Vectors and Matrices

Some familiarity with vectors and matrices is essential to understand quantum computing. We provide a brief introduction below and interested readers are recommended to read a standard reference on linear algebra such as Strang, G. (1993). Introduction to linear algebra (Vol. 3). Wellesley, MA: Wellesley-Cambridge Press or an online reference such as Linear Algebra.

A column vector (or simply vector) $v$ of dimension (or size) $n$ is a collection of $n$ complex numbers $(v_1,v_2,\ldots,v_n)$ arranged as a column:

$$v =\begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{bmatrix}$$

The norm of a vector $v$ is defined as $\sqrt{\sum_i |v_i|^2}$. A vector is said to be of unit norm (or alternatively it is called a unit vector) if its norm is $1$. The adjoint of a vector $v$ is denoted $v^\dagger$ and is defined to be the following row vector where $*$ denotes the complex conjugate,

$$\begin{bmatrix}v_1 \\ \vdots \\ v_n \end{bmatrix}^\dagger = \begin{bmatrix}v_1^* & \cdots & v_n^* \end{bmatrix}$$

The most common way to multiply two vectors together is through the inner product, also known as a dot product. The inner product gives the projection of one vector onto another and is invaluable in describing how to express one vector as a sum of other simpler vectors. The inner product between $u$ and $v$, denoted $\left\langle u, v\right\rangle$ is defined as

$$ \langle u, v\rangle = u^\dagger v=u_1^{*} v_1 + \cdots + u_n^{*} v_n. $$

This notation also allows the norm of a vector $v$ to be written as $\sqrt{\langle v, v\rangle}$.

We can multiply a vector with a number $c$ to form a new vector whose entries are multiplied by $c$. We can also add two vectors $u$ and $v$ to form a new vector whose entries are the sum of the entries of $u$ and $v$. These operations are depicted below:

$$\mathrm{If}~u =\begin{bmatrix} u_1\\ u_2\\ \vdots\\ u_n \end{bmatrix}~\mathrm{and}~ v =\begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{bmatrix},~\mathrm{then}~ au+bv =\begin{bmatrix} au_1+bv_1\\ au_2+bv_2\\ \vdots\\ au_n+bv_n \end{bmatrix}. $$

A matrix of size $m \times n$ is a collection of $mn$ complex numbers arranged in $m$ rows and $n$ columns as shown below:

$$M = \begin{bmatrix} M_{11} ~~ M_{12} ~~ \cdots ~~ M_{1n}\\ M_{21} ~~ M_{22} ~~ \cdots ~~ M_{2n}\\ \ddots\\ M_{m1} ~~ M_{m2} ~~ \cdots ~~ M_{mn}\\ \end{bmatrix}.$$

Note that a vector of dimension $n$ is simply a matrix of size $n \times 1$. As with vectors, we can multiply a matrix with a number $c$ to obtain a new matrix where every entry is multiplied with $c$, and we can add two matrices of the same size to produce a new matrix whose entries are the sum of the respective entries of the two matrices.

Matrix Multiplication and Tensor Products

We can also multiply two matrices $M$ of dimension $m\times n$ and $N$ of dimension $n \times p$ to get a new matrix $P$ of dimension $m \times p$ as follows:

\begin{align} &\begin{bmatrix} M_{11} ~~ M_{12} ~~ \cdots ~~ M_{1n}\\ M_{21} ~~ M_{22} ~~ \cdots ~~ M_{2n}\\ \ddots\\ M_{m1} ~~ M_{m2} ~~ \cdots ~~ M_{mn} \end{bmatrix} \begin{bmatrix} N_{11} ~~ N_{12} ~~ \cdots ~~ N_{1p}\\ N_{21} ~~ N_{22} ~~ \cdots ~~ N_{2p}\\ \ddots\\ N_{n1} ~~ N_{n2} ~~ \cdots ~~ N_{np} \end{bmatrix}=\begin{bmatrix} P_{11} ~~ P_{12} ~~ \cdots ~~ P_{1p}\\ P_{21} ~~ P_{22} ~~ \cdots ~~ P_{2p}\\ \ddots\\ P_{m1} ~~ P_{m2} ~~ \cdots ~~ P_{mp} \end{bmatrix} \end{align}

where the entries of $P$ are $P_{ik} = \sum_j M_{ij}N_{jk}$. For example, the entry $P_{11}$ is the inner product of the first row of $M$ with the first column of $N$. Note that since a vector is simply a special case of a matrix, this definition extends to matrix-vector multiplication.

All the matrices we consider will either be square matrices, where the number of rows and columns are equal, or vectors, which corresponds to only $1$ column. One special square matrix is the identity matrix, denoted $\boldone$, which has all its diagonal elements equal to $1$ and the remaining elements equal to $0$:

$$\boldone=\begin{bmatrix} 1 ~~ 0 ~~ \cdots ~~ 0\\ 0 ~~ 1 ~~ \cdots ~~ 0\\ ~~ \ddots\\ 0 ~~ 0 ~~ \cdots ~~ 1 \end{bmatrix}.$$

For a square matrix $A$, we say a matrix $B$ is its inverse if $AB = BA = \boldone$. The inverse of a matrix need not exist, but when it exists it is unique and we denote it $A^{-1}$.

