1. Notes on Quantum Computation

• Topics
• Motivation
• Commentary on References
  • Researcher, Group and Institute ~ John Preskill ~ Alexei Kitaev ~ Andrew Childs ~ Scott Aaronson
    • Institute
  • Conferences & Workshops
    • course
  • General References
    • Book
    • Lecture
  • References on special topics
    • Algorithms
    • AI ~ QNN
    • Topological Quantum Computation
    • Quantum Advantage ~ Quantum-inspired algorithms
    • Linear system
    • Error correction & stabilizer codes ~ Experiment Realizing
    • stabilizer codes ~ Finite Temperature
    • Programming ~ Library

2. Topics

Traditional textbooks on quantum computing usually start with qubits and the quantum gates, and then use quantum gates to build quantum algorithms. This process actually belongs to the quantum circuit model. There are other models that are equivalent to the quantum circuit model, such as

1. Topological quantum computing
2. Quantum adiabatic/annealing computations
3. Measurement based quantum computation
4. Quantum walk

These models proved to be universal quantum computing. They are equivalent to the quantum Turing machine. It should be noted that the models of quantum computing mentioned here refer to theoretical models, and the same model can be implemented with different physical systems. Candidates for physical realizations of quantum compution:

1. Superconducting circuits
2. Trapped ions
3. Neutral atoms in optical lattices
4. Optical (Linear or Nonlinear)
5. Nuclear magnetic resonance(NMR)
6. Colour centres (e.g., NV-centers in diamond)

In this note, I will make introductions to these models and physical realizations, some just briefly and some in more detail. One of the challenges in constructing quantum computers is that the noise in the environment will destroy the coherence of quantum states. Quantum error correction will be necessary to build reliable quantum computers.

• Quantum error correction

A quantum computer is of little use without an algorithm to run on it. I will also introduce some quantum algorithms and their potential applications.

• Quantum algorithms in traditional textbooks
  1. Deutsch-Josza Algorithm
  2. Quantum Fourier Transform
  3. Quantum Phase Estimation
  4. Shor algorithm
  5. Grover algorithm
• Advanced algorithms
  1. QAOA (Quantum Approximate Optimization Algorithm)
  2. HHL algorithm for linear systems
  3. VQA (variational Quantum Algorithm)
  4. QNN (Quantum Neural Networks)

Potential applications

1. Simulation of quantum systems
2. Search problems
3. Machine learning
4. Cryptography

3. Motivation

This note mainly records the process of my learning quantum computing. This is an unfinished, ongoing note. Part of it comes from my participation in Professor Lin's group meetings, the presentation I gave at the group meetings and listening to other people's speeches.

The concept of 'More is different' by Anderson broken my high school naive thinking in reductionism, which was one of my motivation to choose physics as career. However, it also re-built my new motivation in condensed matter physics to explore the barrier between microcosmic and mesoscopic. While quantum computation is a powerful tool in this exploration.

The teaching philosophy to be used later to develop notes to lectures

"For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples."

—— John B. Conway

4. Commentary on References

In this part, I will introduce some scholars and references that I am interested in in the field of quantum computing.

1. Researchers, Groups and Institutes
2. Conferences & Workshops
3. General References
4. References on special topics

4.1. Researcher, Group and Institute

1. Hannes Pichler

   • homepage
   • his research focusses on trapped neutral (Rydberg) atoms, quantum computing and simulation.
2. Patrick Coles
   • homepage
   • quantum machine learning, Variational quantum algorithms

3. Han Hsuan Lin

   • Intro to Quantum Computation
   • Google Scholar

4. Man-Hong Yung

   • HomePage and Google Scholar
   • course 和 Group Meeting

5. Deng dongling

   • Google Scholar QML
6. Ewin Tang
   • quantum-inspired classical algorithm
     • Home page

7. Duan luming

   • Google Scholar

8. Umesh Vazirani

   • Home Page

   • defined a model of quantum Turing machines which was amenable to complexity based analysis arXiv:quant-ph/9701001

   • quantum computers cannot solve NP-complete problems in polynomial time

9. David Deutsch

   • Home Page
   • Deutsch–Jozsa algorithm
   • formulating a description for a quantum Turing machine 1985 Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer
   • Quantum Circuits (1989) Quantum Computational Networks

John Preskill

• Home Page

Lecture

1. Ph219/CS219 Quantum Computation Lecture
   • vidos

Review paper

1. Quantum Computing in the NISQ era and beyond
   • link: ArXiv
2. Quantum computing 40 years later
   • link: ArXiv

Alexei Kitaev

Part of Kitaev's work in the field of quantum computing

1. The concept of the topological quantum computer

   • Fault-tolerant quantum computation by anyons link
   • Topological phases and quantum computation link

2. How to encode quantum information and protect it from noise using “topological” many-body systems, for example, the Majorana chain.

