On tensor category

Posted on 16 Mar 2019 by Zhih-Ahn Jia (贾治安)

Recently, I have been undertaking some projects about topological ordered phases, topological quantum field theories and related topics, many category theoretical concepts are needed. I would like to collect some useful materials which I benefit from in learning the subject.

General category theory:

Categories for the Working Mathematician, Mac Lane. This is a classic book of the field, I really enjoy reading the book.
Basic Category Theory, Tom Leinster. The exposition of the book is very clear and is very friendly to beginners.
Category theory in context, Emily Riehl. A little harder than the above two books.
The Stacks project, online notes, famous reference of algebraic geometry.
The CRing Project, online notes about the commutative ring.
Foundations of Algebraic Geometry, Ravi Vakil’s famous notes about algebraic geometry.

Tensor category theory:

Quantum Invariants of Knots and 3-Manifolds, V. Turaev. This is a classic book about the tensor category and TQFT, I learn very much from this book.
Lectures on tensor categories and modular functor, Bojko Bakalov and Alexander Kirillov, Jr. This is also a very good book to learn tensor category and TQFT.
Tensor categories, Pavel Etingof. Shlomo Gelaki. Dmitri Nikshych. Victor Ostrik. This is now a standard reference of the field.

Quantum discrete integral transform

Posted on 1 Jan 2019 by Zhih-Ahn Jia (贾治安)

Integral transforms such as Fourier transform, Laplace transform and so on, are powerful tools, they are ubiquitous in mathematics, engineering, physics and many other scientific areas. Mathematically, an integral transform ${T}$ is an operator which maps a function ${f}$ into another function ${Tf}$ , and two functions do not necessarily have the same domain and range. Usually an integral transform ${T}$ can be written via a corresponding integral kernel ${K(x,y)}$ as

$\displaystyle Tf(x)=\int K(x,y)f(y)dy.$

This integral form is also the origin of the name of integral transform. There are numerous crucial integral transforms, for example, the most famous Fourier transform

$\displaystyle \mathcal{F}(\psi)(p)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{2\pi i xp}\psi(x)dx$

have the integral kernel ${e^{2\pi i xp}/\sqrt{2\pi}}$ .

Note that a function ${f(x)}$ can be regarded as a sequence of numbers index by ${x}$ in a continuous index set. The integral transform ${Tf}$ thus looks like a matrix transform with continuous indices. In the discrete context, it turns out that an integral transform becomes a matrix transform.

Definition 1 (Discrete integral transform) Let ${K_{ij}=:K(i,j)}$ be a matrix with indices ${i,j=0,\cdots,N-1}$ and ${\mathbf{x}=(x_0,\cdots,x_{N-1})^T=:(x(0),\cdots,x(N-1))^{T}}$ be a vector (discrete function). Then the discrete integral transform is defined as ${y(i)=(Kx)(i)=\sum_{j}K(i,j)x(j)}$ which is in fact a matrix transform

$\displaystyle \mathbf{y}=K\mathbf{x}. \ \ \ \ \ (1)$

Note that we always assume vectors are column vectors. And for the convenience of the generalization we will also assume that ${K}$ is unitary, for which case the invertible discrete integral transform kernel is given by the Hermitian conjugate ${K^{\dagger}}$ of ${K}$ .

For instance, the discrete Fourier transform determined by the kernel matrix ${K_{jk}=e^{2\pi i jk/N}/\sqrt{N}}$ is of the form

$\displaystyle y_k=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}e^{2\pi i jk/N}x_j. \ \ \ \ \ (2)$

Here we want to mention that in the large ${N}$ limit, the discrete Fourier transform kernel will become the usual Fourier integral kernel and sum is replaced with the integral. This is the reason why we choose ${K_{jk}=e^{2\pi i jk/N}/\sqrt{N}}$ as discrete Fourier transform kernel.