For any matrix $M$, the adjoint or conjugate transpose of $M$ is a matrix $N$ such that $N_{ij} = M^*_{ji}$. The adjoint of $M$ is usually denoted $M^\dagger$. We say a matrix $U$ is unitary if $UU^\dagger = U^\dagger U = \boldone$ or equivalently, $U^{-1} = U^\dagger$. Perhaps the most important property of unitary matrices is that they preserve the norm of a vector. This happens because

$$\langle v,v \rangle=v^\dagger v = v^\dagger U^{-1} U v = \langle U v, U v\rangle.$$

A matrix $M$ is said to be Hermitian if $M=M^\dagger$.

Finally, the tensor product (or Kronecker product) of two matrices $M$ of size $m\times n$ and $N$ of size $p \times q$ is a larger matrix $P=M\otimes N$ of size $mp \times nq$, and is obtained from $M$ and $N$ as follows:

\begin{align} M \otimes N &= \begin{bmatrix} M_{11} ~~ \cdots ~~ M_{1n} \\ \ddots\\ M_{m1} ~~ \cdots ~~ M_{mn} \end{bmatrix} \otimes \begin{bmatrix} N_{11} ~~ \cdots ~~ N_{1q}\\ \ddots\\ N_{p1} ~~ \cdots ~~ N_{pq} \end{bmatrix}\\ &= \begin{bmatrix} M_{11} \begin{bmatrix} N_{11} ~~ \cdots ~~ N_{1q}\\ \ddots\\ N_{p1} ~~ \cdots ~~ N_{pq} \end{bmatrix}~~ \cdots ~~ M_{1n} \begin{bmatrix} N_{11} ~~ \cdots ~~ N_{1q}\\ \ddots\\ N_{p1} ~~ \cdots ~~ N_{pq} \end{bmatrix}\\ \ddots\\ M_{m1} \begin{bmatrix} N_{11} ~~ \cdots ~~ N_{1q}\\ \ddots\\ N_{p1} ~~ \cdots ~~ N_{pq} \end{bmatrix}~~ \cdots ~~ M_{mn} \begin{bmatrix} N_{11} ~~ \cdots ~~ N_{1q}\\ \ddots\\ N_{p1} ~~ \cdots ~~ N_{pq} \end{bmatrix} \end{bmatrix}. \end{align}

This is better demonstrated with some examples:

$$ \begin{bmatrix} a \\ b \end{bmatrix} \otimes \begin{bmatrix} c \\ d \\ e \end{bmatrix} = \begin{bmatrix} a \begin{bmatrix} c \\ d \\ e \end{bmatrix} \\[1.5em] b \begin{bmatrix} c \\ d \\ e\end{bmatrix} \end{bmatrix} = \begin{bmatrix} a c \\ a d \\ a e \\ b c \\ b d \\ be\end{bmatrix} $$

and

$$ \begin{bmatrix} a\ b \\ c\ d \end{bmatrix} \otimes \begin{bmatrix} e\ f\\g\ h \end{bmatrix} = \begin{bmatrix} a\begin{bmatrix} e\ f\\ g\ h \end{bmatrix} b\begin{bmatrix} e\ f\\ g\ h \end{bmatrix} \\[1em] c\begin{bmatrix} e\ f\\ g\ h \end{bmatrix} d\begin{bmatrix} e\ f\\ g\ h \end{bmatrix} \end{bmatrix} = \begin{bmatrix} ae\ af\ be\ bf \\ ag\ ah\ bg\ bh \\ ce\ cf\ de\ df \\ cg\ ch\ dg\ dh \end{bmatrix}. $$

A final useful notational convention surrounding tensor products is that, for any vector $v$ or matrix $M$, $v^{\otimes n}$ or $M^{\otimes n}$ is short hand for an $n$-fold repeated tensor product. For example:

\begin{align} &\begin{bmatrix} 1 \\ 0 \end{bmatrix}^{\otimes 1} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad\begin{bmatrix} 1 \\ 0 \end{bmatrix}^{\otimes 2} = \begin{bmatrix} 1 \\ 0 \\0 \\0 \end{bmatrix}, \qquad\begin{bmatrix} 1 \\ -1 \end{bmatrix}^{\otimes 2} = \begin{bmatrix} 1 \\ -1 \\-1 \\1 \end{bmatrix}, \\ &\begin{bmatrix} 0 & 1 \\ 1& 0 \end{bmatrix}^{\otimes 1}= \begin{bmatrix} 0& 1 \\ 1& 0 \end{bmatrix}, \qquad\begin{bmatrix} 0 & 1 \\ 1& 0 \end{bmatrix}^{\otimes 2}= \begin{bmatrix} 0 &0&0&1 \\ 0 &0&1&0 \\ 0 &1&0&0\\ 1 &0&0&0\end{bmatrix}. \end{align}