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Introduction

The Open University — Wed, 07 May 2025 14:40:31 GMT

Quantum computing is a rapidly evolving field at the intersection of physics, mathematics, and computer science. Unlike classical computing, which relies on bits as units of information, quantum computing leverages the principles of quantum mechanics – such as superposition and entanglement – to solve certain computational problems more efficiently.

This course begins with an introduction to the basics of ‘classical’ computing, providing a foundation of logic gates and information processing before moving on to quantum computing. You will explore the fundamental principles of quantum computing and be introduced to the qubit (quantum-bit, pronounced kew-bit), the quantum analogue of the classical bit.

You will examine quantum computing processes, from input to output, including single-qubit gates and two-qubit gates. By the end of this section, you will be able to read a quantum circuit, predicting how a series of gates transforms input qubits into output states. Additionally, you will engage in activities, such as designing your own quantum circuit to achieve a specific computational outcome.

To ensure you have the necessary technical background, the course includes dedicated sections on relevant quantum physics and mathematics topics. Whether these sections serve as a review or introduce new concepts, they provide the foundation for grasping quantum computing. Essential concept to pay close attention to are quantum superposition and quantum entanglement, two of the most fascinating and fundamental phenomena in quantum mechanics.

Finally, the course concludes with an overview of the technologies driving real-world quantum computing, exploring how researchers and companies are working to make quantum computers a reality.

By the end of this course, you will have a grasp of quantum computing fundamentals, its computational advantages, and the technological advancements shaping its future.

This OpenLearn course is an adapted extract from the Open University course SM380 Quantum physics: fundamentals and applications.

Learning outcomes

The Open University — Wed, 07 May 2025 14:40:31 GMT

After studying this course, you should be able to:

describe a qubit and understand how it differs from a bit in classical computing
explain how a two-qubit CNOT gate can generate entanglement between two qubits
derive the output qubits of a quantum circuit given the input qubits
describe different ways quantum computing is being implemented in practice.

1 Why quantum computing?

The Open University — Wed, 07 May 2025 14:40:31 GMT

Quantum computers have the potential to revolutionize science and technology by overcoming the limitations of classical computing. Classical computers are limited by processing speed and computational resources so complex problems quickly become impossible to solve. Quantum computers can solve some computational problems significantly faster. This enhanced processing capability enables quantum computing to tackle complex problems more efficiently and unlock new types of applications. In this section, you will explore how quantum computers may outperform classical computers and drive innovation in various fields.

1.1 What can classical computers do?

The Open University — Wed, 07 May 2025 14:40:31 GMT

In this course, the term classical computers is used to refer to computers which use bits to encode information and carry out computations; i.e. they are binary based.

The complexity of tasks that any computer can complete is limited by the available time and computing resources. There are two routes to increasing the power of classical computing: one is to improve the hardware, which means increasing the size of the memory, the number of gates on a processor chip, or improving the speed of those elements. The other approach seeks to improve the software, i.e. the algorithms.

Figure 1 A classical computer

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A picture of a laptop on a desk. There is a plant to the right of the laptop and a lamp positioned behind. In the back ground is a bookcase.

Figure 1 A classical computer

Thinking about the hardware, the improvement in computing power over the past 50 years has been enormous. The number of transistors on a microprocessor chip has doubled approximately every two years, a fact known as Moore’s law. Unfortunately, this trend cannot continue indefinitely, because one of the main ways that these improvements are realised is by shrinking the size of the circuit elements.

Improving the algorithms is the second possibility. The power of an algorithm can be expressed by stating how the ‘run-time’ which is the number of steps required to implement the algorithm, scales with the size of the task. Some algorithms have a polynomial run-time meaning that if a procedure is to be carried out on a number, n, of elements, the run-time scales as n^x where x is typically a small integer. Other algorithms have an exponential run-time, scaling as eⁿ.

The algorithms with polynomial run-time are considered to be much more useful than algorithms with exponential run-time, because exponential scaling means that only modest increases in the task size can exhaust available resources.

Throughout this course there are a series of exercises for you to work through. Some of these exercises you can supply your answer to in the response boxes provided. Others will require you to work through your calculations on paper.

Exercise 1

Consider some algorithms that are carried out on two data sets – one with n = 10 elements and another with n = 100 elements. The run-time of one algorithm scales as n³ and the run-time of another algorithm scales as eⁿ. How do the run-times of the two algorithms compare for the smaller data set? How do the run-times compare for the larger data set?

Answer

For the smaller data set, the run-times for the two algorithms are in the ratio , whereas for the larger data set, the run-times for the two algorithms are in the ratio . The time to carry out the algorithm with the exponential run-time soon becomes unfeasibly long as the size of the data set increases.

1.2 What can quantum computers do?

The Open University — Wed, 07 May 2025 14:40:31 GMT

Quantum computing offers a new computing approach that is based on the idea that the rules of quantum mechanics can allow shortcuts to solutions for certain tasks. There are quantum algorithms with polynomial run-times that can solve problems where the only known classical algorithms have exponential run-times. Quantum computers make use of the fact that quantum states can be made of linear combinations of individual states and when a measurement is taken, one of the individual states will be measured with a given probability. You will see that entanglement is another fundamental resource for quantum computing.