   • Unpaired Majorana fermions in quantum wires link

3. introducing the quantum phase estimation algorithm

   • Quantum measurements and the Abelian Stabilizer Problem link

Andrew Childs

• Home page

• quantum walk

  • for spatial search
    • Spatial search by quantum walk arXiv:quant-ph/0306054
  • universal computation
    • Universal computation by quantum walk arXiv:0806.1972
• developed quantum algorithms
  • algebraic problems
    • Quantum algorithms for algebraic problems arXiv:0812.0380
  • adiabatic quantum computation
    • Robustness of adiabatic quantum computation arXiv:quant-ph/0108048
  • Hamiltonian simulation

Scott Aaronson

• Home page
• computational complexity theory
  • Complexity Zoo
• Blog
• Book《Quantum Computing since Democritus》

4.1.1. Institute

1. Institute for Quantum Information at the California Institute of Technology
   • IQI Faculty Jeff Kimble; Alexei Kitaev; John Preskill; Eric Rains; Gil Refael; Leonard Schulman
   • IQIM Institute for Quantum Information and Matter, a National Science Foundation Physics Frontiers Center
2. Institute for Quantum Computing (University of Waterloo) Canada
   • YT
   • Home
3. Tsinghua University Center for Quantum Information
4. Quantum Information Society (Oxford University)
   • YT

5. Department of Computer Science and Technology at Nanjing University

   • Home

4.2. Conferences & Workshops

1. QIP Conferences (Quantum Information Processing)
   • QIP2022 Youtube list
   • QIP2021 Youtube list
   • QIP2020 koushare list
   • QIP2019 Youtube list
2. Q2B: QUANTUM FOR BUSINESS 2018
3. The 10th International Workshop on Solid-State Quantum Computing
   • 2021 in CityU
   • videos
4. 量子组会一起开(SUSTech)
5. AMSS-UTS量子计算前沿研讨会
6. 2022量子计算基础与前沿讲习班

4.2.1. course

1. Quantum Computation Ph219/CS219 John Preskill
   • home page
   • vidos
2. Advanced Quantum Algorithms 2019
   • (WACQT) Wallenberg Centre for Quantum Technology

4.3. General References

4.3.1. Book

1. Mikio Nakahara, Tetsuo Ohmi.《Quantum Computing. From Linear Algebra to Physical Realizations 》
2. David McMahon. 《Quantum Computing Explained 》
3. Michael A Nielsen and Isaac Chuang.《Quantum computation and quantum information》, 2002
4. Giuliano Benenti.《Principles of Quantum Computation and Information: Basic Tools and Special Topics》
5. Ivan B. Djordjevic《Quantum Information Processing, Quantum Computing, and Quantum Error Correction: An Engineering Approach》

4.3.2. Lecture

1. Lecture Notes on Quantum Algorithms Andrew M. Childs
2. John Preskill (Caltech): Physics 219 Quantum Computation
3. Advanced Quantum Algorithms Lecture by Giulia Ferrini, Anton Frisk Kockum, Laura García-Álvarez, Pontus Vikstål (2020)
4. Scott Aaronson
5. Quantum Algorithm Implementations for Beginners arXiv:1804.03719 Los Alamos National Laboratory

4.4. References on special topics

4.4.1. Algorithms

1. Lecture Notes on Quantum Algorithms Andrew M. Childs
2. Quantum Algorithm Implementations for Beginners arXiv:1804.03719 Los Alamos National Laboratory
3. Quantum Algorithm Zoo
   • 中文版

4.4.2. AI

1. M. Schuld and F. Petruccione.《Supervised learning with quantum computers. Springer》, 2018
2. Santanu Pattanayak 《Quantum Machine Learning With Python Using Cirq from Google Research and IBM Qiskit》2020

QNN

What is QNN

1. Kak, S. (1995). "On quantum neural computing". Advances in Imaging and Electron Physics. 94: 259–313. doi):10.1016/S1076-5670(08)70147-270147-2)
2. Efficient Learning for Deep Quantum Neural Networks (2020) arXiv:1902.10445
   • Training deep quantum neural networks Nat Commun 11, 808 (2020)
3. On quantum neural networks 2021 arXiv:2104.07106
   • early definition of QNN & modern definition of QNN

What problem QNN advantage in?

1. The Power of Quantum Neural Networks 2020 arXiv:2011.00027
2. Power of data in quantum machine learning 2021 arXiv:2011.01938

What is （Barren Plateau）in QNN ?

1. Barren plateaus in quantum neural network training landscapes
2. Cost function dependent barren plateaus in shallow parametrized quantum circuits Nature Communications 12, 1791 (2021)
3. Trainability of Dissipative Perceptron-Based Quantum Neural Networks arxiv.2005.12458

What make（Barren Plateau）?

