Category: Quantum computation

PC5228(2024): Quantum Information and Computation

This fall semester, I have been serving as a teaching assistant for PC5228: Quantum Information and Computation. As many students have requested additional reading materials, I will now provide further resources related to the course.

Descripition of the course

The course provides an introduction to quantum information and quantum computation. In addition to physics majors, the course addresses students with a good background in discrete mathematics or computer science.The following topics will be covered:

(1) Introduction: a brief review of basic notions of information science (Shannon entropy, channel capacity) and of basic quantum kinematics with emphasis on the description of multi-qubit systems and their discrete dynamics.

(2) Quantum information: Entanglement and its numerical measures, separability of multi-partite states, quantum channels, standard protocols for quantum cryptography and entanglement purification, physical implementations.

(3) Quantum computation: single-qubit gates, two-qubit gates and their physical realization in optical networks, ion traps, quantum dots, Universality theorem, quantum networks and their design, simple quantum algorithms (Jozsa-Deutsch decision algorithm, Grover search algorithm, Shor factorization algorithm). The course is tightly integrated with IBM quantum computer hands-on experience via IBM Q Experience cloud services. Students will learn fundamentals of Qiskit, a modern and rapidly developing quantum computer programming language, by directly implementing concepts learnt in the classroom.

Syllabus

The course is divided into four chapters(PDF lecture notes are available on Canvas):

  • chapter 1 Brief introduction to discrete quantum mechanics
  • Chapter 2 Elements of quantum information
  • Chapter 3 Elements of quantum computation
  • Chapter 4 Quantum algorithms

Some general references

The following are some book that may be helpful for you to learn quantum information and quantum computation:

  • John Preskill, Course Information for Physics 219/Computer Science 219 Quantum Computation (Formerly Physics 229). This page is a collection of wonderful lecture notes by Preskill on quantum information theory. In Fall 2020 his course is recoded, see the online videos Ph/CS 219A Quantum Computation.
  • Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information. This a standard textbook for quantum information and quantum computation theory.
  • Mark Wilde, Quantum Information Theory. This book provides a comprehensive discussion of quantum Shannon theory, covering a wide range of topics in depth.
  • John Watrous, The Theory of Quantum Information. This is a book that contains many mathematical details on techniques used in quantum informtaion theory. His Theory of Quantum Information and Advanced topics in Quantum Information Theory are also very good reading materials.
  • Vlako Vedral, Introduction to quantum information science, this book covers the basics of quantum information theory and quantum computation theory.
  • Dénes Petz, Quantum Information Theory and Quantum Statistics, leans more towards a mathematician-oriented approach. It covers several topics that are typically not addressed in standard quantum information textbooks.
  • Sumeet Khatri and Mark M. Wilde, Principles of Quantum Communication Theory: A Modern Approach, the pdf version is avaliable on arxiv: 2011.04672. This 1,240-page book covers numerous topics on quantum communications that are rarely found elsewhere.
  • Gregg Jaeger, Quantum Information-An Overview. This is a concise book that serves as a good introduction for beginners.
  • Giuliano Benenti, Giulio Casati, and Giuliano Strini’s Principles of Quantum Computation and Information Volumes I & II are excellent resources for beginners.
  • Karl Kraus, ‘States, Effects, and Operations: Fundamental Notions of Quantum Theory’, published in 1983, may not be considered an ideal textbook today. However, I will mention it for readers interested in the history of quantum information.
  • Asher Peres, ‘Quantum Theory: Concepts and Methods’ is another classic book that explores many fascinating topics in the foundations of quantum theory.

Don’t be intimidated by the long list of materials. You can start with just one as your textbook, and the mathematical tools will naturally be learned as you dive into research. Remember, this is a life-long learning process.

If you like mathematics related to quantum information theory, I would like to recommend (notice that to read these books, you should have some background on analysis and topology first):

  • Guillaume Aubrun, Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory. This book is a discussion of quantum information theory from functional analysis perspective.
  • Vern Paulsen, Completely Bounded Maps and Operator Algebras. This book covers many crurical results for us to understand CP maps, channels, etc., from the mathematical perspective.
  • Charalambos D. Aliprantis , Owen Burkinshaw, Positive Operators. As it name indicated, this is a book about positive operators.
  • R. Tyrrell Rockafellar, Convex Analysis. This is a classics about convex analysis, you will encounter convex analysis in many places of quantum information theory.
  • Rajendra Bhatia, Matrix analysi, this from famous GMT series. Matrix analysis plays a crucial role in quantum information theory.

