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arXiv:1709.01236

Early days following Grover's quantum search algorithm

Fang Song

Published Sep 5, 2017• 1 citation
arXivHTMLPDF

Abstract

This is a note accompanying "CS 410/510: INTRO TO QUANTUM COMPUTING" I taught at Portland State University in Spring 2017. It is a review and summary of some early results related to Grover's quantum search algorithm in a consistent way. I had to go back and forth among several books, notes and original papers to sort out various details when preparing the lectures, which was a pain. This is the motivation behind this note. I would like to thank Peter Høyer for valuable feedback on this note.

DOI
10.48550/arxiv.1709.01236

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This is a note accompanying "CS 410/510: INTRO TO QUANTUM COMPUTING" I taught at Portland State University in Spring 2017. It is a review and summary of some early results related to Grover's quantum search algorithm in a consistent way. I had to go back and forth among several books, notes and original papers to sort out various details when preparing the lectures, which was a pain. This is the motivation behind this note. I would like to thank Peter Høyer for valuable feedback on this note.
These are the lecture notes of the master's course "Quantum Computing", taught at Chalmers University of Technology every fall since 2020, with participation of students from RWTH Aachen and Delft University of Technology. The aim of this course is to provide a theoretical overview of quantum computing, excluding specific hardware implementations. Topics covered in these notes include quantum algorithms (such as Grover's algorithm, the quantum Fourier transform, phase estimation, and Shor's algorithm), variational quantum algorithms that utilise an interplay between classical and quantum computers [such as the variational quantum eigensolver (VQE) and the quantum approximate optimisation algorithm (QAOA), among others], quantum error correction, various versions of quantum computing (such as measurement-based quantum computation, adiabatic quantum computation, and the continuous-variable approach to quantum information), the intersection of quantum computing and machine learning, and quantum complexity theory. Lectures on these topics are compiled into 12 chapters, most of which contain a few suggested exercises at the end, and interspersed with four tutorials, which provide practical exercises as well as further details. At Chalmers, the course is taught in seven weeks, with three two-hour lectures or tutorials per week. It is recommended that the students taking the course have some previous experience with quantum physics, but not strictly necessary.
The purpose of this lecture note is to introduce lattice based cryptography, which is thought to be a cryptosystem of post-quantum age. We have tried to give as many details possible specially for novice on the subject. Something may be trivial to an expert but not to a novice. Many fundamental problems about lattice are thought to be hard even against quantum computer, compared to factorization problem which can be solved easily with quantum computer, via the celebrated Shor factorization quantum algorithm. The first part of our presentation is based on slides of Christ Peikert 2013 Bonn lecture (crypt@b-it2013). We, more or less, give somewhat detailed explanation of Professor Peikert’s lecture slides. We unfortunately could not attend his Bonn class. We are afraid that there are many mistakes in this note; if any, they are due to our misunderstanding of the material. Part II of our lecture note is on ring LWE, based on the paper “A tool-kit for Ring-LWE Cryptography” by Lyubashevsky, Peikert and Regev. Part III is about multilinear maps together with cryptanalysis of GGH map due to Hu and Jia. Our presentation follows professor Steinfeld’s lecture slides on GGHLite, and the paper by Yupu Hu and Huiwen Jia. When you read this lecture note, the corresponding original paper should be accompanied. We thank professor Jung Hee Cheon for introducing the subject and asking Dong Pyo Chi to give a lecture on the subject at the department of mathematics in Seoul National University. We also thank Hyeongkwan Kim for many helps, especially many corrections and improvements of the manuscript during the 2015 Summer session at UNIST. We also thank the students who took the classes at SNU and UNIST. The lecture was given by a novice for novice, so many mistakes are unavoidable. If the reader lets us know any errors, we will very much appreciate it.