Quantum Field Theories, Topological Materials,

and Topological Quantum Computing

Muhammad Ilyas

A dissertation submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

Applied Physics

Dissertation Committee:

Marek Perkowski, Chair

Jack Straton, Co-Chair

Steven Bleiler

Peter Leung

Aslam Khalil

Portland State University

A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-Abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-Abelian anyons. Such topological phases of matter have a  ground state degeneracy. The fusion of two or more non-Abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-Abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this dissertation, we provide a comprehensive review of the fundamentals of logic design in topological quantum computing. The braid group and knot invariants in the skein theory are discussed. The physical insight behind the braiding is explained by the geometric phases and the gauge transformation. The mathematical models for the fusion and braiding are presented in terms of the category theory and the quantum deformation of the recoupling theory. The topological phases of matter are described by the topology of band structure. The wave function of quasiparticles in the quantum Hall effect and the theory of Majorana fermions in topological superconductors are also discussed. The dynamics of the charge-flux composites and their Hilbert space are expressed through the Chern-Simons theory and the two-dimensional topological quantum field theory. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. We reduced the quantum cost of the existing ternary quantum arithmetic gates and proposed that these gates can be implemented with the metaplectic anyons.

tocsectionTable of Contents
tocsectionList of Figures
tocsectionList of Tables

CHAPTER: INTRODUCTION

Two of the greatest revolutions of the twentieth century were the discovery of quantum mechanics and the invention of computers. At the end of the twentieth century, these two fields merged and a new field of quantum information was born. The quantum information science ends Moore's law, according to which the computing power would double every eighteen months. This law governed silicon chip-based computers, for which the density of chips can be increased. Such computers obey the laws of classical mechanics. But we cannot reduce the physical size of chips infinitely. At the atomic level, particles behave according to the laws of quantum mechanics rather than the laws of classical mechanics.

In 1982, Richard Feynman pointed out that there is a fundamental limit with the ability of classical computers to efficiently simulate a quantum system [1]. He showed that some problems can be solved exponentially faster on a quantum computer using exponentially large-sized Hilbert space than they could be solved on a classical computer. David Deutsch showed that classical computers cannot efficiently simulate a quantum computer [2]. Hence, a quantum computer is important for two reasons; it can perform faster, and it can answer questions about nature.

The building blocks of a classical computer are bits. These bits are based on classical logic that has values of either $0$ or $1$. Operations on these bits are performed by a series of gates. These gates change their values and answer the operations. The classical circuits are composed in space from gates connected by wires. But quantum computers are based on quantum logic, which has values in the superposition of $0$ and $1$. The quantum gates manipulate the quantum superposition and give outputs with some probability. A ternary quantum gate is a three-valued logic design, based on the superposition of $0$,$1$, and $2$.

Many methods of encryption on a classical computer are based on difficulty in finding the prime factors of a large number. Peter Shor <cit.> invented an algorithm to find the prime factors of a number on a quantum computer with an exponential speed up. This algorithm created widespread interest in quantum computers. Many other quantum algorithms have already been proposed. Grover's search algorithm for an unstructured search [3, 4] offers a quadratic speed up compared with a classical counterpart. These algorithms are implemented on a particular model of quantum computation.

Building a quantum computer is a great challenge due to its susceptibility to errors. The quantum superposition is destroyed due to its interaction with the environment. This process is called decoherence. Moreover, we cannot measure the state and look for errors. In doing so we would kill the superposition. Errors can also be in the phase of a state. There are quantum error correction codes [5, 6, 7, 8], but a quantum system needs to be completely isolated from the environment. In 1997, Alexei Kitaev proposed a model for the fault-tolerant quantum computation [9]. Information is encoded in some non-local degrees of freedom of particles, hence inaccessible to local perturbations [10, 11, 12]. This is done using the systems which are topological in nature.
The topology is a study of spaces that are continuously deformable to each other. Such spaces are called manifolds. A manifold is a space that is Euclidean flat space locally when a small patch is taken, but it has some non-Euclidean structure globally. To compare two spaces, some properties of the spaces are computed. These properties remain invariant under the continuous deformation of one space to the other. Such properties are called topological invariants. We will discuss topology in Chapter <ref>.