The integer factorisation problem, which seeks to find the prime number factors of an n-digit integer, is of particular interest because it is the basis of a lot of information security. The run-time of the classical algorithm scales exponentially so as n increases a classical computer takes longer and longer to factorise the integers as n increases. The integer factorisation problem is an example of a a problem that quantum computers can solve much more efficiently. If quantum computing was able to solve the integer factorisation problem with a polynomial run-time then there would be a major problem for information security.

Another example of a problem with an exponential run-time is the travelling salesperson problem, which seeks to find the shortest route between a number of cities subject to visiting each city only once and returning to the starting city at the end of the journey (see Figure 2). This is a typical optimisation problem which can be used to demonstrate the power of quantum computing as well as being a problem which delivery companies would like to be able to solve quickly.

Figure 2 An illustration of the travelling salesperson problem. The image on the left shows 7 cities (in blue) and a trial of all 360 possible routes between them (in red). The image on the right shows the optimum distance after trialling all possible routes.

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The image on the left shows shows 7 blue dots in a grid with red lines connecting them. The red lines change position as different connections are trialled. The image on the right shows the same 7 blue dots with a green line connecting the shortest possible distance between them.

Figure 2 An illustration of the travelling salesperson problem. The image on the left shows 7 cities (in blue) and a trial of all 360 ...

Exercise 2

Write a sentence or two to summarise in general terms the context in which quantum computers are considered to be an improvement on classical computers.

Interactive feature not available in single page view (see it in standard view).

Answer

Your answer will not be the same but a possible answer is:

Quantum computers may be able solve some problems more quickly than classical computers if problem solving algorithms which have exponential run-times on a classical computer can be written to have polynomial run-times on a quantum computer.

Your answer should include the same conclusions.

2 Background mathematics and terminology

The Open University — Wed, 07 May 2025 14:40:31 GMT

In this section, some mathematics and terminology which will be needed to learn about quantum computing are introduced. This includes matrices, eigenvalue equations and complex numbers.

2.1 Matrix multiplication

The Open University — Wed, 07 May 2025 14:40:31 GMT

A matrix, , is a rectangular array of numbers arranged in rows and columns. (Matrices is the plural of the word matrix.) You will concentrate on two-dimensional situations, and so consider matrices of the form:

In order to multiply two matrices, the number of columns of first matrix should be equal to the number of rows in the second matrix. Matrices of the right shape can be multiplied together as follows:

Equation label: (1)

To calculate , where is the element in the i th row and j th column of , you go along the th row of and down the th column of , multiplying corresponding elements and adding the results. For two matrices, this pattern may be visualized as follows:

where the indicates a matrix element in the new matrix, , and the arrows show how matrix elements in the old matrices are processed to obtain this. In order to multiply two matrices, the number of columns of first matrix should be equal to the number of rows in the second matrix. In other words, each term is given by .

Exercise 3

a.What shape is the result of multiplying a row matrix by a square matrix?
b.What shape is the result of multiplying a square matrix by a row matrix?
c.What shape is the result of multiplying a square matrix by a column matrix?
d.What shape is the result of multiplying a row matrix by a column matrix?

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Answer

a.The result is a row matrix.
b.This operation cannot be performed because the number of columns of first matrix is not equal to the number of rows in the second matrix.
c.The result is a column matrix.
d.The result is a matrix (i.e. a scalar number).

Exercise 4

Evaluate the following combination of square matrices:

Answer

The product of two matrices is given by:

So, to find the matrix element in row and column of the product , we multiply the elements in row of with the corresponding elements in column of , and add the results together.

To find the product of three matrices, , we first evaluate and then form the product , taking care to preserve the order of the matrices.

So the solution is

2.2 Finding the eigenvalues and eigenvectors of a two multiplication two matrix

The Open University — Wed, 07 May 2025 14:40:31 GMT

For a given square matrix, , it is possible to solve the equation

Equation label: (2)

where are column vectors known as eigenvectors and is a scalar called an eigenvalue.

A procedure to find the eigenvectors and eigenvalues of a square matrix is

Solve the quadratic equation to find the two values of which are the required eigenvalues.
For each eigenvalue found, write down the eigenvector equations
This pair of equations usually reduces to a single equation that is readily solved for and . The eigenvector is given by with and replaced by their solved values.
It is often useful to normalise by writing it as a unit vector. It this case, the unit vector is given by

Exercise 5

Find the eigenvalues and eigenvectors of the following matrix:

Answer

Following the prescription described above: , , and . So we first need to solve the quadratic equation

which is simply

This can be written as

So it has solutions and . These are the two eigenvalues.

We now write the two eigenvector equations:

For eigenvalue these reduce to

Both equations imply that , so and and the first eigenvector is

For eigenvalue these reduce to

Both equations imply that , so and and the second eigenvector is

2.3 Complex numbers

The Open University — Wed, 07 May 2025 14:40:31 GMT

A complex number may be written in the form:

Equation label: (3)

where and are real numbers and i is a special quantity with the property that .

Each complex number has a real part, Re, and an imaginary part, Im.

Complex numbers can be added,

and multiplied,

using the usual rules of algebra along with .

The complex conjugate of is (pronounced "z star").