There are numerous online PDF notes and video courses available. Here are a few notable ones:

Some useful numerical tools

Many tools are now available for simulating various quantum information and quantum computation tasks. Below is a brief selection:

  • QuTip, a python-based open-source software for simulating the dynamics of open quantum systems.
  • Qiskit, IBM package for quantum information and quantum computation.

Course: Quantum computation theory

This is a page for quantum computation theory.

Syllabus

The topics will cover

Part I quantum computation model

  • Quantum Turing machine
  • Quantum Circuit model
  • Measurement-based quantum computation

Part II Quantum complexity

Part III Quantum algorithms

Quantum computation theory on the web

There are many wonderful materials discussing the basics of quantum computation theory, here are some:

Reading seminar: Measurement-based quantum computation

Starting this June, there will be a reading seminar on measurement-based quantum computation (or one-way quantum computation, which will be called MBQC for simplicity hereinafter). The topics will include aspects of MBQC:

  • Cluster state and graph state;
  • Entanglement and nonlocality properties of graph state;
  • Local unitary symmetry of graph state;
  • Computationally universal many-body quantum states;
  • SPT and SET in MQBC
  • etc.

We will begin will some classics in this field, mainly focus on graph state. All the information will be updated along the way, and I hope that I can make a pdf note on this topic during the seminar.

Here are some papers and reviews that will be discussed in the first stage (mainly about the properties of graph state and their application in MBQC):

Quantum Discrete Integral Transform

The integral transform is a powerful mathematical tool that maps a function from its original domain into a new function space through an integration operation (or summation in the discrete case). Typical examples includs the Fourier transform, Laplace transform, Mellin transform, and many others, are indispensable across mathematics, physics, engineering, signal processing, quantum mechanics, and beyond. These transformation often reveals properties of the original function—such as frequency content, growth behavior, or analytic structure—that are more easily analyzed or manipulated in the transformed space. In most cases, the original function can be recovered exactly via an inverse transform.

Mathematically, an integral transform \mathcal{T} is an operator which maps a function {f} into another function \mathcal{T}[f], 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(p,x)} as

\displaystyle \mathcal{T}[f](p)=\int K(p,x)f(x) dx

Among the vast family of integral transforms, the Fourier transform stands out for its elegance and ubiquity. In the natural units where \hbar = 1, physicists typically define the position-space wave function \psi(x) from its momentum-space counterpart \psi(p) by the (inverse) Fourier transform (chosen to avoid an overall minus sign in the exponent, which is a common convention in quantum Fourier transform):

\mathcal{F}[\psi](x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{ipx} \psi(p) dp

have the integral kernel {e^{2\pi i xp}/\sqrt{2\pi}}. This single transformation lies at the very heart of quantum mechanics. It converts differential equations into algebraic ones, turns convolutions into products, and recasts purely local position-space information as global momentum content. Momentum-space methods often render seemingly intractable operator equations manifestly solvable.

The discussion that follows extends these classical ideas into the realm of quantum theory through the notion of quantum integral transforms. Since we are concerned with a discrete qubit system, our focus will be on discrete integral transforms. 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 \mathcal{T}[f] thus looks like a matrix transform with continuous indices. In the discrete context, it turns out that an integral transform becomes a matrix transform, which is us discrete integral transform or simply discrete transform.

Definition 1 (Discrete integral transform) Let {K_{ij}=:K(i,j)} be a matrix with indices {i,j=0,\cdots,N-1} and {\vec{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 \vec{y}=K\vec{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 implement a quantum discrete integral transform algorithm, one must design a quantum circuit that realizes the unitary operator $U_K$. This step is generally highly nontrivial and requires careful thought and insightful design. Moreover, the structure of the quantum circuit depends sensitively on the choice of kernel K; different kernels typically lead to very different circuit constructions.

 

Useful materials for learning quantum computation

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:

John Preskill,

John Watrous,

Umesh Vazirani,

Andrew Childs,

Scott Aaronson

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.