The topological nature of particles can be studied through their exchange statistics. Let $\psi(\bm{r}_i,\bm{r}_j)$ be the wave function of two particles at positions $\bm{r}_i$ and $\bm{r}_j$. In three dimensions, when two particles exchange their places, the wave function gets multiplied with a phase factor. That is,
ψ(r_i,r_j) = e^iθψ(r_j,r_i) ,
where the values $\theta = 0, 2\pi$ correspond to the exchange of bosons and $\theta = \pi$ corresponds to the exchange of fermions. The phase acquired by the wave function is $+1$ for bosons and $-1$ for fermions. Bosons are integral spin particles and obey Bose-Einstein statistics, whereas fermions are half-integer spin particles and obey Fermi-Dirac statistics. A double exchange of these particles is equivalent to no exchange. If the particles are distinguishable, then their statistics is described by the permutation group $\mathcal{S}_N$. A group is a mathematical structure to study symmetries. The permutation group is used to study the exchange symmetry. See Appendix <ref> for the definition of a group.

Jon Magne Leinaas in 1977  [13] suggested that in a two-dimensional space, another statistic may occur, called fractional statistics. For this new kind of statistics, $\theta$ has an arbitrary value between $0$ and $\pi$. Bosons and fermions do not obey fractional statistics even in two-dimensional space. Wilczek [14, 15] proposed a model for the realization of the fractional statistics. He also named these particles as anyons (neither boson nor fermion but any on) [15].

An anyon is not an elementary particle, but a collective phenomenon or a local disturbance in two-dimensional topological materials in a high magnetic field and at a very low temperature. A large number of elementary particles behave in a coordinated way to make quasiparticles. These particles can exist only inside a material, not in free space. Magnetic fluxes are attached to quasiparticles and make charge-flux composites. These quasiparticles obey fractional statistics. The quasiparticles will be discussed in Chapters <ref> and <ref>. The Chern-Simons gauge theory is used as an effective field theory to describe these materials. Quasiparticles have a topological charge which is a topological quantum number and is a generalization of the conventional charge. It is a topological invariant and changes on topological phase transition. We will discuss the topological invariants in Chapter <ref> and the topological phase transition in Chapter <ref>.

The anyons are quasiparticles in quantum Hall states [16, 17, 18] and as Majorana fermions in topological superconductors [19].
Anyons are detected in laboratory [20, 21, 22, 23, 24, 25], and more recently [23, 24]. The measurement of an anyon is done by interference as described in [26, 11].

The fundamental difference between 2D and 3D is the difference in the topology of spacetime. The motion of particles makes knots and links in spacetime. Two paths are topologically equivalent if one can be deformed to the other. In two dimensions, in general, we cannot transform one path to the other without cutting, as shown in Fig. <ref>. All smoothly deformable trajectories are in the same equivalence class. Fermions and bosons do not obey the fractional statistics even in 2D, but the change in the wave function of the system in two-dimensional topological materials, when two quasiparticles are exchanged, is independent of the distance and speed of exchange. In contrast, the evolution may depend on some global characteristic of the path. Therefore, the statistics of anyons are topological. Instead of the permutation group, the double exchange of anyons is not equal to the identity. The exchange statistics of anyons is described by a braid group in $(2+1)$-dimensional spacetime. The braid group is defined in Chapter <ref>. When the order of composition of two elements of a group does not matter then the group is Abelian, otherwise, it is non-Abelian.

[fill=green!20,thick] (-0.3,1.1) – (4.4,1.1) – (3.3,-0.2) – (-1.5,-0.2)– cycle;
[ultra thick,opacity=0.8,red] (0.5,0.5) ellipse (0.8 and 0.2);
[ultra thick,opacity=0.8,blue] (1.2,0.5) ellipse (1.5 and 0.4);
[blue] (0.5,0.5) circle (3pt);
[red] (2,0.5) circle (3pt);
at (1.5,0.3) $C_1$;
at (2.9,0.3) $C_2$;

[fill=green!20,thick] (-0.3,1.1) – (4.4,1.1) – (3.3,-0.2) – (-1.5,-0.2)– cycle;
[ultra thick,opacity=0.8,red] (0.5,0.5) ellipse (0.8 and 0.2);
[ultra thick,opacity=0.8,blue,rotate=45] (1.5,0.55) ellipse (1.4 and 0.3);
[blue] (0.5,0.5) circle (3pt);
[red] (2,0.5) circle (3pt);
at (1.5,0.3) $C_1$;
at (2,2) $C_2$;

[In general, two closed curves $C_1$ and $C_2$ are topologically distinct in two dimensions]Two closed paths $C_1$ and $C_2$ are topologically distinct in two dimensions, but they can be deformed to each other in three dimensions.