This results in showing that is a positive real number, (unless ).

The modulus of the complex number is defined as

Equation label: (4)

which is a real, non-negative quantity.

Complex numbers can also be written in polar form,

Equation label: (5)

where and are real numbers. The relationship between and , and is shown in Figure 3. Here is the modulus of as defined in Equation (4); is known as the phase, and is a phase factor.

Figure 3 A diagram showing the relationship between Cartesian coordinates and , and polar coordinates and

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The fighure shows a graph with two axes, labelled x-axis and y-axis, which intersect near the middle of the diagram, marked 0. An arrow is drawn from the origin at 0 into the upper right quadrant of the graph. The length of the arrow is labelled r and the angle it makes with the horizontal x-axis is labelled theta. A horizontal dashed line from the end of this arrow intersects the y-axis at a point marked y. A vertical dashed line from the end of this arrow intersects the x-axis at a point marked x.

Figure 3 A diagram showing the relationship between Cartesian coordinates and , and polar coordinates and

Exercise 6

Consider the complex number .

a.Write down its complex conjugate .
b.Calculate the modulus of .
c.Write in polar form.

Answer

a.The complex conjugate is .
b.The modulus of is
c.In polar form where and , so radians. Therefore we can write .

2.4 Operators and superposition

The Open University — Wed, 07 May 2025 14:40:31 GMT

In quantum mechanics, an operator is a mathematical entity which converts one function into another function and is written with a ‘hat’on, for example , (pronounced ‘A hat’). Given an operator , the eigenvalue equation for that operator is

Equation label: (6)

Here, is a constant known as an eigenvalue, which may be complex, and is function known as an eigenfunction. There may be more than one eigenvalue and corresponding eigenfunction associated with each eigenvalue equation. The eigenvalue matrix equation, Equation (2) as described in Section 2.2 is an example of this type of equation with the operator written as a matrix and the eigenfunction as a column vector.

Consider an operator with two eigenvectors and two corresponding eigenvalues so that

Since and are both eigenfunctions or solutions of the eigenvalue equation any linear combination of and is also a solution. Such a linear combination, known as a superposition, is

where and are complex numbers.

Exercise 7

The wave function of a free particle in quantum mechanics may be written as

Confirm that is an eigenfunction of (where indicates a partial derivative) and show that the eigenvalue is the energy of the free particle, . ( is the reduced Planck’s constant.)

Answer

Operating on with we find that

Thus the free-particle wave function is an eigenfunction of and the corresponding eigenvalue is the energy, , associated with this wave.

3 Setting the scene in quantum physics

The Open University — Wed, 07 May 2025 14:40:31 GMT

In this section, some quantum physics needed to study quantum computing is introduced. You will learn about spin-½ particles, which have two fundamental basis states but can also exist in a superposition of these states. This concept is central to quantum mechanics and directly relates to quantum computing, where quantum bits (qubits) similarly have two basis states and can exist in any superpositions of these two states. In the context of spin-½ particles the two states are called spin-up and spin-down; in quantum computing the two states are referred to as logical states. At the end of this section, the essential concept for quantum computing of entanglement is illustrated using spin-states.

3.1 Spin-½ particles

The Open University — Wed, 07 May 2025 14:40:31 GMT

Experiments show that electrons have an intrinsic property which is called spin. (Mass and charge are other examples of intrinsic properties of particles.) Spin is a type of angular momentum with a quantum number of ½ which means that a measurement of spin along an axis can only have values of or as an outcome. (Here where h is Planck’s constant, 6.626 × 10^-34 J s.) Thus, electrons are referred to as spin-½ particles.

The most important spin operators are the component of spin angular momentum in the z-direction, and the total spin angular momentum, . The eigenvalue equations for these operators are

where the angled bracket, is used and is called a ket. represents the eigenfunction, known as an eigenstate with a spin quantum number of and a spin magnetic quantum number of . For a single electron and . The state with is the spin-up state and the state with is the spin-down state.

3.2 Representing a general spin state

The Open University — Wed, 07 May 2025 14:40:31 GMT

A ket can also be considered as a vector. The spin-up state is often represented by the symbol and the spin-down state by the symbol . The spin-up state obeys

Equation label: (7)

and the spin-down state obeys

Equation label: (8)

The spin vectors are normalised and orthogonal to one another, as represented by the following relations:

where these equations use additional notation. is known as a bra and the combination of the bra and ket together is an inner product, .

A general spin state can be written as a linear combination of and , thus

Equation label: (9)

where and are complex numbers referred to as probability amplitudes.

In other words, and provide an orthonormal basis for spin space. The vectors and are called basis vectors. Because any spin state can be written as a linear combination of just two basis vectors, the spin space of a spin-½ particle is two-dimensional.

For an atom in any spin state, , as given in Equation 9 the probability of the outcome of a measurement indicating spin-up is and for spin-down is . Since these are the only possible outcomes, the corresponding probabilities must sum to one, therefore

Matrices can be used as an alternative representation of spin states to simplify calculations. and are represented by the following column vectors:

These matrices have two elements because spin space is two-dimensional. Spin states that do not have definite values of are expressed as linear combinations of and .

