Alexander polynomial (1)
Algebra (1)
Anyons (1)
Berry's Phase (1)
Braiding (2)
Chern number (1)
Classical simulation of Quantum computers (1)
Coding (3)
Complexity (1)
Diagrams (1)
Differential Forms (1)
Euler Characteristic (1)
Exponential map (1)
Fiber (1)
Fiber bundles (1)
First Principle (1)
Fortran (4)
Fundamental Group (1)
Generator (1)
Genus (1)
Gram Schmidt (1)
Homology (1)
Homotopy (1)
Ising anyons (1)
Jekyll (1)
Jones polynomial (1)
Kauffman bracket (1)
Kitaev's Toric Code (1)
Knot theory (1)
Lie algebra (4)
Lie group (4)
Mathematica (1)
Meta (1)
One form (2)
PIE (1)
Pentagon and Hexagon equation (2)
Pullback (1)
Pushforward (1)
Python (1)
QFT (1)
Quantum computation (1)
SSH model (1)
Statistical mechanics (1)
Stokes' Theorem (1)
Tangent space (1)
Tensor (1)
Thoughts (1)
Tight binding (1)
Topological quantum computation (5)
Topology (5)
Triangulation (1)
VPN (1)
Vector (1)
Wild thoughts (1)
Winding number (1)
de Rham Cohomology (1)
pitfall (1)
representation (1)
study notes (1)
topology (1)
word cloud (1)
20 Mar 2019
Lie Algebra as Generator
Paused
In this post generator of Lie groups as well as its subgroup are considered. This short post is preparation for the application of Mathematical theory into QM and CM. This post is the third of a series of posts that start from Lie group and Lie algebra.
11 Mar 2019
Lie Group as Differential Manifold
Completed
In this post Lie groups are regarded as a differential manifold, and one-parameter subgroups are introduced. This post is the second of a series of posts that start from Lie group and Lie algebra.
12 Jan 2019
Lie Groups and "Actions"
Completed
In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.
06 Jun 2018
李群和李群的李代数
Archived
My understanding of Lie-group and its Lie-algebra. The statement “Lie-algebra is approximation of Lie-group” is inaccurate since they essentially live in different spaces. This post is not finished. Check posts under Lie Group
and/or Lie Algebra
where I rewrote and added more aspects of Lie group and Lie algebra.
20 Mar 2019
Lie Algebra as Generator
Paused
In this post generator of Lie groups as well as its subgroup are considered. This short post is preparation for the application of Mathematical theory into QM and CM. This post is the third of a series of posts that start from Lie group and Lie algebra.
11 Mar 2019
Lie Group as Differential Manifold
Completed
In this post Lie groups are regarded as a differential manifold, and one-parameter subgroups are introduced. This post is the second of a series of posts that start from Lie group and Lie algebra.
12 Jan 2019
Lie Groups and "Actions"
Completed
In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.
06 Jun 2018
李群和李群的李代数
Archived
My understanding of Lie-group and its Lie-algebra. The statement “Lie-algebra is approximation of Lie-group” is inaccurate since they essentially live in different spaces. This post is not finished. Check posts under Lie Group
and/or Lie Algebra
where I rewrote and added more aspects of Lie group and Lie algebra.
07 Jun 2018
量子场论
Archived
My Study notes on QFT lesson for the final. Each step of deduction is present in this post. This post covers K-G equations and Dirac equations.
07 Jun 2018
量子场论
Archived
My Study notes on QFT lesson for the final. Each step of deduction is present in this post. This post covers K-G equations and Dirac equations.
08 Jun 2018
第一性原理热统
Archived
Class project. My attempt at building Statistical Mechanics from First Principles. I think Statistical Mechanics can be applied to distinguishable particles such as millions of footballs. This post tries to develop such Statistical Mechanics, especially on the entropy. This post is not finished.
08 Jun 2018
第一性原理热统
Archived
Class project. My attempt at building Statistical Mechanics from First Principles. I think Statistical Mechanics can be applied to distinguishable particles such as millions of footballs. This post tries to develop such Statistical Mechanics, especially on the entropy. This post is not finished.