One of the properties of the topological phases of matter is the existence of ground state degeneracy. The degenerate ground states have a large energy gap to the excited states. The degeneracy depends on the topology of the two-dimensional system and the types of anyons present. The ground state is unique for trivial topology.
For Abelian anyons, the braid operators commute and the ground state is unique, but for non-Abelian anyons, the braiding corresponds to the evolution of the system in the degenerate ground state. The change of the system from one ground state to the other is studied using the Berry phase [27] as described in Chapter <ref>.
Let $g$ be degenerate states $\psi_a$ with $a=1,2,...,g$ of particles at positions $x_1,x_2,...,x_n$. Exchanging particles 1 and 2 may not just change the phase but may rotate it into a different state $\psi_b$. Braiding of 1 and 2 and that of 2 and 3 are given as
ψ_a →M_abψ_b,     ψ_a →N_ab ψ_b ,
where $M_{ab}$ and $N_{ab}$ are $g\cross g$ dimensional unitary matrices. For Abelian anyons, the $\theta$ in Eq. <ref> is arbitrary and clockwise and anticlockwise exchanges commute. Which means that even the clockwise and anticlockwise exchanges may not be the same, but if we exchange particles clockwise then anticlockwise, it will be the same as if we perform anticlockwise exchange first then clockwise. In contrast, $M_{ab}$ and $N_{ab}$ in Eq. <ref> do not commute in general, that is $M_{ab}N_{ab}- N_{ab}M_{ab} \ne 0$ and particles obey non-Abelian statistics.

Since the unitary evolution only depends on the topology of the path, wiggles of the path would not affect the outcome. No local perturbation can split the degeneracy, hence the system is decoherence-free. The topological nature of anyons is the source of the fault tolerance in a quantum computer.
A topological quantum computer is based on three steps; the creation of anyon-antianyon pairs from the vacuum, braiding, and fusion [28, 11]. Anyons can be combined by bringing them close to each other. This is called fusion. The fusion is an inverse of the creation of the particles. The fusion of an anyon with its antiparticle gives the total topological charge zero, but the fusion of an anyon with another different type of anyon or antianyon may give the third particle or a superposition of a collection of several particles. The resultant types of particles depend on the fusion rules. The topological charge of an anyon is assigned with respect to its fusion with other particles to get a vacuum.
There might not be a unique way to combine anyons. Different ways of the fusion of multiple anyons to get an outcome are called the fusion channels. These fusion channels provide the basis states of the Hilbert space for quantum gates. The dimension of the Hilbert space is equal to the degeneracy of the ground state. The transformation between different fusion channels is given by $F$-matrices.
The internal degrees of freedom of the anyons are changed by braiding and can put the system in another ground state. The phases acquired by anyons during the braiding are computed through R-matrices.
A quantum superposition of states can be created by a suitable combination of $F$ and $R$ matrices.

From the path integral point of view, the anyon's trajectories make knots whose invariants are the probability amplitudes from an initial to a final configuration of the system of anyons. The orientations of knots correspond to the direction of particle trajectories, and the twist in a ribbon knot corresponds to the topological spin. The topological spin is the phase due to the rotation of a topological charge around its magnetic flux attached to it.
To specify the braiding statistics, we need the data such as; particle species, fusion rules, F-matrices, R-matrices, and topological spin.
The mathematical model for such data is the category theory and the quantum deformation of the recoupling theory of angular momenta.

The ternary logic gates and circuits, or the ones that consist of a combination of binary and ternary, are more compact than their binary counterparts [29]. There are some non-Abelian anyons for which the ternary structures naturally arise. Such anyons are called metaplectic anyons discussed in Chapter <ref>. The $F$ and $R$ matrices are computed using the recoupling theory in Chapter <ref>. In this dissertation, we proposed improved ternary arithmetic circuit designs that can be implemented with the metaplectic anyons.