Any vector in spin space may be written as a linear combination of and . This means that , as defined in Equation 9 becomes:

In this way any spin state of a spin-½ particle can be represented as a two-element matrix, which is called a spinor.

The inner product in spin space of two vectors in matrix form is written

which is consistent with the matrix multiplication:

and thus the inner product of two spin vectors can be written in the matrix form:

We can therefore identify the separate bra and ket vectors as follows:

A ket spin vector is represented by a column spinor, and a bra spin vector is represented by a row spinor. To convert a column spinor into the corresponding row spinor, the rule is to turn the column into a row and take the complex conjugate of all elements. So

When using a matrix representation, the spin operators are also represented as matrices, for example,

Rewriting Equation 7 using matrices gives,

where you can see that carrying out the matrix multiplication gives the expected result.

The operators for a spin-½ particle are each represented by matrices. Along the three axes these are:

It is common to define the so called Pauli operators, , as such that we have the following Pauli operator matrices:

These will be useful in the context of quantum computing where they can be used to represent the action of quantum gates.

Exercise 8

Show that the spin vectors

are normalized and orthogonal, i.e. and .

Answer

Using matrix multiplication

The first two equations show that the vectors and are normalised; the third that they are orthogonal to each other. Hence, they are orthonormal.

3.3 Spin observables

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In quantum mechanics measurable quantities are called observables. Spin is an example of an observable because it can be measured in an experiment. (Position and orbital angular momentum are other examples of observables.) Each observable is associated with an operator and, in general, the only possible outcomes of a measurement of an observable are any of the eigenvalues.

When a measurement is performed on a quantum system with spin, the wavefunction collapses into one of the eigenstates of the observable being measured. For example, if we measure the spin of an electron along the z-axis, the quantum state collapses into one of the two basis states: or with probabilities determined by the initial state before measurement (see Equation 9). This collapse means and any superposition that existed before the measurement is lost.

After the measurement, the electron will be in a new well-defined spin state, either spin-up or spin-down depending on the result of the measurement. The general spin state has collapsed into one of the eigenstates due to being measured. As long as the initial general spin state is not an eigenstate, the spin state after the measurement will be different from the spin state before the measurement.

Exercise 9

Particles are prepared in the spin state

a.If a single particle is prepared in the state , what prediction can be made about the result of measuring for this particle?
b.If a million particles are prepared identically, all in the state , what prediction can be made about the results of measuring for this collection of particles?

Answer

a.No definite prediction can be made for a single particle in the given state, but a measurement of will give either or ; see Equations 7 and 8. In given state and using Equation 9, and so the probability of getting is and the probability of getting is . As expected these two probabilities sum to unity because for any measurement either one or the other outcome will be obtained. This shows that the value is more likely, but the value would not be that surprising.
b.For a million particles, the expected outcome is that close to three-quarters or 750,000 measurements will give , and the remainder will give .

3.4 Two-particle spin states

The Open University — Wed, 07 May 2025 14:40:31 GMT

If we have two indistinguishable^{Footnote 1} electrons, we can define a two-particle spin state. Due to symmetry and the rules of quantum mechanical addition of angular momentum, there are four possible spin states in total. These spin states are represented using the quantum numbers and , as introduced in Section 3.1, but now the quantum numbers are the sum of the values for the individual electrons. Therefore the spin quantum number is or and, for the case when , the spin magnetic quantum is , while for the case when , we only have .

Such a two-particle spin state therefore can only have an overall spin function which is either symmetric or antisymmetric with respect to exchange of the electrons. The symmetric spin state is referred to as a triplet because there are three possible symmetric combinations:

where the first arrow in each ket refers to particle 1 and the second to particle 2. The antisymmetric spin state is referred to as a singlet because there is only one possible combination:

Equation label: (10)

You can see that the states and can be factorised into and respectively, where the subscripts label each particle. In contrast, the states and cannot be factorised into the product of a particle 1 state multiplied by a particle 2 state. Two-particle states which cannot be factorised are known as entangled states and said to exhibit entanglement.

Exercise 10

Verify that the three spin kets

are symmetric with respect to swapping the labels of the particles.

Answer

Starting with

exchanging the particle labels and then rearranging gives

Since this final expression is identical to the initial expression, this shows is symmetric to swapping particle labels.

For and the particle labels are interchanged and re-ordered (perfectly acceptable!) to get the same expressions as required.

3.5 Entanglement

The Open University — Wed, 07 May 2025 14:40:31 GMT

Entanglement is a consequence of quantum theory and leads to a correlation between the outcomes of measurements which cannot be explained by classical physics. Two-particle states which show such non-classical correlations are known as entangled states.

Consider the following thought experiment. Suppose you have a two-particle system in the spin state as given in Equation 10. Before the experiment you know that particle 1 can be either spin-up or spin-down with equal probability. However, if you measure particle 1 to be spin-up then you know that particle 2 is spin-down as the measurement means that the two-particle state has collapsed into the arrangement of spins. In contrast, if you measure particle 1 to be spin-down then you know that particle 2 is spin-up as the two-particle state has collapsed into the arrangement of spins. This type of prediction is quite puzzling because the two entangled particles can be as far apart as possible and when a measurement is made on particle 1 then it is known simultaneously what the outcome of a measurement on particle 2 will be.