21 Aug 2018
N-Forms and Tensors on Manifold
Completed
We are going to generalize the concept of vectors and one-forms to tensors and differential forms. In the meantime, the wedge product and exterior derivative were introduced. Exterior derivative lies the foundation for cohomology.
20 Aug 2018
Vectors and One-Forms on Manifold
Completed
Since in general there is no way to define a “straight arrow” connecting two points, Vectors can only be “tangent vectors” on manifolds. In this post, tangent vectors are introduced heuristically, with emphasis on how and why should we define vectors as operators. Co-vectors, also called one-forms, are introduced as the dual. The reason for defining one-forms as differentials are introduced heuristically. This post also addresses the problem of inconsistency when the basis of vector act on that of one form.
16 Jan 2019
Fiber Bundles
Paused
Fiber bundle is introduced with intuitive examples of pasta and pancakes, Berry phase and Calabi-Yau space. Structure group is introduced as a natural consequence of transition functions.
23 Nov 2018
Introduction to de Rham Cohomology
Completed
Cohomology is viewed as a natural dual space of homology in this post. The bilinear map (i.e., the inner product) between these two spaces is just integration. At the end of this post, the cohomology group as an indicator of “holes” in space is discussed.
01 Nov 2018
Introduction to Homology
Completed
Euler characteristics is a topological invariant, and can be interpreted as a “hole”-indicator. Homology is just a natural way of defining Euler characteristics in topological spaces. With triangulation of a manifold, we can define cycles and boundaries and combine them to homology groups. We see that the group is trivial for trivial spaces, and can distinguish manifolds in terms “holes” in them.
21 Aug 2018
N-Forms and Tensors on Manifold
Completed
We are going to generalize the concept of vectors and one-forms to tensors and differential forms. In the meantime, the wedge product and exterior derivative were introduced. Exterior derivative lies the foundation for cohomology.
20 Aug 2018
Vectors and One-Forms on Manifold
Completed
Since in general there is no way to define a “straight arrow” connecting two points, Vectors can only be “tangent vectors” on manifolds. In this post, tangent vectors are introduced heuristically, with emphasis on how and why should we define vectors as operators. Co-vectors, also called one-forms, are introduced as the dual. The reason for defining one-forms as differentials are introduced heuristically. This post also addresses the problem of inconsistency when the basis of vector act on that of one form.
20 Aug 2018
Vectors and One-Forms on Manifold
Completed
Since in general there is no way to define a “straight arrow” connecting two points, Vectors can only be “tangent vectors” on manifolds. In this post, tangent vectors are introduced heuristically, with emphasis on how and why should we define vectors as operators. Co-vectors, also called one-forms, are introduced as the dual. The reason for defining one-forms as differentials are introduced heuristically. This post also addresses the problem of inconsistency when the basis of vector act on that of one form.
20 Aug 2018
Vectors and One-Forms on Manifold
Completed
Since in general there is no way to define a “straight arrow” connecting two points, Vectors can only be “tangent vectors” on manifolds. In this post, tangent vectors are introduced heuristically, with emphasis on how and why should we define vectors as operators. Co-vectors, also called one-forms, are introduced as the dual. The reason for defining one-forms as differentials are introduced heuristically. This post also addresses the problem of inconsistency when the basis of vector act on that of one form.
21 Aug 2018
N-Forms and Tensors on Manifold
Completed
We are going to generalize the concept of vectors and one-forms to tensors and differential forms. In the meantime, the wedge product and exterior derivative were introduced. Exterior derivative lies the foundation for cohomology.
21 Aug 2018
N-Forms and Tensors on Manifold
Completed
We are going to generalize the concept of vectors and one-forms to tensors and differential forms. In the meantime, the wedge product and exterior derivative were introduced. Exterior derivative lies the foundation for cohomology.
30 Dec 2019
Computer Code For Toric Code
Completed
The post is my initial set up for Kitaev’s toric code. The work is published on arXiv:1912.12964.
