Introduction

• A typical byte in a computer is represented as follows:

  $01010101$

• The very same byte can be represented in a quantum computer (called a Qubyte) as follows:

  $\ket{0},\ket{1},\ket{0},\ket{1},\ket{0},\ket{1},\ket{0},\ket{1}$

  or

  $\left[\begin{array}{ll}1 \\ 0\end{array}\right] , \left[\begin{array}{ll}0 \\ 1\end{array}\right] , \left[\begin{array}{ll}1 \\ 0\end{array}\right] ,\left[\begin{array}{ll}0 \\ 1\end{array}\right] ,\left[\begin{array}{ll}1 \\ 0\end{array}\right] ,\left[\begin{array}{ll}0 \\ 1\end{array}\right] ,\left[\begin{array}{ll}1 \\ 0\end{array}\right] , \left[\begin{array}{ll}0 \\ 1\end{array}\right]$

• We now, need a way to combine these quantum systems together into one quantity.

VIA TENSOR PRODUCT

• For this purpose, we use the Tensor Product, which is denoted by $\otimes$.

  <aside> ✏️ Mathematically, the tensor product of matrices, is a function:

  $\otimes : \mathbb{C}^{m \times m^{'}} \times \mathbb{C}^{n \times n^{'}} \longrightarrow \mathbb{C}^{mn \times m^{'}n^{'}}$

  and it is defined as:

  $(A \otimes B)[j, k] = A[j/n,k/m] \times B[j \% n, k \% m]$

  it is derived, that for vectors:

  $\otimes : \mathbb{C}^{m} \times \mathbb{C}^{n} \longrightarrow \mathbb{C}^{m\times n}$

  </aside>

  • General Example (Matrix)
  • General Example (Vector)
  • Bits as a vector

• Via the tensor product, we now write our byte $01010101$ as:

  $\ket{0}\otimes\ket{1}\otimes\ket{0}\otimes\ket{1}\otimes\ket{0}\otimes\ket{1}\otimes\ket{0}\otimes\ket{1}$

  This representation is, then condensed and written as:

  $\ket{01010101}$

• Similarly, any byte can now be represented as an element of:

  $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes\mathbb{C}^2 \otimes \mathbb{C}^2 = (\mathbb{C}^2)^{\otimes 8} \sim \mathbb{C}^{256}$

  $\begin{matrix}\ket{00000000}\\\ket{00000001}\\\ket{00000010}\\\vdots\\\ket{11111110}\\\ket{11111111}\end{matrix} \in (\mathbb{C^2})^{\otimes 8}$

COLUMN VECTOR REPRESENTATION

• A byte can also be represented by a column vector as follows:

  $\begin{matrix}00000000\\00000001\\00000010\\\vdots\\11111110\\11111111\end{matrix}\left[\begin{matrix}0\\0\\0\\\vdots\\1\\0\end{matrix}\right] = \ket{11111110}$

• Clearly, such a column vector (doesn’t permit superposition) is only applicable for classical computers, thus for a quantum computer the same column vector will be written as:

  $\begin{matrix}00000000\\00000001\\00000010\\\vdots\\11111110\\11111111\end{matrix}\left[\begin{matrix}\alpha_1\\\alpha_2\\\alpha_3\\\vdots\\\alpha_{255}\\\alpha_{256}\end{matrix}\right]$ where, $\sum_{i=1}^{256} |\alpha_i|^2 = 1$

  This, implies that indicating the state of eight qubits requires 256 complex numbers.

  Emulating a quantum computer, as small as that with a 64-qubit register, would require storing $2^{64}$ on a classical computer, which is way beyond the current storage capacity.

PAIR OF QUBITS

• A pair of qubits can we written in the following different ways:

  $\begin{matrix}\ket{0} \otimes \ket{1} & \ket{0 \otimes 1} & \ket{0}\ket{1} & \ket{0,1} &\ket{01}\end{matrix}$

• Yet another way to represent this qubit pair is via column vector representation:

  $\begin{matrix}00\\01\\10\\11\end{matrix}\left[\begin{matrix}0\\1\\0\\0\end{matrix}\right]$

• A general state of a two-qubit system can be written as

  $\ket{\psi} = c_{0,0}\ket{00} + c_{0,1}\ket{01} + c_{1,0}\ket{10} + c_{1,1}\ket{11} = \begin{matrix}00\\01\\10\\11\end{matrix}\left[\begin{matrix}c_{0,0}\\c_{0,1}\\c_{1,0}\\c_{1,1}\end{matrix}\right]$

• The tensor product of two states is not commutative:

  $\ket{0 \otimes 1} \ne \ket{1 \otimes 0}$

Formal Definition

Given, two Hilbert Spaces $H_1$ & $H_2$ of dimentions $N_1$ & $N_2$ respectively, we can put them together to construct a larger Hilbert Space $H = H_1 \otimes H_2$ of dimension $N_1.N_2$.

Properties