This dissertation is organized as follows. In Chapter <ref>, we will discuss the basics of binary and ternary logic gates in quantum computing. Chapter <ref> is on the preliminaries of knot theory and the braid group. Anyons' motion in spacetime makes braids, and the outcomes of the braids are obtained through geometric and topological phases. The geometric phases are discussed in Chapter <ref>. One of the most popular physical systems for topological quantum computation is the quantum Hall effect. The existence of anyons as quasiparticles is described by Chern-Simons theory. The quantum Hall effect and topological materials are explained in Chapter <ref> and <ref>, whereas the Chern-Simons theory is given in Chapter <ref>. This chapter also includes the two-dimensional topological quantum field theory, which describes the Hilbert space of the anyon on a two-dimensional topological manifold. The knot invariant is also obtained using the partition function of the Chern-Simons theory on a topological manifold.
The concept of quantum dimensions, topological spin, F-matrices, R-matrices, and the Hilbert space can alternatively be described by the category theory as explained in Chapter <ref>. The unitary modular category and the quantum group, are also used as the mathematical models for the topological orders and topological quantum computing.

The elementary binary logic gates design in topological quantum computing is given in Chapter <ref>. It will provide a prototype theoretical model for topological quantum computing.
The ternary gates are formed as the quantum deformation, also called q-analog, of the $F$ and $R$ matrices in the recoupling theory. The quantum deformation and the recoupling theory are discussed in Chapter <ref>. The ternary logic gates are designed using the fusion and braiding of the metaplectic anyons. These anyons are the ones with quantum dimensions equal to the square root of integers. The metaplectic anyon and ternary logic design, using the recoupling theory, are discussed in Chapter <ref>. In this chapter, we also presented our designs of the ternary adder, ternary subtractor, and ternary multiplier, and their implementation with the braiding and fusion of metaplectic anyon.

The preliminaries from the abstract algebra and topology are given in Appendices <ref> and <ref>. The quantum group is a deformation of the Hopf algebra. The knot invariant is calculated using the quantum group method and topological quantum field theory in Appendix <ref>.
Some topics from quantum field theory are used in the main dissertation, like the concept of fields, second quantization, path integral, and gauge theory. Appendix <ref> is the basics of quantum field theory.

For this dissertation, basic textbook knowledge of the fundamentals of quantum mechanics and condensed matter physics is assumed.

CHAPTER: QUANTUM COMPUTING

The $\ket{\psi}$ can be written in matrix form as
|ψ⟩ =

Such bases are known as computational bases.

The inner product also obeys the rule $\bra{a_1b_1}\ket{a_2b_2}= \bra{a_1}\ket{a_2}\bra{b_1}\ket{b_2}$.

|ψ⟩=∑_i=1^n a_i |i⟩
∑_i=1^n |i⟩⟨i|=I

Sum is identity. this is completeness.

|ψ⟩ =

, |ϕ⟩ =

|ϕ⟩⟨ϕ| =

b_1^*     b^*_2
a_1b_1^*     a_1b_1^*

a_2b_1^*     a_2b_2^*

The CNOT operation can be written as
U_CNOT    = |0⟩⟨0|⊗I + |1⟩⟨1|⊗X

$= \ket{0}\bra{0}\otimes I + \ket{1}\bra{1}\otimes U$

$\ket{0}\bra{1} + \ket{1}\bra{0} = \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}$
$I=\ket{0}\bra{0}+\ket{1}\bra{1}$, $X=\ket{0}\bra{1}+\ket{1}\bra{0}$,  $Y=-i\ket{0}\bra{1}+ i\ket{1}\bra{0}$, $Z=\ket{0}\bra{0}-\ket{1}\bra{1}$

$H=\frac{1}{\sqrt{2}}(\ket{0}\bra{0}+\ket{1}\bra{0}+\ket{0}\bra{1}-\ket{1}\bra{1})$ $=\frac{1}{\sqrt{2}}\sum_{x,y=0}^1 (-1)^{xy}\ket{x}\bra{y}$

$P(\phi)=\ket{0}\bra{0} + e^{i\phi}\ket{1}\bra{1}$

In quantum mechanics, a system is described by a state vector or a wave function that is written in the Dirac notation as
|ψ⟩ = a_1|1⟩+a_2|2⟩+...+a_n|n⟩ = ∑_i=1^n a_i |i⟩,
where $a_i$'s are complex numbers. The state vector $\ket{\psi}$ represents a system in a superposition of the eigenstates $\ket{i}$ of the system with probability amplitudes $a_i$. On measurement, the system will be found in an eigenstate $\ket{i}$ with a probability $\abs{a_i}^2$. The state vector is assumed to be normalized, such that $\sum_i\abs{a_i}^2=1$.