18 Apr 2019
The Fortran Gram-Schmidt Process Pitfall
Completed
I tried using Gram-Schmidt Process the other day and just couldn’t get the correct result. It took me a few days trying to figure out why. Turns out that it’s just a simple precision error problem.
06 Oct 2018
You Should Try Fortran
Completed
I found Fortran incredibly fast compared to pure Python. Fortran’s reputation for hard to maintain is true only if you read the code without an understanding of the Physics. It’s still an active language. You should give it a try.
01 Sep 2018
Fortran with VS2017
Completed
Instructions on using Fortran with VS2017
18 Apr 2019
The Fortran Gram-Schmidt Process Pitfall
Completed
I tried using Gram-Schmidt Process the other day and just couldn’t get the correct result. It took me a few days trying to figure out why. Turns out that it’s just a simple precision error problem.
06 Oct 2018
You Should Try Fortran
Completed
I found Fortran incredibly fast compared to pure Python. Fortran’s reputation for hard to maintain is true only if you read the code without an understanding of the Physics. It’s still an active language. You should give it a try.
01 Sep 2018
Fortran with VS2017
Completed
Instructions on using Fortran with VS2017
06 Oct 2018
You Should Try Fortran
Completed
I found Fortran incredibly fast compared to pure Python. Fortran’s reputation for hard to maintain is true only if you read the code without an understanding of the Physics. It’s still an active language. You should give it a try.
06 Oct 2018
You Should Try Fortran
Completed
I found Fortran incredibly fast compared to pure Python. Fortran’s reputation for hard to maintain is true only if you read the code without an understanding of the Physics. It’s still an active language. You should give it a try.
01 Nov 2018
Introduction to Homology
Completed
Euler characteristics is a topological invariant, and can be interpreted as a “hole”-indicator. Homology is just a natural way of defining Euler characteristics in topological spaces. With triangulation of a manifold, we can define cycles and boundaries and combine them to homology groups. We see that the group is trivial for trivial spaces, and can distinguish manifolds in terms “holes” in them.
01 Nov 2018
Introduction to Homology
Completed
Euler characteristics is a topological invariant, and can be interpreted as a “hole”-indicator. Homology is just a natural way of defining Euler characteristics in topological spaces. With triangulation of a manifold, we can define cycles and boundaries and combine them to homology groups. We see that the group is trivial for trivial spaces, and can distinguish manifolds in terms “holes” in them.
23 Nov 2018
Introduction to de Rham Cohomology
Completed
Cohomology is viewed as a natural dual space of homology in this post. The bilinear map (i.e., the inner product) between these two spaces is just integration. At the end of this post, the cohomology group as an indicator of “holes” in space is discussed.
23 Nov 2018
Introduction to de Rham Cohomology
Completed
Cohomology is viewed as a natural dual space of homology in this post. The bilinear map (i.e., the inner product) between these two spaces is just integration. At the end of this post, the cohomology group as an indicator of “holes” in space is discussed.
23 Nov 2018
Thank You Word Cloud with Mathematica
Completed
Code for generating thank-you word cloud with Mathematica. It’s perfect for thank-you page of your slide-shows.
23 Nov 2018
Thank You Word Cloud with Mathematica
Completed
Code for generating thank-you word cloud with Mathematica. It’s perfect for thank-you page of your slide-shows.
01 Dec 2018
Build Your Blog with GitHub Pages
Paused
Build Your Blog with GitHub Pages. This is a sketch of how to use my theme PointingToTheMoon
to write your blog. This theme is great for academic use, for it features simple post page with mathjax support and a side bar with toc. The main page on the other hand is somewhat fancy.
09 Dec 2018
Shadowsocks + Digital Ocean = VPN of 1TB/month
Completed
Code for using digital ocean service for VPN. It is cheap for .edu-e-mail owners (free 50$ on GitHub Student Pack).
12 Jan 2019
Lie Groups and "Actions"
Completed
In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.
12 Jan 2019
Lie Groups and "Actions"
Completed
In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.
