CSE5313 Final Project

Description

Write a report which presents the paper in detail and criticizes it constructively. Below are some suggestions to guide a proper analysis of the paper:

What is the problem setting? What is the motivation behind this problem?
Which tools are being used? How are these tools related to the topics we have seen in class/HW?
Is the paper technically sound, easy to read, and does the problem make sense?
Are there any issues with the paper? If so, suggest ways these could be fixed.
What is the state of the art in this topic? Were there any major follow-up works? What are major open problems or new directions?
Suggest directions for further study and discuss how should those be addressed.

Please typeset your work using a program such as Word or LaTeX, and submit via Gradescope by Dec. 10th (EOD).

Reports will be judged by the depth and breadth of the analysis. Special attention will be given to clarity and organization, including proper abstract, introduction, sections, and citation of previous works.

There is no page limit, but good reports should contain 4-6 single-column pages. Please refer to the syllabus for our policy regarding the use of GenAI.

Paper selection

Good quantum error-correcting codes exist

Tip

I will build the self-contained report that is readable for me 7 months ago, assuming no knowledge in quantum computing and quantum information theory.

We will use notation defined in class and $[n]=\{1,\cdots,n-1,n\}$ , (yes, we use 1 indexed in computer science) each in natural number. And $\mathbb{F}_q$ is the finite field with $q$ elements.

Warning

This notation system is annoying since in mathematics, $A^*$ is the transpose of $A$ , but since we are using literatures in physics, we keep the notation of $A^*$ . In this report, I will try to make the notation consistent as possible and follows the physics convention in this report. So every vector you see will be in $\ket{\psi}$ form. And we will avoid using the $\langle v,w\rangle$ notation for inner product as it used in math, we will use $\langle v|w\rangle$ or $\langle v,w\rangle$ to denote the inner product.

A quantum error-correcting code is defined to be a unitary mapping (encoding) of $k$ qubits (two-state quantum systems) into a subspace of the quantum state space of $n$ qubuits such that if any $t$ of the qubits undergo arbitary decoherence, not necessarily independently, the resulting $n$ qubit state can be used to faithfully reconstruct the original quantum state of the $k$ encoded qubits.

Asymptotic rate $k/n=1-2H_2(2t/n)$ , where $H_2$ is the binary entropy function

H_2=-p\log_2(p)-(1-p)\log_2(1-p)

Problem setting and motivation

Linear algebra 102

The main vector space we are interested in is $\mathbb{C}^n$ , therefore, all the linear operator we defined are from $\mathbb{C}^n$ to $\mathbb{C}^n$ .

We denote a vector in vector space as $\ket{\psi}=(z_1,\cdots,z_n)$ (might also be infinite dimensional, and $z_i\in\mathbb{C}$ ).

A natural inner product space defined on $\mathbb{C}^n$ is given by the Hermitian inner product:

\langle\psi|\varphi\rangle=\sum_{i=1}^n z_i\bar{z}_i

This satisfies the following properties:

$\bra{\psi}\sum_i \lambda_i\ket{\varphi}=\sum_i \lambda_i \langle\psi|\varphi\rangle$ (linear on the second argument)
$\langle\varphi|\psi\rangle=(\langle\psi|\varphi\rangle)^*$
$\langle\psi|\psi\rangle\geq 0$ with equality if and only if $\ket{\psi}=0$

Here $\psi$ is just a label for the vector and you don’t need to worry about it too much. This is also called the ket, where the counterpart:

$\langle\psi\rangle$ is called the bra, used to denote the vector dual to $\psi$ , such element is a linear functional if you really wants to know what that is.
$\langle\psi|\varphi\rangle$ is the inner product between two vectors, and $\bra{\psi} A\ket{\varphi}$ is the inner product between $A\ket{\varphi}$ and $\bra{\psi}$ , or equivalently $A^\dagger \bra{\psi}$ and $\ket{\varphi}$ .
Given a complex matrix $A=\mathbb{C}^{n\times n}$ $A = C^{n \times n}$ ,
- $A^*$ $A^{*}$ is the complex conjugate of $A$ $A$ .
  - i.e., $A=\begin{bmatrix}1+i & 2+i & 3+i\\4+i & 5+i & 6+i\\7+i & 8+i & 9+i\end{bmatrix}$ , $A^*=\begin{bmatrix}1-i & 2-i & 3-i\\4-i & 5-i & 6-i\\7-i & 8-i & 9-i\end{bmatrix}$
- $A^\top$ $A^{⊤}$ is the transpose of $A$ $A$ .
  - i.e., $A=\begin{bmatrix}1+i & 2+i & 3+i\\4+i & 5+i & 6+i\\7+i & 8+i & 9+i\end{bmatrix}$ , $A^\top=\begin{bmatrix}1+i & 4+i & 7+i\\2+i & 5+i & 8+i\\3+i & 6+i & 9+i\end{bmatrix}$
- $A^\dagger=(A^*)^\top$ $A^{†} = (A^{*})^{⊤}$ is the complex conjugate transpose, referred to as the adjoint, or Hermitian conjugate of $A$ $A$ .
  - i.e., $A=\begin{bmatrix}1+i & 2+i & 3+i\\4+i & 5+i & 6+i\\7+i & 8+i & 9+i\end{bmatrix}$ , $A^\dagger=\begin{bmatrix}1-i & 4-i & 7-i\\2-i & 5-i & 8-i\\3-i & 6-i & 9-i\end{bmatrix}$
- $A$ is unitary if $A^\dagger A=AA^\dagger=I$ .
- $A$ is hermitian (self-adjoint in mathematics literatures) if $A^\dagger=A$ .