Entanglement is essential for quantum computing. Entangled states are generated as part of the workings of a quantum computer, as you will see later.

Exercise 11

Confirm that the following states are normalised and determine whether the states are entangled.

Answer

The normalisation condition for a general two-particle state is that the sum of the squares of the probability amplitudes is equal to 1.

The states are entangled if the two-particle state cannot be factorised into the product of a particle 1 state multiplied by a particle 2 state.

a.Checking the normalisation of state :
showing state is normalised.
can also be factorised:
which is a particle 1 state multiplied by a particle 2 state so state is not entangled.
b.Checking the normalisation of state :
showing state is normalised.
State cannot be factorised and so is an entangled state.

4 Classical computing

The Open University — Wed, 07 May 2025 14:40:31 GMT

In this section, you will be introduced to a few aspects of classical computing to give you a reference frame for discussing quantum computing. All computers work by taking input information and processing it using gates to give output information. By the end of the section, you will be familiar with the classical NOT and CNOT gates including their truth tables, so that you can compare them with the non-classical output of the quantum versions of these gates described in Section 5.

4.1 Classical bits and logic gates

The Open University — Wed, 07 May 2025 14:40:31 GMT

In classical computing, the smallest piece of information is called a bit. A bit may take only one of two logical values: either 0 or 1. Strings of bits are used to represent information as numbers, which can be stored, copied and processed by the computer. The processing of information is accomplished by logic gates, which take strings of bits as their input and produce an output value for each bit that they act on. A diagram showing how a logic gate works is given in Figure 4.

Figure 4 A diagram to illustrate how a logic gate works

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The figure shows a schematic of a logic gate with a string of binary numbers as input and a different string of binary numbers as output.

Figure 4 A diagram to illustrate how a logic gate works

4.2 Classical Boolean gates

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A single-bit gate acts on one bit at a time. In classical computing, there are two universal single-bit gates, the NOT gate and the Reset gate which, either acting alone or in a sequence, can generate all possible transformations of a single bit.

The NOT gate (Figure 5) simply flips the value of the bit to the alternative value, so a 0 becomes a 1, and a 1 becomes a 0.

Figure 5 The symbol for a NOT gate

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The figure shows a horizontal line labelled as input leading to the base of a right-pointing equilateral triangle. At the right hand tip of the triangle is a small circle, followed by another horizontal line labelled as output.

Figure 5 The symbol for a NOT gate

A truth table is a convenient way of summarising the action of a gate. The truth table for a NOT gate is given in Table 1.

**Table 1** Classical NOT gate truth table
Input	Output
0	1
1	0

The Reset gate sets a bit to value 0, regardless of the input state.

Single-bit gates are not sufficient to perform computing: it is also necessary have conditional gates in which an operation on a target bit depends on the state of one or more control bits. It will be helpful to write the two input bits as an ordered pair of values, CT (for example, 01 has C = 0 and T = 1), where C represents the control bit and T the target bit.

An important two-bit gate is the CNOT gate (controlled NOT gate), which performs a NOT operation on the target bit conditional on the state of the control bit being 1; if the control bit is 0, then no operation is applied. The state of the control bit is unchanged by the CNOT operation. Table 2 is the truth table for the CNOT gate.

**Table 2** Classical CNOT gate truth table
Input	Output
0 0	0 0
0 1	0 1
1 0	1 1
1 1	1 0

To perform classical computing, the bits are set to some initial values and the gates are applied to the bits in an ordered sequence to make an algorithm.

Quantum computing has different sets of universal gates, which include quantum versions of the NOT and CNOT gates.

5 Qubits and quantum gates

The Open University — Wed, 07 May 2025 14:40:31 GMT

Quantum computing is based on units of information called qubits (quantum bits, and pronounced kew-bits), which obey the laws of quantum mechanics. A qubit is the quantum analogue of a classical bit. The classical bit values 0 and 1 are replaced by the orthonormal basis states of the quantum-mechanical qubit and . The basis states are given the name logical states, since they correspond to the classical bits upon which the logic gates operate. The key difference between qubits and classical bits is that qubits can exist in a superposition of the and states, which means qubits can be prepared in the superposition state:

Remarkably, you can take logic gates similar to the Boolean logic gates of classical computing and apply them to the qubits. In doing so, the input state of the qubits is transformed into the output state. To obtain the result of the computation, the value (0 or 1) of each qubit is measured. As you will see, the resulting outputs can include entangled states of two or more qubits.

Quantum entanglement is a fundamental resource for quantum computing as it involves the distribution of information in a fundamentally non-classical way.

In this section, you will learn the definition of a qubit and be introduced to some single-qubit and two-qubit logic gates. The quantum CNOT gate is an important gate as it can entangle and disentangle a pair of qubits. By the end of this section you will have been introduced to quantum circuits and there is a final activity to test your understanding.

5.1 Defining a qubit

The Open University — Wed, 07 May 2025 14:40:31 GMT

A qubit is defined by the equation

Equation label: (11)

where is a two-state quantum system and and are logical states. Note that has a logical value of 0 and has a logical value of 1. and are the probability amplitudes which may be complex numbers and satisfy the normalisation condition .