12 Jan 2019
Lie Groups and "Actions"
Completed
In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.
16 Jan 2019
Fiber Bundles
Paused
Fiber bundle is introduced with intuitive examples of pasta and pancakes, Berry phase and Calabi-Yau space. Structure group is introduced as a natural consequence of transition functions.
16 Jan 2019
Fiber Bundles
Paused
Fiber bundle is introduced with intuitive examples of pasta and pancakes, Berry phase and Calabi-Yau space. Structure group is introduced as a natural consequence of transition functions.
16 Jan 2019
Fiber Bundles
Paused
Fiber bundle is introduced with intuitive examples of pasta and pancakes, Berry phase and Calabi-Yau space. Structure group is introduced as a natural consequence of transition functions.
11 Mar 2019
Lie Group as Differential Manifold
Completed
In this post Lie groups are regarded as a differential manifold, and one-parameter subgroups are introduced. This post is the second of a series of posts that start from Lie group and Lie algebra.
20 Mar 2019
Lie Algebra as Generator
Paused
In this post generator of Lie groups as well as its subgroup are considered. This short post is preparation for the application of Mathematical theory into QM and CM. This post is the third of a series of posts that start from Lie group and Lie algebra.
11 Apr 2019
Introduction to Topological Quantum Computation: Basics of Quantum Computation
Completed
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post establishes the barebone basics of quantum computations.
23 May 2019
Introduction to Topological Quantum Computation: Crash Course on Knots Theory
Completed
This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.
14 May 2019
Introduction to Topological Quantum Computation: Ising Anyons Case Study
Completed
This is a series of posts on topological quantum computations. In this post, the most promising candidate for TQC, Ising anyons, are discussed. A theoretical topological quantum computer is realized via Ising anyons’ initialization, braiding, and fusion. F and R matrices are calculated from the consistency requirement, i.e. Hexagon and Pentagon equations. Braiding matrices are introduced heuristically. A set of Clifford gates is implemented as the result of braiding. This post features lots of diagrams.
01 May 2019
Introduction to Topological Quantum Computation: Side Notes on Simulations
Writing
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post started as introducing a pitfall of using Stern-Gerlach experiment as quantum computers, and end with a discussion on simulations of QC and TQC using classical computers.
13 Apr 2019
Introduction to Topological Quantum Computation: Anyons Model
Completed
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post discusses anyon model in general. Fusion diagram and hexagon and pentagon identities are introduced.
11 Apr 2019
Introduction to Topological Quantum Computation: Basics of Quantum Computation
Completed
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post establishes the barebone basics of quantum computations.
13 Apr 2019
Introduction to Topological Quantum Computation: Anyons Model
Completed
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post discusses anyon model in general. Fusion diagram and hexagon and pentagon identities are introduced.
14 May 2019
Introduction to Topological Quantum Computation: Ising Anyons Case Study
Completed
This is a series of posts on topological quantum computations. In this post, the most promising candidate for TQC, Ising anyons, are discussed. A theoretical topological quantum computer is realized via Ising anyons’ initialization, braiding, and fusion. F and R matrices are calculated from the consistency requirement, i.e. Hexagon and Pentagon equations. Braiding matrices are introduced heuristically. A set of Clifford gates is implemented as the result of braiding. This post features lots of diagrams.
13 Apr 2019
Introduction to Topological Quantum Computation: Anyons Model
Completed
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post discusses anyon model in general. Fusion diagram and hexagon and pentagon identities are introduced.
14 May 2019
Introduction to Topological Quantum Computation: Ising Anyons Case Study
Completed
This is a series of posts on topological quantum computations. In this post, the most promising candidate for TQC, Ising anyons, are discussed. A theoretical topological quantum computer is realized via Ising anyons’ initialization, braiding, and fusion. F and R matrices are calculated from the consistency requirement, i.e. Hexagon and Pentagon equations. Braiding matrices are introduced heuristically. A set of Clifford gates is implemented as the result of braiding. This post features lots of diagrams.