The quantum discrete integral transform is the quantum analogue of discrete integral transform, actually they are exactly the same transform as we will see. Suppose that we have a Hilbert space ${\mathcal{H}}$ with basis states ${|0\rangle,\cdots,|N-1\rangle}$ , for a given discrete integral transform kernel ${K_{ij}}$ , the corresponding quantum discrete integral transform on the basis states is defined as

$\displaystyle |j\rangle\overset{QDIT}{\rightarrow}U_K|j\rangle=\sum_{k=0}^{N-1}K_{kj}|k\rangle. \ \ \ \ \ (3)$

Then for a state ${|\psi\rangle=\sum_{j=0}^{N-1}x_j|j\rangle/\|\mathbf{x}\|}$ ,

$\displaystyle U_K|\psi\rangle=\frac{1}{\|\mathbf{x}\|}\sum_{j=0}x_jU_K|j\rangle=\frac{1}{\|\mathbf{x}\|}\sum_{k=0}^{N-1}(\sum_{j=0}K_{kj}x_j)|k\rangle=\frac{1}{\|\mathbf{y}\|}\sum_{k=0}^{N-1}y_k|k\rangle \ \ \ \ \ (4)$

calculate the discrete integral transform ${y_k=\sum_{j=0}^{N-1}K_{kj}x_j}$ . Notice that ${U_K}$ is unitary matrix, thus ${\|\mathbf{x}\|=\|\mathbf{y}\|}$ . In summary, we have

Definition 2 (Quantum discrete integral transform) Quantum discrete integral transform corresponding to a unitary kernel ${K_{kj}}$ is a unitary transformation ${U_K=(K_{kj})}$

$\displaystyle |j\rangle\overset{QDIT}{\rightarrow}U_K|j\rangle=\sum_{k=0}^{N-1}K_{kj}|k\rangle. \ \ \ \ \ (5)$

To carry out the discrete integral transform ${y_k=\sum_{j=0}^{N-1}K_{jk}x_j}$ , we first prepare the state ${|\psi\rangle=\sum_{j=0}^{N-1}x_j|j\rangle/\|\mathbf{x}\|}$ and then apply ${U_K}$ to get ${U_K|\psi\rangle}$ , finally we read out the value ${y_k}$ by measuring in the basis state ${|k\rangle}$ .

To construct a quantum discrete integral transform algorithm, we will need to construct a algorithm which realize ${U_K}$ . From next section on, we will focus on quantum Fourier transform (QFT) which is the most crucial example of quantum discrete integral transform algorithm. The unitary matrix corresponding to QFT is ${U_{QFT}=(K_{jk})}$ with ${K_{jk}=e^{2\pi i jk/N}/\sqrt{N}}$ .

QC book: useful materials

Posted on 1 Jan 2019 by Zhih-Ahn Jia (贾治安)

Recently, I will be giving a series of lectures on quantum computation theory and related topics from the perspective of theoretical physics. The lecture notes will be posted in this blog under the label QCbook, since I hope eventually the material can be organized into a book. Roughly speaking, these lectures can be categorized into several groups:

Generalities of quantum computation theory: including classical computation theory, Turing machine, computational complexity; then quantum mechanics, and quantization of the notion of computation.
Quantum algorithms: including algorithms based on quantum Fourier transform, and algorithm based on Grover search.
Quantum error correcting codes
Quantum machine learning
The physical realization of a quantum computer

My plan for the time duration of the project is about one year as I stay in UCSB.

Books on classical and quantum computation theory

There are many elegant and readable books on different topics of classical and quantum computation theory. Here I list some of them which I feel suit for learning the subject.

Computational Complexity: A Modern Approach, by S. Arora and B. Barak, the internet draft can be found here.
Lecture Notes for Physics 229: Quantum Information and Computation, by J. Preskill, the online notes can be found here.
Quantum Computation and Quantum Information, by M. A. Nielsen and I. Chuang. The online file of the book can be found here.
Introduction to the Theory of Computing, by John Watrous, the lecture note can be found here.

Collection of useful materials on the web:

There are many useful materials to learn quantum computation theory.

Lecture notes:

Collection of quantum algorithms:

Quantum algorithm zoo: this is a webpage for quantum algorithms and their corresponding computational complexities.