1. explain
2. arXiv: 2010.15968

What is QCNN

2019 Harve

• Nature Physics 15,1273-1278(2019) Quantum convolutional neural networks
• A Tutorial on Quantum Convolutional Neural Networks (QCNN) arxiv.2009.09423
• TensorFlow implements QCNN

absence Barren Plateau in QCNN

1. Absence of Barren Plateaus in Quantum Convolutional Neural Networks

4.4.3. Topological Quantum Computation

1. 《Introduction To Topological Quantum Computation》Pachos, Jiannis K.
2. 《Quantum Computation with Topological Codes From Qubit to Topological Fault-Tolerance》 Keisuke Fujii
3. Introduction to topological quantum matter quantum computation by Stanescu, Tudor D
4. Topological Orders with Spins and Fermions Quantum Phases and Computation by Laura Ortiz Martín
5. Topological Quantum Computation 2008 Conference Board of the Mathematical Sciences

Review

1. 《Introduction to topological quantum computation with non-Abelian anyons》 (2018) Bernard Field and Tapio Simula
2. 《A Short Introduction to Topological Quantum Computation》 (2017) V.T. Lahtinen, J.K. Pachos
3. 《Non-Abelian Anyons and Topological Quantum Computation》（2007）Chetan Nayak, Sankar Das Sarma

braiding

1. Topologically protected gates for quantum computation with non-Abelian anyons in the Pfaffian quantum Hall state 2006 Lachezar S. Georgiev Phys. Rev. B 74, 235112
   • How braiding is equivalent to quantum gate
2. Universal quantum computation with ideal Clifford gates and noisy ancillas 2005 Sergey Bravyi and Alexei Kitaev Phys. Rev. A 71, 022316
   • Universal

4.4.4. Quantum Advantage

1. Scott Aaronson. 2015. Read the fine print
   • New quantum algorithms promise an exponential speed-up for machine learning, clustering and finding patterns in big data. But to achieve a real speed-up, we need to delve into the details.
2. Demonstration of quantum advantage in machine learning 2017

Quantum-inspired algorithms

1. An overview of quantum-inspired classical sampling Ewin Tang
2. A quantum-inspired classical algorithm for recommendation systems arXiv:1807.04271

4.4.5. Linear system

1. A survey on HHL algorithm: From theory to application in quantum machine learning 2020 Physics Letters A
   • code
2. Step-by-Step HHL Algorithm Walkthrough to Enhance the Understanding of Critical Quantum Computing Concepts 2021 arXiv:2108.09004

4.4.6. Error correction & stabilizer codes

1. Daniel Gottesman 《An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation》
2. Roffe, Joschka. 《Quantum error correction: an introductory guide》 (2019)

Book

1. 《Quantum Error Correction》Lidar D.A., Brun T.A （2013）
2. 《 Quantum Information Processing and Quantum Error Correction. An Engineering Approach》Ivan Djordjevic (2012)
3. 《Quantum Error Correction and Fault Tolerant Quantum Computing》Frank Gaitan (2008)

Experiment Realizing

1. Realizing topologically ordered states on a quantum processor (2021.12) arxic.2104.01180
   • 31 coupled 2D Supercondtor - Sycamore。
   • toric code
2. Probing topological spin liquids on a programmable quantum simulator (2021.12) arxiv.2104.04119

4.4.7. stabilizer codes

1. Topological defect network representations of fracton stabilizer codes (2021.12) arxiv.2112.14717

Finite Temperature

1. Topological order in a 3D toric code at finite temperature (2008) arXiv:0804.3591
2. The Toric Code at Finite Temperature (2018) UC Berkeley Electronic Theses and Dissertations
3. Finite temperature protocols for stabilizer codes with few measurements
   • algorithms for finite temperature stabilizer error correction codes
   • Engineering autonomous error correction in stabilizer codes at finite temperature arXiv:1603.05005

4.4.8. Programming

1. Hidary, J.D. 《Quantum Computing: An Applied Approach》 (2019) code Jupyter
2. Hassi Norlen《Quantum Computing in Practice with Qiskit® and IBM Quantum》(2020) code Python
3. 2019年，松山湖春季学校 Deep Learning and Quantum Programming: A Spring School
4. 《Quantum Computing Solutions: Solving Real-World Problems Using Quantum Computing and Algorithms》 by Bhagvan Kommadi code
5. Hands-On Quantum Information Processing with Python(2021) code
6. Vladimir Silva《Practical Quantum Computing for Developers Programming Quantum Rigs in the Cloud using Python, Quantum Assembly Language and IBM QExperience》
7. Mingsheng Ying《Foundations of Quantum Programming》
8. Tensor Flow Quantum
9. Silq quantum programming language developed by ETH Zurich
10. Quantum Machine Learning with Python by Santanu Pattanayak
11. Mooc in University of Toronto Quantum Machine Learning 代码

Library

1. Qiskit Machine Learning Tutorials
2. HiQ 华为
3. 飞桨 百度