Ch1-Lecture 1

Jupyter

This is a jupyter notebook.

It consists of cells.

  • Markdown cells like this one, which contains “text”

  • Code cells that contain Python code.

  • Each cell can be run independently, regardless of the order (beware of the code flow!)

  • Useful short-cuts:

    • Press shift+Enter to run a cell

    • Press ESC+D to delete a cell

    • Press ESC+A to create a cell above your current cell

    • Press ESC+B to create a cell below your current cell

Installing the Jupyter to your local PC Use a package manager like miniconda!

Markdown and LATEX

Markdown is a formatted text language that makes writing on the web fast and easy. It allows writing formulas using LATEX. We will use it in our exercises and homeworks.

Have you ever asked yourself, how they write complex maths and physics equations using computer? Well, it’s all about LaTeX.

You can insert formulas by placing them in between $

Anaconda

Python package management is messy, do NOT install packages to your user environment directly without a package manager.

Anaconda is a distribution of the Python and R programming languages for scientific computing (data science, machine learning applications, large-scale data processing, predictive analytics, etc.), that aims to simplify package management and deployment. The distribution includes data-science packages suitable for Windows, Linux, and macOS.

Exercise 1:

Download and install miniconda to your local computer. Create an environment, and switch to it. Install Jupyter notebook to that environment.

IBM Quantum Cloud

In this lecture, we will heavily rely on IBM’s Quantum cloud. Please use this form for me to add you to the lecture (you will receive an invitation to create one, if you don’t have an account yet).

Introduction

The field of quantum computing is nowadays far too large to be captured in a single course. Here, we shall focus on a broad introduction which aims to cover topics such as: What is quantum mechanics, and how can it be harnessed to build a computer? What kind of computational problems can such a computer solve? Are there problems which are hard even for a quantum computer? And finally, what does the study of quantum computing tell us about nature itself?

Even if this course is the last time you encounter the topic of quantum computing, I hope it will leave you with a deeper understanding of Quantum concepts, and develop your skills as a basic scientist.

In order to represent Quantum phenomena, we will need the mathematical framework of Linear Algebra. As a scientist, you should already be familiar with the content, here I will give you a refresher. These notes contain many exercises intended to help the reader; it is strongly recommended for you to work on these as you follow.

The basic objects we need are complex vectors \( \left|\psi\right\rangle \in \mathbb{C}^d \)

\[\begin{split} \left|\psi\right\rangle = \begin{pmatrix} \psi_1 \\ \vdots \\ \psi_d \end{pmatrix} \end{split}\]

for \(\psi_i\in\mathbb{C}\) (i.e. complex number). In this course, we will mostly use the polar form for complex numbers : \(re^{i\theta}\).

The notation \(\left|\cdot\right\rangle\) is called Dirac notation, named after physicist Paul Dirac. \(\left|\psi\right\rangle\) is pronounced “ket \(\psi\)”.

Complex vectors are used to represent quantum states. (The difference between a representation and an actual state is similar to the difference between a movie star, and the poster of a movie star).

Some further basic properties of vectors:

The conjugate transpose of \( \left|\psi\right\rangle\) is given by

\[ \left\langle\psi\right| = \left(\psi_1^*,\psi_2^*\dots\psi_d^*\right)\]

where \( \left\langle\psi\right|\) is a row vector. The term \( \left\langle\psi\right|\) is pronounced “bra \(\psi\)”.

How much two vectors “overlap” is given by the inner product function:

\[ \left\langle\left.\psi \right|\phi \right\rangle = \sum_{i=1}^d\psi_i^d\phi_i \]

The inner product satisfies \(\left(\left\langle\left.\psi \right|\phi \right\rangle\right)^*=\left\langle\left.\phi \right|\psi \right\rangle\). The “length” of a vector \(\left|\psi\right\rangle\) can now be quantified by measuring the overlap of \(\left|\psi\right\rangle\) with itself, which yields the Euclidean norm, \(\left\lVert\left|\psi\right\rangle \right\lVert_2 =\sqrt{\left\langle\left.\psi \right|\phi \right\rangle}\).

Send it after class 1

Let \( \left|\psi\right\rangle =\frac{1}{\sqrt{2}}\left(1,i\right)^T\), where T denotes the transpose. What are \( \left\langle\psi\right|\) and \(\left\lVert\left|\psi\right\rangle \right\lVert_2\)?

With a norm in hand, we can define a notion of distance between vectors \(\left|\psi\right\rangle\),\(\left|\phi\right\rangle\) called the Euclidean distance: \(\left\lVert\left|\psi\right\rangle-\left|\phi\right\rangle \right\lVert_2\). This distance will play the important role of quantifying how well two quantum states \(\left|\psi\right\rangle\) and \(\left|\phi\right\rangle\) can be “distinguished” via measurements. Two useful properties of the Euclidean norm are:

  1. (Positive scalability) \(\left\lVert a \left|\psi\right\rangle \right\lVert_2 = \left|a\right|\left\lVert \left|\psi\right\rangle \right\lVert_2\) for \(a \in \mathbb{C} \)

  2. (Triangle inequality) For any \(\left|\psi\right\rangle\) and \(\left|\phi\right\rangle\) one has \(\left\lVert\left|\psi\right\rangle+\left|\phi\right\rangle \right\lVert_2 \leq \left\lVert\left|\psi\right\rangle\right\lVert_2 +\left\lVert\left|\phi\right\rangle \right\lVert_2\) .