Motivation of Tensor product

Recall from the traditional notation of product space of two vector spaces $V$ and $W$ , that is, $V\times W$ , is the set of all ordered pairs $(\ket{v},\ket{w})$ where $\ket{v}\in V$ and $\ket{w}\in W$ .

The space has dimension $\dim V+\dim W$ .

We want to define a vector space with notation of multiplication of two vectors from different vector spaces.

That is

(\ket{v_1}+\ket{v_2})\otimes \ket{w}=(\ket{v_1}\otimes \ket{w})+(\ket{v_2}\otimes \ket{w})

\ket{v}\otimes (\ket{w_1}+\ket{w_2})=(\ket{v}\otimes \ket{w_1})+(\ket{v}\otimes \ket{w_2})

and enables scalar multiplication by

\lambda (\ket{v}\otimes \ket{w})=(\lambda \ket{v})\otimes \ket{w}=\ket{v}\otimes (\lambda \ket{w})

And we wish to build a way associates the basis of $V$ and $W$ to the basis of $V\otimes W$ . That makes the tensor product a vector space with dimension $\dim V\times \dim W$ .

Definition of linear functional

Tip

Note the difference between a linear functional and a linear map.

A generalized linear map is a function $f:V\to W$ satisfying the condition

$f(\ket{u}+\ket{v})=f(\ket{u})+f(\ket{v})$
$f(\lambda \ket{v})=\lambda f(\ket{v})$

A linear functional is a linear map from $V$ to $\mathbb{F}$ .

Definition of bilinear functional

A bilinear functional is a bilinear function $\beta:V\times W\to \mathbb{F}$ satisfying the condition that $\ket{v}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{w}\in W$ and $\ket{w}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{v}\in V$ .

The vector space of all bilinear functionals is denoted by $\mathcal{B}(V,W)$ .

Definition of tensor product

Let $V,W$ be two vector spaces.

Let $V'$ and $W'$ be the dual spaces of $V$ and $W$ , respectively, that is $V'=\{\psi:V\to \mathbb{F}\}$ and $W'=\{\phi:W\to \mathbb{F}\}$ , $\psi, \phi$ are linear functionals.

The tensor product of vectors $v\in V$ and $w\in W$ is the bilinear functional defined by $\forall (\psi,\phi)\in V'\times W'$ given by the notation

(v\otimes w)(\psi,\phi)\coloneqq\psi(v)\phi(w)

The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$

Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$ , that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$ , then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V',W')$ .

That is, every element of $\mathcal{B}(V',W')$ can be written as a linear combination of the basis.

Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$ , respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$ .

Here $\delta_{ij}=\begin{cases} 1 & \text{if } i=j \\ 0 & \text{otherwise} \end{cases}$ is the Kronecker delta.

V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\}

Note that $\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w)$ is a bilinear functional that maps $V'\times W'$ to $\mathbb{F}$ .

This enables basis free construction of vector spaces with proper multiplication and scalar multiplication.

This vector space is equipped with the unique inner product $\langle v\otimes w, u\otimes x\rangle_{V\otimes W}$ defined by

\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle_V\langle w,x\rangle_W

In practice, we ignore the subscript of the vector space and just write $\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle\langle w,x\rangle$ .

Note

All those definitions and proofs can be found in Linear Algebra Done Right by Sheldon Axler.

Definition of two-state quantum system

The finite dimensional Hilbert space $\mathcscr{H}

Definition of Coherent states from the view of physics

Side node: Why quantum error-correcting code is hard

Decoherence process

No-cloning theorem

Reference from P.532 of the book

Suppose we have a quantum system with two slots $A$ , and $B$ , the data slot, starts out in an unknown but pure quantum state $\ket{\psi}$ . This is the state which is to be copied into slot $B$ m the target slot. We assume that the target slot starts out in some standard pure state $\ket{s}$ . Thus the initial state of the copying machine is $\ket{\psi}\otimes \ket{s}$ .