Useful software:

Q-Kit: a graphical tool for simulating quantum algorithms.
Q-circuit: the package for drawing quantum circuit in latex.

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will be updated

Seminar: Conformal field theory

Posted on 1 Sep 2018 by Zhih-Ahn Jia (贾治安)

Recently, I am about to organize a learning seminar on conformal field theory, the purpose is to get familiar with the basic language of the conformal field theory and related topics.

The canonical reference for learning the field is Ginsparg’s applied conformal field theory. I find D. Tong’s string theory note is also very readable. Another one is the book by M. Schottenloher, A Mathematical Introduction to Conformal Field Theory.

Conformal field theory is a filed theory which is invariant under conformal transformations. This means that physics looks the same at all length scales. Conformal field theory plays an important role in statistical mechanics, where it offers a desciption of critical phenomena; in string theory; and in gravity-gauge theory duality (AdS/CFT duality). Our aim here it ot understand its role in understanding anyon theory and some exotic phases in condensed matter physics.

Some other materials will be added as the seminar progresses.

Latex for wordpress

Posted on 19 Aug 2018 by Zhih-Ahn Jia (贾治安)

Although wordpress supports latex, I find it difficult to write latex code on blog in wordpress-latex format. I try to collect some useful tools for wordpress latex here.

Firstly, I recommand Luca Trevisan’s poster on this topic. By and large, LaTeX2WP is really something I am searching for, which is a program that converts a LaTeX file into something that is ready to be cut and pasted into WordPress..

——To be completed——

It’s not uncommon for technical books to include an admonition from the author that readers must do the exercises and problems. I always feel a little peculiar when I read such warnings. Will something bad happen to me if I don’t do the exercises and problems? Of course not. I’ll gain some time, but at the expense of depth of understanding. Sometimes that’s worth it. Sometimes it’s not.

— Michael Nielsen, Neural Networks and Deep Learning

一些不存在的计划

Posted on 16 Jun 2018 by Zhih-Ahn Jia (贾治安)

追忆我逝去的人生，我倍感惋惜，惋惜之余我又无可奈何。故此，我特意将我要研究的方向罗列下来，时时提醒我，我的人生在流逝，一切都无可追回。

首先我要学习一些语言，去更多的地方，认识更多的人。是谁说要去看江海湖泊，最后却囿于厨房和爱情？

语言学习

最大的问题，也是我最懒于学习的方面就是语言。我需要多学几门语言。

英语

日语

标准日本语上下

法语

简明法语教程上下

德语

简明德语教程上下

理论物理

===================

量子信息与量子计算

这是我最主要也是最初的研究方向，也是我了解得比较多的领域，下面是我的一些计划

一、量子信息

量子关联的刻画与分类：quantum discord, entanglement, steering, Bell nonlocality;
量子力学基本原理的寻找：exclusivity principle, measurement sharpness, causuality, contextuality;
量子信息理论的研究演：开放系统，PT对称破缺，quantum channel与quantum operation；
量子信息的各种应用：QKD，密集编码

需要阅读的书籍：

M. A. Nielsen& I. L. Chuang, Quantum computation and quantum information(有中文版)
J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation
M. M. Wilde, Quantum Information Theory
G. Benenti, G. Casati & G. Strini, Principles of Quantum Computation and Information Volume I: Basic Concepts
G. Benenti, G. Casati & G. Strini, Principles of Quantum Computation and Information Volume II: Basic Tools and Special Topics

二、量子计算

量子计算的基本概念研究：布尔函数，可计算性，复杂性理论等
量子计算的模型研究：回路模型，图灵机及其变种
量子算法：量子傅立叶，相位估计，Shor算法，Grover算法
量子机器学习：神经网络态表示，张量网络态表示，机器学习算法
量子码的研究
量子计算的实现：拓扑量子计算，绝热量子计算
量子计算与度量几何的联系
计算复杂性与物相：computationally universal phase