Examples of qubits are the general spin states of spin-½ particles as described in Section 3.2. You can see that the spin-up state has been replaced by logical state and the spin-down state by logical state and you can see that Equation 9 and Equation 11 have a similiar form.

To fully specify the two complex amplitudes or four real numbers are required: the real and imaginary parts of each. However, the number of values can be reduced by two, one because of the normalisation condition, and one because the phase of one basis state can be set to zero without changing anything.

This leads to the equation,

where and are real numbers with and .

A qubit state can therefore also be represented as a column vector

This representation is like the spinor representation introduced in Section 3.2. The column vector representation is useful because the operators corresponding to single-qubit gates and observables may be written as 2 × 2 matrices. The qubit basis states are defined as

Equation label: (12)

The column vectors of Equations 12 are eigenstates of a 2 × 2 matrix operator known as the Pauli-Z operator or , which as mentioned earlier is defined as

Exercise 12

Show that the qubit basis states and are eigenvectors of the Pauli-Z operator and find the corresponding eigenvalues. Determine the relationship between the eigenvalues and the logical values of the basis states. Use the symbol, m to represent the logical value.

Answer

Noting and has logical value, m = 0; and and has logical value, m = 1 and that

First, for basis state, the eigenvalue equation (Equation 2) becomes

Doing the matrix multiplication gives

showing that

Similarly, for basis state

showing that

Comparing the eigenvalues with the logical values: when , m = 0, and when , m = 1. The relationship between the eigenvalue and the logical value is therefore

5.2 Single qubit gates

The Open University — Wed, 07 May 2025 14:40:31 GMT

In quantum computing, a gate is a reversible transformation of a qubit state to another qubit state , represented by an operator . It is convenient to write the single-qubit gate operators as 2 × 2 matrices. Therefore, the action of the gate can be written as follows:

The word reversible is important because it is a reminder that a gate operation is of a different nature from a measurement. The operation of the gate can be reversed so that it is possible to get back to the state , whereas, in general, a measurement makes an irreversible change to the qubit state. is defined as the operator needed to reverse the gate action and transform back to :

therefore

which means that

Equation label: (13)

where is the identity operator, represented by the 2 × 2 matrix appearing in Equation 13. An operator, that obeys Equation 13 is called a unitary operator, hence gates are represented by unitary operators. The identity operator is itself a gate, denoted , and its symbol is shown in Figure ‘6.

Figure 6 The symbol used in a quantum circuit for an identity gate,

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The figure shows a horizontal line on the left leading to a square box with the capital letter I in it, then a horizontal line leading from the box to the right.

Figure 6 The symbol used in a quantum circuit for an identity gate,

You will now look at some gates, starting with the quantum NOT gate. (From now on the prefix quantum will be omitted as long as it is obvious the gates are quantum gates and not classical gates from the context.)

5.2.1 The NOT gate

The Open University — Wed, 07 May 2025 14:40:31 GMT

The NOT gate is denoted by the operator, and, in the basis of the logical qubits and , is represented by the matrix

which is also the Pauli-X operator, , mentioned earlier. In a circuit diagram representing a quantum algorithm, the symbol for the NOT gate is shown in Figure 7.

Figure 7 The symbol used in a quantum circuit for a NOT gate,

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The figure shows a horizontal line on the left leading to a circle with a large plus sign in it, then a horizontal line leading from the box to the right.

Figure 7 The symbol used in a quantum circuit for a NOT gate,

In quantum computing, the truth table specifies outcomes for the logical states and . In Table 3 the general superposition state is also included for reference.

**Table 3** Quantum NOT gate truth table
Input	Output

5.2.2 The Hadamard gate

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The Hadamard gate is a single-qubit gate defined by a matrix

In a circuit diagram representing a quantum algorithm, the symbol for the Hadamard gate is shown in Figure 8.

Figure 8 The symbol used in a quantum circuit for a Hadamard gate,

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The figure shows a horizontal line on the left leading to a square box with a capital letter H in it, then a horizontal line leading from the box to the right.

Figure 8 The symbol used in a quantum circuit for a Hadamard gate,

Consider the action of a Hadamard gate on logical state .

First, the gate and logical state are written as matrices,

Next, the matrices are multiplied to obtain the final state matrix

Finally, the final state matrix is rewritten in terms of the logical states and :

You can see that the final output state, is a superposition state. This calculation shows that a Hadamard gate allows the transformation of the logical qubit state into a superposition state.

Exercise 13

Use matrices to work out the action of a Hadamard gate on logical state .

Answer

Noting, logical state and the Hadamard gate is Using matrices gives

You can see that the Hadamard gate has transformed logical state into a superposition state.

The effect on a general state can be determined by combining the results of the action of a Hadamard gate on logical states and .

All these results are summarised in the truth table for the Hadamard gate in Table 4.

**Table 4** Hadamard gate truth table
Input	Output

5.2.3 Sequences of gates

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The next step towards quantum computing is to combine gates sequentially to give a quantum circuit. In a circuit, successive operations are applied to a single qubit.