Assume there exists some unitary operator $U$ such that $U(\ket{\psi}\otimes \ket{s})=\ket{\psi}\otimes \ket{\psi}$ .

Consider two pure states $\ket{\psi}$ and $\ket{\varphi}$ , such that $U(\ket{\psi}\otimes \ket{s})=\ket{\psi}\otimes \ket{\psi}$ and $U(\ket{\varphi}\otimes \ket{s})=\ket{\varphi}\otimes \ket{\varphi}$ . The inner product of the two equation yields:

\langle \psi|\varphi\rangle =(\langle \psi|\varphi\rangle)^2

This equation has only two solutions, either $\langle \psi|\varphi\rangle=0$ or $\langle \psi|\varphi\rangle=1$ .

If $\langle \psi|\varphi\rangle=0$ , then $\ket{\psi}=\ket{\varphi}$ , no cloning for trivial case.

If $\langle \psi|\varphi\rangle=1$ , then $\ket{\psi}$ and $\ket{\varphi}$ are orthogonal.

Proposition: Encoding 8 to 9 that correct 1 errors

Recover 1 qubit from a 9 qubit quantum system. (Shor code, 1995)

Shore code

Theoretical upper bound for quantum error-correcting code

From quantum information capacity of a quantum channel

\min\{1-H_2(2t/3n),H_2(\frac{1}{2}+\sqrt{(1-t/n)t/n})\}

Definition of quantum error-correcting code from binary linear error-correcting code

All the operations will be done in $\mathbb{F}_2=\{0,1\}$ .

Consider two binary vectors $v=[v_1,...,v_n],v_i\in\{0,1\}$ and $w=[w_1,...,w_n],w_i\in\{0,1\}$ with size $n$ .

Recall from our lecture that

$d$ denotes the Hamming weight of a vector.

$d_H(v,w)=\sum_{i=1}^{n}\begin{cases} 0 & \text{if } v_i=w_i \\ 1 & \text{if } v_i\neq w_i \end{cases}$ denotes the Hamming distance between $v$ and $w$ .

$\operatorname{supp}(v)=\{i\in[n]:v_i\neq 0\}$ denotes the support of $v$ .

$v|_S$ denotes the projection of $v$ onto the subspace $S$ , we usually denote the $S$ by a set of coordinates, that is $S\subseteq[n]$ .

When projecting a vector $v$ onto a another vector $w$ , we usually write $v|_E\coloneqq v|_{\operatorname{supp} w}$ .

When we have two vector we may use $v\leqslant w$ (Note that this is different than $\leq$ sign) to mean $\operatorname{supp}(v)\subseteq \operatorname{supp}(w)$ .

Example

Let $v=[1,0,0,1,1,1,1]$ and $w=[1,0,0,1,0,0,1]$ , then $\operatorname{supp}(v)=\{1,4,5,6,7\}$ , $\operatorname{supp}(w)=\{1,4,7\}$ . Therefore $w\leqslant v$ .

$v|_w=[v_1,v_4,v_7]=[1,1,0]$

$\mathcal{C}$ denotes the code, a set of arbitrary binary vectors with length $n$ .

$d(\mathcal{C})=\{d(v,w)|v,w\in\mathcal{C}\}$ denotes the minimum distance of the code.

If $\mathcal{C}$ is linear then the minimum distance is the minimum Hamming weight of a non-zero codeword.

A $[n,k,d]$ linear code is a linear code of $n$ bits codeword with $k$ message bits that can correct $d$ errors.

$R\coloneqq\frac{\operatorname{dim}\mathcal{C}}{n}$ is the rate of code $\mathcal{C}$ .

$\mathcal{C}^{\perp}\coloneqq\{v\in\mathbb{F}_2^n:v\cdot w=0\text{ for all }w\in\mathcal{C}\}$ is the dual code of a code $\mathcal{C}$ . From linear algebra, we know that $\dim\mathcal{C}^{\perp}+\dim\mathcal{C}=n$ .

Example used in the paper

Consider the $[7,4,3]$ Hamming code with generator matrix $G$ .

Proposition: Encoding $k$ to $n$ that correct $t$ errors

Evaluation of paper

Limitation and suggestions

Further direction and research

Toric code, surface code

This is the topic I really want to dig into.

This method gives a [2nm+n+m+1, 1, min(n,m)] error correcting code with only needs local stabilizer checks and really interests me.