The $\ket{\psi}$ is a vector in a Hilbert space and is written as a linear combination of the basis vectors $\ket{i}$. The Hilbert space is defined in Appendix <ref>.
For each ket $\ket{i}$, there is a dual vector bra $\bra{i}$ in the Hilbert space. The purpose of this dual vector is to find the linear dependence of two states by computing the projections. The projections are found by inner product or orthonormality condition as $\bra{i}\ket{j}=\delta_{ij}$, where $\delta_{ij}=1$ when $i=j$ and $\delta_{ij}=0$ when $i\ne j$. Also, $\bra{i}\ket{\psi}=a_i$ and $\bra{\psi}\ket{\psi}=1$. The inner product works the same way as a dot product in vector analysis.
Another product, known as the outer product, is written as $\ket{i}\bra{i}$, such that it obeys the completeness relation $\sum_{i=1}^{n} \ket{i}\bra{i}=I$. The outer product $\ket{i}\bra{i}$ is an operator called a projection operator or a projector. Since $\ket{i}\bra{i}\ket{j}=\bra{i}\ket{j}\ket{i}$, a projector projects $\ket{j}$ along $\ket{i}$. The length of the projection is $\bra{i}\ket{j}$. In the next section, we will see that the state vector and eigenstates can be written as column vectors, so that the linear transformations on the Hilbert space are square matrices that act on the vectors as operators, see Ref. [30].

The physical quantities or dynamical variables like position, momentum, and energy are called observables. There is an operator corresponding to each physical quantity. The values of observables we obtain on measurement are called eigenvalues. Mathematically, the eigenvalues $\lambda_i$ are obtained by the action of the operator on the eigenstate as
Â|i⟩ = λ_i |i⟩.
A system in an eigenstate has a definite value of a physical quantity. But in general, a system is in a state $\ket{\psi}$ which is a superposition of the eigenstates. In this case, the eigenvalues are probabilistic or expected values in a measurement on the system. The expectation value of an operator $A$ is written as
⟨Â⟩= ⟨ψ|Â|ψ⟩=∑_i λ_i a_i^2.
This equation tells that the probability of getting a particular eigenvalue $\lambda_i$ on a measurement is $\abs{\bra{i}\ket{\psi}}^2=\abs{a_i}^2$. The measurement process collapses the state vector to one of the eigenstates. Since, the eigenvalues $\lambda_i$ must be real, $\bra{\psi}\hat{A}\ket{\psi} = (\bra{\psi}\hat{A}\ket{\psi})^*$. It implies that $\hat{A}=\hat{A}^\dagger$, and $\hat{A}$ is a self-adjoint or a Hermitian operator. The adjoint operator $\hat{A}^\dagger$ is obtained by taking the transpose and complex conjugate of the matrix $\hat{A}$. Therefore, the operation $\hat{A}\ket{i}$ is equivalent to the operation $\bra{i}\hat{A}^\dagger$, where $\bra{i}$ is the adjoint of $\ket{i}$. (Compare the operators to the random variables in statistics, and computing the expectation value to finding the mean or expected value of a probability distribution.)

According to the Heisenberg uncertainty principle, complementary or conjugate variables cannot be measured simultaneously with an absolute certainty. For conjugate variables, the eigenstates of one observable are not the eigenstates of the other observable simultaneously. So there can be a superposition of eigenvalues of one observables corresponding to each value of the other observable and vice versa. Measuring one of the conjugate variables, the other one is perturbed in an unpredictable way. Therefore, the outcomes are affected by the order in which we are observing the two quantities $A$ and $B$. The commutator quantifies the uncertainty and is written for two operators $\hat{A}$ and $\hat{B}$ as
Observables corresponding to the operators involved in the commutator are also called incompatible observables. The uncertainty principles is closely linked with the wave-particle duality and complementarity. A quantum object can behave as a particle in some situation and as a wave in some other situations, but it does not show both characters in the same situation. The wave function is the solution of wave equation describing the particle. This wave equation is the Scrödinger equation in Schrödinger's mechanics. The wavelength of a matter wave associated with a quantum object is the de Broglie wavelength given as $\lambda=h/p$, where $p$ is the momentum of the particle and $h$ is the Planck constant. In general, the wave funciton is not a single wave but many waves added to make a wave packet.

