New type of quantum algorithm for determining the 2N mappings of a function 7.5 Relation between set-theoretic atoms and the result in section 7.2 CH008.pdf Chapter 8 Some multi-quantum computing on quantum gating computers beyond a von Neumann architecture 8.1 Quantum algorithm for determining all the mappings of two logical functions 8.2 Overview of the quantum algorithm 8.3 Orthogonal pairs 8.4 Quantum algorithm for determining all the mappings of all 16 two-variable functions CH009.pdf Chapter 9 Quantum cryptography based on an algorithm for determining simultaneously all the mappings of a logical function 9.1 Quantum algorithm for determining all the two mappings of a logical function 9.2 Concrete example 9.3 Quantum algorithm for determining all the three mappings of a logical function 9.4 Concrete example 9.5 Quantum algorithm for determining all the 22 mappings of a logical function 9.6 Concrete example CH010.pdf Chapter 10 Quantum cryptography based on an algorithm for determining a function using qudit systems 10.1 Quantum cryptography based on an algorithm for determining a function using qudit systems 10.2 Concrete example CH011.pdf Chapter 11 Continuous-variable quantum computing and its applications to cryptography 11.1 Quantum cryptography based on an algorithm for determining a function using continuous-variable entangled states 11.2 Concrete example CH012.pdf Chapter 12 Various new forms of the Bernstein–Vazirani algorithm beyond qubit systems 12.1 Algorithm for determining a bit string 12.2 Extension to a natural number string 12.3 Extension to an integer string 12.4 Extension to a complex number string 12.5 Extension to a matrix string CH013.pdf Chapter 13 Creating genuine quantum algorithms for quantum energy-based computing 13.1 Quantum algorithm for determining a homogeneous linear function 13.2 Quantum algorithm for determining M homogeneous linear functions CH014.pdf Chapter 14 Quantum algorithms for finding the roots of a polynomial function 14.1 Finding the roots of a polynomial function by using a bit string 14.2 Finding the roots of a polynomial function by using a natural number string 14.3 Finding the roots of a polynomial function by using an integer string CH015.pdf Chapter 15 Quantum algorithm for rapidly plotting a function 15.1 Description of the algorithm CH016.pdf Chapter 16 Efficient exact quantum algorithm for the parity problem of a function 16.1 Description of the algorithm CH017.pdf Chapter 17 Necessary and sufficient condition for quantum computing 17.1 Necessary and sufficient condition for quantum computing CH018.pdf Chapter 18 Toward practical quantum-gated computers 18.1 Quantum algorithm for storing all the mappings of a logical function 18.2 Toward practically mathematical evaluations 18.3 Concrete quantum circuits for addition of any two numbers Reference CH019.pdf Chapter 19 Computational complexity in quantum computing 19.1 Quantum algorithm for storing simultaneously all the mappings of three logical functions 19.2 Typical arithmetic calculations CH020.pdf Chapter 20 Measurement theory in Deutsch’s algorithm based on the truth values 20.1 The new measurement theory can satisfy observability 20.2 Wave function analysis 20.3 New measurement theory 20.4 The new measurement theory can satisfy controllability CH021.pdf Chapter 21 Conclusions