These two properties can be used to show that for all \( \left|\psi\right\rangle \in \mathbb{C}^d \), \(\left\lVert\left|\psi\right\rangle\right\lVert_2\geq 0\)

Send it after class 2

Let \( \left|\psi\right\rangle =\frac{1}{\sqrt{2}}\left(1,i\right)^T\) and \( \left|\phi\right\rangle =\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)^T\). What is \(\left\lVert\left|\psi\right\rangle-\left|\phi\right\rangle \right\lVert_2\)?

Orthonormal bases

A much more natural way to represent vectors in this course shall be in terms of orthonormal bases. Recall that a set of vectors \( \left \{\left|\psi\right\rangle\right\} \subseteq \mathbb{C}^d \) is orthogonal if for all \(i\neq j\), \(\left\langle\left.\psi_i \right|\psi_j \right\rangle=0\), and orthonormal if \(\left\langle\left.\psi_i \right|\psi_j \right\rangle=\delta_{ij}\). Here, \(\delta_{ij}\) is the Kroenecker delta, whose value is 1 if \(i = j\) and 0 otherwise. For the vector space \(\mathbb{C}^d\) , which has dimension \(d\), it is necessary and sufficient to use \(d\) orthonormal vectors in order to form an orthonormal basis.

One of the most common bases we use is the computational basis, defined for \(\mathbb{C}^2\) as

\[\begin{split} \left|0\right\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \qquad \left|1\right\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{split}\]

since \( \left \{\left|0\right\rangle,\left|1\right\rangle\right\}\) is an orthonormal basis, any vector \( \left|\psi\right\rangle \in \mathbb{C}^2 \) can be written as \( \left|\psi\right\rangle= \alpha\left|0\right\rangle+\beta\left|1\right\rangle \) for some \(\alpha,\beta\in \mathbb{C}\). We say \(\left|\psi\right\rangle\) is normalized when it has length 1, i.e. \(\left\lVert\left|\psi\right\rangle\right\lVert_2=1\) or equivalently \(\left|\alpha\right|^2+\left|\beta\right|^2=1\)

Send it after class 3

Let \( \left|\psi\right\rangle =\frac{1}{\sqrt{2}}\left(1,1\right)^T\in\mathbb{C}^2\). Write \( \left\langle\psi\right|\) in terms of computational basis for \(\mathbb{C}^2\). Is it normalized?

The computational basis is easily extended to \(d\)-dimensional vectors by defining \(\left|i\right\rangle\in\mathbb{C}^d\) as having a 1 in position i and 0 elsewhere (here, \(0 \leq i \leq d − 1\)). In this course, vectors labelled by integers (e.g. \(\left|1\right\rangle\), \(\left|3\right\rangle\in\mathbb{C}^d\) ) will be assumed to be \(d\)-dimensional computational basis vectors

Linear maps.

Given a vector \( \left|\psi\right\rangle \in \mathbb{C}^d \) , we are interested in how \(\left|\psi\right\rangle\) can be “mapped” to other vectors. The maps are linear, which by definition means that for map \(\Phi : \mathbb{C}^d \mapsto \mathbb{C}^d\) and arbitrary \(\sum_i\alpha_i\left|\psi_i\right\rangle\in \mathbb{C}^d \),

\[ \Phi\left(\sum_i\alpha_i\left|\psi_i\right\rangle\right)=\sum_i\alpha_i\Phi\left( \left|\psi_i\right\rangle\right) \]

The set of linear maps from vector space \(\mathcal{X}\) to \(\mathcal{Y}\) is denoted \(\mathcal{L}(\mathcal{X} , \mathcal{Y})\). For brevity, we use the shorthand \(\mathcal{L}(\mathcal{X})\) to mean \(\mathcal{L}(\mathcal{X} , \mathcal{Y})\)

Send it after class 4

Consider the linear map \(\Phi : \mathbb{C}^d \mapsto \mathbb{C}^d\) with action \(\Phi\left(\left|0\right\rangle\right)=\left|1\right\rangle \) and \( \Phi\left(\left|1\right\rangle\right)=\left|0\right\rangle \). If \( \left|\psi\right\rangle= \alpha\left|0\right\rangle+\beta\left|1\right\rangle \), what is \( \Phi\left(\left|\psi\right\rangle\right) \)?

the action of a linear map \(\Phi \in \mathcal{L}\left(\mathbb{C}^d\right)\) is fully characterized by understanding how \(\Phi\) acts on a basis for \(\mathbb{C}^d\) . This leads to a natural representation for \(\Phi\) in terms of a matrix.

Recall that a \(d\times d\) matrix \(A\) is a two-dimensional array of complex numbers whose \((i, j)\) th entry is denoted \(A(i, j) \in \mathbb{C}\) for \(i, j \in \left[ d \right]\). To represent a linear map \(\Phi : \mathbb{C}^d \mapsto \mathbb{C}^d\) as an \(d \times d\) matrix \(A_\Phi\) , we use its action on a basis for \(\mathbb{C}^d\) . Specifically, define the ith column of \( A_\Phi \) as \( \Phi\left(\left|i\right\rangle\right) \) for \( \left\{\left|i\right\rangle\right\} \) the standard basis for \( \mathbb{C}^d \) , or

\[ A_\Phi = \left[ \Phi\left(\left|0\right\rangle\right), \Phi\left(\left|1\right\rangle\right), \dots , \Phi\left(\left|d-1\right\rangle\right) \right ] \]

In this course, we use both the matrix and linear map views interchangeably, with the application notion clear from context.

Send it after class 5

What is the \(2 \times 2\) complex matrix tepresenting the linear map \(\Phi\) from “send it it after class 4” ? What is the linear map whose matrix (with respect to the computational basis) is the identity matrix

\[\begin{split} I= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \end{split}\]