For example, a freely moving electron makes a wave packet of width $\Delta x$ spread in space, and have possible momentum values in the range $p\pm \Delta p$. As an electron can also behave as a wave, there is a probability of the electron on multiple points in the wave packet and have multiple momentum values at a single point. Measuring the position of the electron at some point gives the momentum values chosen from many values in the range $p\pm \Delta p$. On the other hand, measuring the momentum of the electron, the value of the position will be measured with an uncertainty equal to the width $\Delta x$ of the wave packet. The uncertainty for conjugate variables position and momentum is computed as $[\hat{x},\hat{p}]=i\hbar$, with $\Delta x \Delta p \ge \frac{1}{2}\abs{\langle [\hat{A},\hat{B}]\rangle}$. The values of momentum are related to the values of the position by the Fourier transform.

Due to the wave nature of quantum objects, the wave function and eigenstates of a matter particles can also have phases associated with them, and the wave properties like diffraction and interference are observed for a particle like electron. For example, in the famous double slit experiment, the wave function of an electron can be splitted and later interfered constructively or destructively. The global phase of a wave function is canceled out while computing the probabilities and expectation values, so only the relative phases of the eigenstates have physical significance. More than one quantum objects can also be entangled. We will discuss the quantum entanglement in the next section.

The time evolution of a state is represented by the unitary time evolution operator $U(t)$ such that $U^{-1}=U^\dagger$ and $U(t)U^\dagger(t)= U^\dagger(t)U(t)=I$. It is written as $U(t)=\exp(-iHt)$, where $H$ is the Hamiltonian operator corresponds to the energy eigenvalues. The unitary evolution relates the changes in the state to the energy of the system. These changes are reversible so that the inner product is preserved. That means, the phase can be changed but the amplitude remains the same with time.
When the initial state $\ket{\psi_i}$ evolves unitarily to the final state $\ket{\psi_f}$, it is written as
A quantum computation model involves three steps; initialization, unitary evolution, and measurement [31]. The initial state is an input state and the final state is an output state [32] of a quantum gate. The evolution operator $U(t)$ corresponds to a quantum gate. The readout is a measurement in certain bases that gives a classical result. See Ref. [33, 34] for the theoretical minimum of quantum mechanics, [35] for conceptual understanding, and [36, 37] for more in depth theoretical framework. For a basic study on quantum computing, see the books [38, 39, 40], and for technical details, see [32, 41].

§ BINARY QUANTUM GATES

A classical bit has a value 0 or 1. A qubit is a superposition of 0 and 1, written as
|ψ⟩= α|0⟩ + β|1⟩.
A qubit can be written in a matrix form as
|ψ⟩ = α

+ β

where $\alpha$ and $\beta$ are complex numbers. The sum of their squares is one, $\abs{\alpha}^2+\abs{\beta}^2 =1$, which means that the sum of probabilities is equal to one. A qubit can be made by any two-level quantum mechanical system. For example, a spin-half particle can be in a superposition state of spin-up state $\ket{0}$ and spin down spin state $\ket{1}$, a photon can be in a superposition of two polarization states, or an atom can be in a superposition of the ground state and the excited state. The states $\ket{0}$ and $\ket{1}$ are the eigenvectors of a system. These eigenvectors $\ket{0}$ and $\ket{1}$ provide the bases for a qubit state. These bases are orthonormal, that is $\bra{i}\ket{j}=\delta_{ij}$, where $i,j=\{0,1\}$. When $\ket{i}$ is a column vector, $\bra{i}$ is a row vector. The superposition allows us to do many calculations in parallel. For $n$ qubits, a state is written as a $2^n$-dimensional vector in a Hilbert space $\cal H$ and the qubits can be entangled.

