Qubits, Quantum Gates and Quantum Circuits – a Computer Science Perspective: Part 1 Qubits and their Representations

Context

This material is part of the three‑part unit Qubits, Quantum Gates and Quantum Circuits – a Computer Science Perspective.
Part 1 is the introductory part and focuses on the concept of qubits as the basic units of quantum information. Students learn how qubits are defined and how they can be represented using bra–ket notation and vector representations, providing the mathematical foundation for the following parts of the unit.

Content of Part 1

1. What is a qubit?
2. Qubits and how they may be represented

1. What is a qubit?

The following exercises deal with the basics of computing: computers somehow have to handle and process “information”.

Exercise 1

Define the words “bit” and “byte”. Also research alternative systems of storing data like a “trit” or decimal computers.

Solution Solution

A “bit” stands for “binary digit”. A bit can only have the values 0 or 1.

A “byte” is a group of usually 8 bits, which can be used to represent an integer from 0 to 255, for example.

A “trit”, on the other hand, stands for “ternary digit”, i.e. a digit from the ternary system. A special fea-ture of these digits is that they are not 0, 1 and 2, but −1, 0 and 1. This system could easily be implemented in computers, for example by “negative voltage”, “no voltage” and “positive voltage”, which would lead to a greater efficiency for certain circuits. Nevertheless, apart from a few prototypes (e.g. the Russian computer “Setun”21), no ternary computers have been built.

Many of the first computers and mechanical calculating machines of previous centuries did not use the binary system like today's computers, but rather the decimal system. This means that the basic unit of storage was not a bit (“0” or “1”) but a decimal digit (“0” to “9”). Although this representation is more familiar to us humans, the processes in the machine itself for calculating additions or multiplications, for example, are much more complex (e.g. the entire 10x10 multiplication table has to be replicated in hardware, whereas in the binary system only 0 · 0, 0 · 1, 1 · 0 and 1 · 1 needs to be built). The well-known ENIAC computer, for example, worked with the decimal system, while Konrad Zuse's Z3 already worked with the binary system.

Exercise 2

Research possibilities and limitations of digital representation of information. In particular, research how natural numbers, decimal numbers and characters are stored in a computer.

Solution Solution

Natural numbers are stored as binary numbers. This means that a number $x$ requires $[\log_{2} x]$ bits.

A decimal number could theoretically also be stored as a binary number, whereby the digits after the decimal point could have the values $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ , …, $\frac{1}{2^{n}}$ in binary format.

However, this leads to unexpected problems because, for example, the number 0.3 would have to be represented as an infinite periodic binary number, such as $0, 010011 \bar{0011}$ , which would require an infinite number of binary digits for the decimals.

For this reason, a rounded version of the binary number is used. The IEEE 754 standard, which is generally used for binary numbers, therefore defines a decimal number in binary format as consisting of three components: one bit for a sign, 23 bits for the mantissa (the most accurate representation of the number with exactly one digit before the decimal point) and eight bits for the exponent (as a positive or negative number), which is used to raise $2^{n}$ to a power.

Characters are stored using a character table. This assigns a character to a binary number. The following table is the ASCII table, which is commonly used for the first 128 characters today (the first 32 characters are control characters, which have been left blank here):

Bits 1–3 →	010	011	100	101	110	111
↓ Bits 4–7 0000	␣	0	@	P	‘	p
0001	!	1	A	Q	a	q
0010	"	2	B	R	b	r
0011	#	3	C	S	c	s
0100	$	4	D	T	d	t
0101	%	5	E	U	e	u
0110	&	6	F	V	f	v
0111	'	7	G	W	g	w
1000	(	8	H	X	h	x
1001	)	9	I	Y	i	y
1010	*	:	J	Z	j	z
1011	+	;	K	[	k	{
1100	,	<	L	\	l	\|
1101	-	=	M	]	m	}
1110	.	>	N	^	n	~
1111	/	?	O	_	o

2. Qubits and how they may be represented

From bits to qubits

Unlike a bit (short for binary digit), which can only be 0 or 1, a qubit (short for quantum bit) is a quantum system which can be in two basis states or any combination of these two states. A combination of the two basis states is also called a superposition. So, a qubit can be in any of an infinite number of possible states. When a measurement is performed on the qubit, the result will be one of the basis states. Only probabilities can be given as to whether the qubit will be measured to be in the one or the other basis state.

A qubit can be created by different methods. A qubit could be, for example, a photon with two polarisation states (for example left or right circularly polarised), or an electron with its spin (spin up or spin down). Although the actual implementations differ, any of these implementations can represent the same abstract concept of a qubit.

Vector and bra-ket notation of a qubit

A qubit may be expressed by using the bra-ket notation (introduced by Paul A. M. Dirac, 1902–1984, see Mathematical Introduction: Matrices and how they can be used in quantum computing). Usually, the basis states of the qubit are written as $| 0 ⟩$ and $| 1 ⟩$ .

Instead of using the bra-ket notation, a qubit can also be represented as a vector of magnitude 1: $| 0 ⟩ = [\begin{matrix} 1 \\ 0 \end{matrix}]$ and $| 1 ⟩ = [\begin{matrix} 0 \\ 1 \end{matrix}]$ . The bra-ket notation and the vector representation are equivalent.

The bra-ket and the vector notation can also be used to represent multiple qubits. For instance, for two qubits $| 0 ⟩$ and $| 1 ⟩$ , the possible basis states are:

$| 00 ⟩ = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], | 01 ⟩ = [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}], | 10 ⟩ = [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}], | 11 ⟩ = [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}]$

Number of qubits	Bra-ket notation	Vector notation
1 qubit	$\| 0 ⟩$	$[\begin{matrix} 1 \\ 0 \end{matrix}]$
1 qubit	$\| 1 ⟩$	$[\begin{matrix} 0 \\ 1 \end{matrix}]$
2 qubits	$\| 00 ⟩$	$[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$
	$\| 01 ⟩$	$[\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}]$
	$\| 10 ⟩$	$[\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}]$
	$\| 11 ⟩$	$[\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}]$

Table 1: Basis states of systems with one or two qubits in bra-ket and vector notation.

Go to Part 2 Quantum Gates and Quantum Circuits

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