If you first apply the NOT gate, and then apply the Hadamard gate, the mathematical expression representing this is written as a sequence of operators, operating from right to left on the qubit :

Equation label: (14)

This ordering of operators should not come as a surprise; reading the expression on the right hand side of Equation 14, first gate is applied to qubit to give an intermediate qubit, say , and then gate is applied to to give the final resultant . Alternatively, if calculating the outcome of on qubit , the matrices representing and can be multiplied together to give a resultant matrix, which can be considered a new operator, . Then operator, can be thought to act on to give the resultant .

The diagram representing this sequence is shown in Figure 9.

Figure 9 A circuit for applying a NOT gate and a Hadamard gate in sequence

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The figure shows a horizontal line on the left leading to a circle with a large plus sign in it, then a further horizontal line from the right of the circle leading to a square box with a capital letter H in it, then a horizontal line leading from the box to the right.

Figure 9 A circuit for applying a NOT gate and a Hadamard gate in sequence

A circuit diagram represents the logical flow of the circuit from the left (the initial state) to the right (the final state) of the diagram. Therefore, in the circuit shown in Figure 8, the elements are ordered from left to right: the NOT, which acts first, is on the left.

Note that the ordering of gates in the circuit diagram is the opposite to the ordering of gates when the circuit is written as a sequence of operators acting on a ket.

By matrix multiplication, any sequence of single-qubit gates can be represented by a single 2 × 2 matrix found by forming an ordered product of the matrices representing the gates. You have to be careful because, in general, the single-qubit operators do not commute. In the next exercise you will see that the method using matrix multiplication to combine gates is equivalent to applying the gates sequentially.

Exercise 14

Given the initial qubit state , show that the sequence of gates in Equation 14 produces the final state

Show that you get the same result by applying either the gates in sequence or by using matrix multiplication to combine gates.

Answer

Applying the gates in sequence from right to left:

Using matrix multiplication to combine gates: First, calculate the product of the matrices :

then apply this matrix to the column vector for :

which is the same final state as applying the gates in sequence, as expected.

5.3 Two-qubit gates

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The gates discussed in Section 5.2 are examples of single-qubit gates, but these are insufficient for quantum computing. Just as the classical computing must include conditional gates that act on two bits, so a working quantum computer needs both single-qubit and two-qubit gates. A quantum CNOT gate is introduced and you will see that it is able to produce entangled states of two qubits.

5.3.1 Two-qubit states

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A straightforward way to define the two-qubit states is to build the two-qubit basis states from product states:

There are four possible product states of the usual single-qubit basis states:

A general two-qubit state, therefore can be expressed in terms the product states as

where is normalised in the usual way:

Equation label: (15)

To specify the state of the two-qubit system, six real numbers must be given. Counting them in the same way as for a single qubit, these are the eight numbers specifying the real and imaginary parts of each complex number (where the indices are mn = 00, 01, 10, 11). Eight is reduced by one because of the normalisation condition, given in Equation 15 and by one more, to six because the phase of one of the basis states can be set to zero without changing anything physical.

In general, an n-qubit system requires real numbers to specify the state, which is an exponential scaling in the number of qubits.

If the two-qubit gate is an operation, an equation can be written which represents the transformation from a two-qubit state into a new two-qubit state :

Multiple qubits are represented using the tensor product, which combines their states into a larger system. The resulting matrices represent the full system and can be used to calculate outputs by multiplying them with quantum state vectors. For instance the example above can be written as

This representation shows that a two-qubit gate can be expressed as a 4 × 4 matrix with elements , which act on a 4 × 1 column vector representing the quantum state. This allows you to compute the output state using matrix multiplication. However, we won’t be using this formalism in our discussions, as we will focus on other intuitive approaches to understanding multi-qubit systems and gates.

5.3.2 How the CNOT gate works

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In the same way as a classical CNOT gate, described in Section 4.2, acts on two bits, a control bit and a target bit; the quantum CNOT gate also acts on two qubits, a control qubit and a target qubit.

The CNOT gate, represented by an operator , acts on a target qubit depending on the state of a control qubit . Note that the single ‘hat’ over the operator tells you that this is one operator in contrast to the sequential operators, e.g. as in Equation 14, which are two operators.

In the following, the two-qubit state is assumed to be the product .

The quantum CNOT gate follows the same rules as the classical CNOT gate: if the state of the control qubit is , then it leaves the target qubit unchanged. If the state of the control qubit is , then it applies the NOT gate to the target qubit. Thus the CNOT gate would act on the state as follows:

and on the state as follows:

The transformations on the kets and can be worked out in the same way. The results of these transformations are collected in the truth table in Table 6 and are identical to the classical rules given in Table 2.

The quantum CNOT gate, however, can also act on superposition states, which is completely beyond the capabilities of the classical CNOT gate. So now consider how the CNOT gate transforms superposition states, starting from the situation where the control state is prepared in the superposition state

Equation label: (16)

and the target qubit is in the state . First, here is the initial two-qubit state:

Then applying the CNOT operator gives:

You can see that the final state is an entangled state because it cannot be factorised. If the control is in the superposition state orthogonal to the state described in Equation 16, i.e. , then the negative sign simply propagates so that:

The quantum CNOT gate is depicted graphically in Figure 10 and the full CNOT truth table including the quantum-mechanical results is given in Table 6.

