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What is Multi Qubit Systems?

  • Difficulty Level : Medium
  • Last Updated : 14 Feb, 2021
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Prerequisite: Qubit Representation

A multi-qubit system is a collection of multiple qubits, treated as a single system. In classical computing, a system of N bits can be in 2N states. For example, for a 2-bit system, there are 4 possible states 00, 01, 10, and 11. In a Quantum system of N qubits, there are 2N basis states and a qubit can be in any of these states or even their superposition. An arbitrary N qubit can be represented as follows:

A=\begin{bmatrix}x_0\\x_1\\.\\.\\.\\x_{n-1} \end{bmatrix}

Computational Basis States

These are the vectors that form an orthonormal computational basis for multi-qubit systems. These are formed by performing outer product or tensor products of individual basis states. For a 2 qubit system following is the computational basis:

\begin{bmatrix}1\\0\\0\\0 \end{bmatrix}\ \begin{bmatrix}0\\1\\0\\0 \end{bmatrix}\ \begin{bmatrix}0\\0\\1\\0 \end{bmatrix}\ \begin{bmatrix}0\\0\\0\\1 \end{bmatrix}



\begin{bmatrix}1\\0\\0\\0 \end{bmatrix}\ = \ \begin{bmatrix}1\\0 \end{bmatrix}\bigotimes \begin{bmatrix}1\\0 \end{bmatrix} =|0\rangle \otimes |0\rangle ,

\begin{bmatrix}0\\1\\0\\0 \end{bmatrix}\ = \ \begin{bmatrix}1\\0 \end{bmatrix}\bigotimes \begin{bmatrix}0\\1 \end{bmatrix} =|0\rangle \otimes |1\rangle ,

\begin{bmatrix}0\\0\\1\\0\ \end{bmatrix}\ = \ \begin{bmatrix}0\\1 \end{bmatrix}\bigotimes \begin{bmatrix}1\\0 \end{bmatrix} = |1\rangle \otimes |0\rangle ,

\begin{bmatrix}0\\0\\0\\1 \end{bmatrix}\ = \ \begin{bmatrix}0\\1 \end{bmatrix}\bigotimes \begin{bmatrix}0\\1 \end{bmatrix} = |1\rangle \otimes |1\rangle

Separable and Non-separable States

Multi-Qubit systems in which possible states of individual qubits can be found are separable states and which can’t be found are said to be non-separable states or entangled states.

When particles become entangled, they form a single system such that the quantum state of any one particle cannot be described independently of the quantum state of the other particles. This means that whatever operation or process you apply to one particle correlates to the other particles as well.

\begin{bmatrix}a\\b \end{bmatrix}\bigotimes \begin{bmatrix}c\\d \end{bmatrix}\ =\ \begin{bmatrix}ac\\ad\\bc\\bd \end{bmatrix}

For a given multi-qubit system state i.e. given the products ac, ad, bc, and bd; If a possible solution for a, b, c, and d can be found, then the system is in the separable state otherwise in the non-separable or entangled state.



Dirac’s Notation

The basis states are expressed as the tensor product of states |0> and |1>. The shorthand notation for a computational basis for 2 qubit system is given as follows:

|0\rangle \otimes |0\rangle = |00\rangle

|0\rangle \otimes |1\rangle = |01\rangle

|1\rangle \otimes |0\rangle = |10\rangle

|1\rangle \otimes |1\rangle = |11\rangle

\begin{bmatrix}x_0\\x_1\\x_2\\x_3 \end{bmatrix}\ = x_0|00\rangle +\ x_1|01\rangle +\ x_2|10\rangle +\ x_3|11\rangle

Endianness

It refers to the order of bits used to represent the binary numbers.

Integer Ket                    |0>      |1>     |2>     |3>

Big Endian Notation    |00>   |01>   |10>   |11>   

Little Endian Notation |00>   |10>   |01>   |11>
 

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