The purpose of quantum gates and circuits is to get the required output with maximum probability. Mathematically, a gate is represented by a matrix that must be unitary. The matrix elements correspond to the probabilities of getting the respective basis state. Two matrices do not commute in general.
Some elementary gates are represented by symbols $I, X, Y, Z$. These are called Pauli matrices in physics and denoted as $\sigma_I,\sigma_x,\sigma_y, \sigma_z$. These matrices have the effect of rotating the qubit about the z-axis by an angle $\theta$.
I =
1     0

0     1
,  X =

0     1

1     0
,  Y =

0     -i

i     0
,  Z =

1     0

0     -1
where $X$ is the NOT gate, $Z$ is the phase gate and $Y$ is the phase and NOT gate together, that is $Y = iXZ$. These matrices have properties that $X^2=Y^2=Z^2=I$ and $XY=iZ, \ YZ=iX, \ ZX=iY$. The superposition is created by the Hadamard gate,
H|ψ⟩ = 1/√(2)
1     1

1     -1

β = 1/√(2)
α+ β

α- β.
The implementation of these gates is shown in Fig. <ref> (a) and (b).

0.52$\alpha\ket{0}+\beta \ket{1}$
3.52$\beta \ket{0} + \alpha \ket{1}$
0.51$\alpha\ket{0}+\beta \ket{1}$
3.51.1$-i\beta \ket{0} + i\alpha \ket{1}$
0.50$\alpha\ket{0}+\beta \ket{1}$
3.50$\alpha\ket{0} - \beta\ket{1}$

21$\frac{\ket{0} + \ket{1}}{\sqrt{2}}$
20$\frac{\ket{0} - \ket{1}}{\sqrt{2}}$

[Implementation of Pauli gates, Hadamard gate, and CNOT gate.]The implementation of (a) Pauli gates, (b) Hadamard gate.

Another example of one-qubit gates is phase gate that can be written as
1     0

0     e^iϕ
Applying on the state ket $\ket{\psi}$, we get
P|ψ⟩ = α|0⟩+e^iϕ β|1⟩=

When $\phi=\pi$ we have the Pauli matrix $Z$, that is $P(\pi)=Z$. Other examples of phase gates are $S$ and $T$ gates written as
1     0

0     i
,     T = e^iπ/8
e^-iπ/8     0

0     e^iπ/8
1     0

0     e^iπ/4
We can see $T^2=S$. $T$ is also known as $\pi/8$ gate.

The one-qubit state can be represented by the Bloch sphere as shown in Fig. <ref>. The general state on the Bloch sphere is given by [32]
|ψ⟩ = cosθ/2 |0⟩ + e^iϕsinθ/2 |1⟩ =

,  where  0 ≤θ≤π,   0 ≤ϕ≤2π.
The operators that rotate the state on the Bloch sphere can be written as
R_x(θ) ≡e^-iθ/2 X = cosθ/2 I + isinθ/2 X =
cosθ/2     -isinθ/2

-isinθ/2     cosθ/2

R_y(θ) ≡e^-iθ/2 Y = cosθ/2 I + isinθ/2 Y =
cosθ/2     -sinθ/2

sinθ/2     cosθ/2

R_z(θ) ≡e^-iθ/2 Z = cosθ/2 I + isinθ/2 Z =
e^-i θ/2     0

0     e^i θ/2
These three rotations on a Bloch sphere are combined into a general rotation as
R_n̂(θ) = exp(-i θ/2 n̂·σ⃗) = cosθ/2 I - isinθ/2(n_xX + n_yY + n_zZ),
where $\hat{n}$ is a unit vector in three dimensions and $\vec{\sigma}$ is the three-component vector of Pauli matrices. $R_{\hat{n}}(\theta)$ is the effect of rotation on the state around the unit vector $\hat{n}$. Let there exist real numbers $\alpha$, $\beta$, $\gamma$ and $\delta$ such that a general unitary operation on a qubit [32] can be written as
U = e^iαR_z(β)R_y(γ)R_z(δ) =
e^i(α- β/2 - δ/2)cosγ/2     -e^i(α- β/2 + δ/2)sinγ/2

e^i(α+ β/2 - δ/2)sinγ/2     e^i(α+ β/2 + δ/2)cosγ/2
The Bloch sphere representation is limited to a single qubit state only.