Figure 10 The symbol for the quantum CNOT gate, with the control (C) and target (T) qubits labelled

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The figure shows two parallel horiontal lines. The upper one is labelled C and the lower one is labelled T. In the middle of the upper line is a large dot. A vertical line leads from this dot to the lower line where it joins a large circle with a large plus sign in it.

Figure 10 The symbol for the quantum CNOT gate, with the control (C) and target (T) qubits labelled

**Table 6** Truth table for the quantum CNOT gate
Input	Output

There are other useful states that can be generated using a CNOT gate; another is introduced in the next exercise.

Exercise 15

Find the two-qubit output state produced by the CNOT operation if the control qubit is prepared in the state , and the target qubit is prepared in the state . State whether the output state is entangled or not.

Answer

First, writing the input two-qubit state

Next, applying the operator:

The output state cannot be factorised so it is an entangled state.

To complete this section, consider the case when one of the entangled outputs from Table 6, is used as an input state.

The final state is now a product state (i.e. it is disentangled), where the control qubit is in the superposition state . Thus the CNOT gate can disentangle a pair of qubits as well as entangle them.

5.4 Quantum circuits

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In this section, you will make use of what you have already learnt about single-qubit and two-qubit gates to construct and interpret quantum circuits. A quantum circuit performs an algorithm in the sense that it executes a predefined sequence of quantum operations that embody an algorithm’s logic. However, unlike classical circuits, quantum circuits rely on principles like superposition and entanglement, and their output is often probabilistic rather than deterministic.

By the end of this section you will be able to predict the outcome of the application of a quantum circuit to a given input state. As well as combining a series of gates, measurements can also be made in quantum circuits when one of the eigenstates of the measurement operator will be obtained with a certain probability. Therefore, the prediction of an output from a quantum circuit may involve more than one possible outcome and the associated probabilities of these outcomes. At the end of this section there is an activity to test your understanding of quantum circuits.

5.4.1 Circuits with multiple gates

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Single-qubit gates and two-qubit gates can be combined in a structured sequence to give a quantum circuit containing multiple quantum gates. Consider as an example the circuit shown in Figure 11, which depicts a sequence of gates acting on two qubits, labelled and . Remember that each qubit enters its circuit from the left; first, a Hadamard gate is applied to qubit . Then a CNOT is applied to and , where the control C is and the target T is . Finally, a NOT gate is applied to .

Figure 11 An example of a quantum circuit with multiple quantum gates

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The figure shows two parallel horiontal lines. The upper one is labelled q1 and the lower one is labelled q2. In the left part of the upper line is a square box with a capital letter H in it. Further along the upper line is a large dot. A vertical line leads from this dot to the lower line where it joins a large circle with a large plus sign in it. to the right of this on the lower line is another large circle with a large plus sign in it.

Figure 11 An example of a quantum circuit with multiple quantum gates

The circuit in Figure 11 is a sequence of operations applied to qubits and can be analysed using the methods already introduced, with extra subscripts labelling the single-qubit gates and operations so that the qubit each operation is acting on is clear. Thus acts only on the qubit , leaving unchanged. Therefore, using as a sample input the calculation becomes:

Now acts only on , so

This means that

Next comes the effect of the CNOT gate, courtesy of and gives

Finally, observe that acts on only, so flip of each ket in the superposition:

5.4.2 Measurements

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Measurements are different from gate operations in a very important way, since rather than transforming a qubit from one definite state to another definite state , the final state after measurement is one of the two eigenstates of the measurement operator, which are obtained with some probability. Therefore, measurements are not reversible. The results of measurements are real numbers, so they can be stored as bits (rather than qubits) in a modern memory cell.

The circuit symbol for the measurement operation is shown in Figure 12.

Figure 12 The symbol for a measurement operation

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The figure shows a horizontal line. In the middle of the line is a square box. In the box is a curved line and an arrow that resemble a measuring meter.

Figure 12 The symbol for a measurement operation

The probabilistic nature of measurement is a feature of quantum computing that must be accounted for when evaluating the performance of a quantum algorithm.

Example

A circuit is set up as shown in Figure 13 and the input qubits are both . Calculate the output qubits and hence the possible results of the measurements and their probabilities.

Figure 13 A circuit incorporating both a single-qubit and a two-qubit gate

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The figure shows two parallel horiontal lines. The upper one is labelled q1 and the lower one is labelled q2. In the middle of the upper line is a large circle with a large plus sign in it. Further along the upper line is a square box with a curved line and an arrow in it. A vertical line leads from the large circle on the upper line to the lower line where it joins a large dot. To the left of this on the lower line a square box with a capital H in it; to the right of the large dot on the lower line is a square box with a curved line and an arrow in it.

Figure 13 A circuit incorporating both a single-qubit and a two-qubit gate

Answer

Writing the sequence of operations applied to the input qubits and using subscripts to label the qubits and the operations to show which qubit the gates are operating on, gives

acts on so

so now

Note that is the control qubit and is the target qubit. Consequently, when operates on , look at to decide whether is flipped. Again, adding subscripts to identify the qubits,

This is the final state which is measured. It is an entangled state. There are two possible outcomes; either is measured as and is measured as or is measured as and is measured as . From the coefficients, the conclusion is that each outcome has a probability of 1/2.