需要阅读的书籍：

M. A. Nielsen& I. L. Chuang, Quantum computation and quantum information(有中文版)
J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation
M. Nakahara, Quantum Computing: From Linear Algebra to Physical Relization
G. Benenti, G. Casati & G. Strini, Principles of Quantum Computation and Information Volume I: Basic Concepts
G. Benenti, G. Casati & G. Strini, Principles of Quantum Computation and Information Volume II: Basic Tools and Special Topics
A. O. Pittenger, An Introduction to Quantum Computing Algorithms
E Rieffel, Quantum Computing: A Gentle Introduction
冯克勤、陈豪, 量子纠错码
S Roman, Coding and Information theory
J.H. van Lint, Introduction to Coding Theory
NIST, Quantum Algorithm Zoo

三、拓扑量子计算

拓扑物态：Kitaev模型，Levin-Wen模型
拓扑量子计算的基本原理
拓扑量子计算所需的数学：扭结理论，辫群，张量范畴极其表示理论
拓扑码：拓扑存储和拓扑解码器

需要阅读的书籍：

M. A. Nielsen& I. L. Chuang, Quantum computation and quantum information(有中文版)
J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation
Zhenghan Wang, Topological Quantum Computation
J. K. Pachos, Introduction to Topological Quantum Computation
V. Turaev, Quantum Invariants of Knots and 3-Manifolds
Alexander Kirillov Jr.，Bojko Bakalov, Lectures on Tensor Categories and Modular Functors
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych & Victor Ostrik, Tensor Categories

四、量子热力学

量子热力学
涨落定理与量子涨落定理
信息、能量与物质的统一
量子信息与统计物理的联系

========================

场论、弦论和凝聚态理论

一、对偶理论

AdS/CFT对偶
边体对偶
开闭弦对偶
黑洞火墙
纠缠及量子关联在场论弦论中的应用
一些重要模型：SYK模型
孤子理论，拓扑孤子理论

二、重整化

实空间重整化
张量网络重整化，纠缠重整化，神经网络重整化
各种场论的可重整化证明

三、离散规范理论

基于张量范畴的离散规范理论
代数化的规范理论
格点规范理论极其量子模拟

四、量子Hall效应

五、高温超导理论

数学

张量范畴及其应用
流形拓扑与几何，量子拓扑与量子代数

计算机理论

人工智能：机器学习，神经网络
算法理论
计算复杂性理论

Tsirelson’s problem and Connes’s conjecture

Posted on 17 Apr 2018 by Zhih-Ahn Jia (贾治安)

In non-relativistic quantum mechanics, spacelike separation is modeled via tensor product of Hilbert spaces. But in algebraic quantum field theory, spacelike separation is modeled via the commutativity of operators. Tsirelson’s problem ask whether these two different definitions give the same set of bipartite correlations.

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The Lost Worldline

A research blog for physics, mathematics and a little computer science. By Zhih-Ahn Jia (贾治安).

Author Archives: Zhih-Ahn Jia (贾治安)

On tensor category

Quantum discrete integral transform

QC book: useful materials

Books on classical and quantum computation theory

Collection of useful materials on the web:

Seminar: Conformal field theory

Latex for wordpress

一些不存在的计划

语言学习

英语

日语

法语

德语

理论物理

量子信息与量子计算

一、量子信息

二、量子计算

三、拓扑量子计算

四、量子热力学

场论、弦论和凝聚态理论

一、对偶理论

二、重整化

三、离散规范理论

四、量子Hall效应

五、高温超导理论

数学

计算机理论

Tsirelson’s problem and Connes’s conjecture

Books on classical and quantum computation theory

Collection of useful materials on the web:

语言学习

英语

日语

法语

德语

理论物理

量子信息与量子计算

一、 量子信息

二、 量子计算

三、 拓扑量子计算

四、 量子热力学

场论、弦论和凝聚态理论

一、对偶理论

二、重整化

三、离散规范理论

四、量子Hall效应

五、高温超导理论

数学

计算机理论

一、量子信息

二、量子计算

三、拓扑量子计算

四、量子热力学