13 Apr 2019
Introduction to Topological Quantum Computation: Anyons Model
Completed
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post discusses anyon model in general. Fusion diagram and hexagon and pentagon identities are introduced.
18 Apr 2019
The Fortran Gram-Schmidt Process Pitfall
Completed
I tried using Gram-Schmidt Process the other day and just couldn’t get the correct result. It took me a few days trying to figure out why. Turns out that it’s just a simple precision error problem.
01 May 2019
Introduction to Topological Quantum Computation: Side Notes on Simulations
Writing
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post started as introducing a pitfall of using Stern-Gerlach experiment as quantum computers, and end with a discussion on simulations of QC and TQC using classical computers.
01 May 2019
Introduction to Topological Quantum Computation: Side Notes on Simulations
Writing
This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post started as introducing a pitfall of using Stern-Gerlach experiment as quantum computers, and end with a discussion on simulations of QC and TQC using classical computers.
07 May 2019
Computation and Currency
Completed
This is my wild thoughts on computation and currency. The story begins when a spaceship appeared near a lake one day…
07 May 2019
Computation and Currency
Completed
This is my wild thoughts on computation and currency. The story begins when a spaceship appeared near a lake one day…
14 May 2019
Introduction to Topological Quantum Computation: Ising Anyons Case Study
Completed
This is a series of posts on topological quantum computations. In this post, the most promising candidate for TQC, Ising anyons, are discussed. A theoretical topological quantum computer is realized via Ising anyons’ initialization, braiding, and fusion. F and R matrices are calculated from the consistency requirement, i.e. Hexagon and Pentagon equations. Braiding matrices are introduced heuristically. A set of Clifford gates is implemented as the result of braiding. This post features lots of diagrams.
23 May 2019
Introduction to Topological Quantum Computation: Crash Course on Knots Theory
Completed
This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.
23 May 2019
Introduction to Topological Quantum Computation: Crash Course on Knots Theory
Completed
This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.
23 May 2019
Introduction to Topological Quantum Computation: Crash Course on Knots Theory
Completed
This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.
23 May 2019
Introduction to Topological Quantum Computation: Crash Course on Knots Theory
Completed
This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.
06 Aug 2019
This Summer and the Site
Completed
After a long absence I decided to recap what happened during the last few months and start updating this blog again.
06 Aug 2019
This Summer and the Site
Completed
After a long absence I decided to recap what happened during the last few months and start updating this blog again.
02 Sep 2019
Introduction To Topological Insulator
Writing
This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.
02 Sep 2019
Introduction To Topological Insulator
Writing
This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.
02 Sep 2019
Introduction To Topological Insulator
Writing
This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.
02 Sep 2019
Introduction To Topological Insulator
Writing
This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.
02 Sep 2019
Introduction To Topological Insulator
Writing
This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.
29 Oct 2019
Introduction To Fundamental Group
Writing
This post is to prepare myself for future study. As the fundamental group plays an important role in many parts of topological insulators.
29 Oct 2019
Introduction To Fundamental Group
Writing
This post is to prepare myself for future study. As the fundamental group plays an important role in many parts of topological insulators.
29 Oct 2019
Introduction To Fundamental Group
Writing
This post is to prepare myself for future study. As the fundamental group plays an important role in many parts of topological insulators.
30 Dec 2019
Computer Code For Toric Code
Completed
The post is my initial set up for Kitaev’s toric code. The work is published on arXiv:1912.12964.
30 Dec 2019
Computer Code For Toric Code
Completed
The post is my initial set up for Kitaev’s toric code. The work is published on arXiv:1912.12964.
05 Mar 2020
The Amazing World Of Diagrams
Completed
This post is the first chapter of a long note that I have been writing and organizing aiming at telling the stories in the languages of mathematical concepts mostly used in condensed matter Physics.
05 Mar 2020
The Amazing World Of Diagrams
Completed
This post is the first chapter of a long note that I have been writing and organizing aiming at telling the stories in the languages of mathematical concepts mostly used in condensed matter Physics.