[very thick,cyan,dashed] (2,0) arc(0:180:2cm and 0.7cm);
[very thick,cyan,dashed] (0,2) arc(90:-270:0.7cm and 2cm);
[ultra thick,-latex[scale=3.0],olive] (0,0)–(0.9,1.6);

[very thick,->,blue] (0,0)–(2.5,0);
[very thick,->,blue] (0,0)–(-2.5,0);
[very thick,->,red] (0,0)–(0,2.5);
[very thick,->,red] (0,0)–(0,-2.5);

[very thick,->,teal] (0,0)–(-1.6,-1.6);
[very thick,cyan] (0,2) arc(90:270:0.7cm and 2cm);
[very thick,cyan] (2,0) arc(0:-180:2cm and 0.7cm);
[very thick,->,teal] (0,0)–(1.6,1.6);
[very thick,cyan] (0,0) ellipse (2 and 2);

[above left,red] at (0,2) $\ket{1}$;
[below left,red] at (0,-2) $\ket{0}$;
[below right,blue] at (1.9,0) $\frac{1}{\sqrt{2}}[\ket{0}+i\ket{1}]$;
[above left,blue] at (-2,0) $\frac{1}{\sqrt{2}}[\ket{0}-i\ket{1}]$;
[above right,teal] at (1.5,1.5) $\frac{1}{\sqrt{2}}[\ket{0}-\ket{1}]$;
[below left,teal] at (-1.5,-1.5) $\frac{1}{\sqrt{2}}[\ket{0}+\ket{1}]$;

[thick,dashed,olive] (0.05,-0.05)–(1.4,-0.5);
[thick,dashed,olive] (0,2) arc(90:-14:1.45cm and 2cm);
[thick] (0,1) arc(90:65:1cm and 2cm);
[above right] at (0,1) $\theta$;
[thick] (-0.15,-0.2) arc(190:350:0.3cm and 0.2cm);
[above right] at (-0.5,-0.8) $\phi$;
[olive] at (1.2,2) $\ket{\psi}$;

One-qubit state can be represented by a Bloch sphere.

Two-qubit states can be separable or inseparable. The separable states are written for independent composite systems, whereas the inseparable states are entangled state. The separable state, also called product state, can be factorized into two separate states and written as the tensor product of the two states as
|ψ⟩ =    |ϕ_1⟩ ⊗|ϕ_2⟩
= (α_1 |0⟩ + β_1 |1⟩ )⊗(α_2 |0⟩ + β_2 |1⟩)

=    α_1 α_2 |0⟩|0⟩ + α_1 β_2|0⟩|1⟩
+β_1 α_2 |1⟩|0⟩ + β_1 β_2 |1⟩|2⟩

=    α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩,
where $\abs{\alpha}^2 + \abs{\beta}^2 + \abs{\gamma}^2 + \abs{\delta}^2 = 1$. The tensor product notation $\otimes$ means that we multiply each term of the first vector to each term of the second vector. The dimension of the Hilbert space of the product state is a product of the dimensions of the two systems, that is $\mathcal{H}=\mathcal{H}_1\otimes \mathcal{H}_2$. Since the gates are operators and we write them as matrices, the tensor product of two operators is expressed in such a way that multiply each entry of the first matrix to all entries of the second matrix. As an example, let us consider two matrices $A$ and $B$ as

a_1 & a_2\\
a_3 & a_4
\end{pmatrix}, \qquad B=\begin{pmatrix}
b_1 & b_2\\
b_3 & b_4
\end{pmatrix}.$$

The tensor product of $A$ and $B$ is given as
A⊗B =
b_1     b_2

b_3     b_4
b_1     b_2

b_3     b_4

b_1     b_2

b_3     b_4
b_1     b_2

b_3     b_4

a_1b_1     a_1b_2     a_2b_1     a_2b_2

a_1b_3     a_1b_4     a_2b_3     a_2b_4

a_3b_1     a_3b_2     a_4b_1     a_4b_2

a_3b_3     a_3b_4     a_4b_3     a_4b_4
When $A\ket{a}=\alpha \ket{a}$ and $B\ket{b}=\beta \ket{b}$ then the following rules are defined for the tensor product
(A⊗B)(|a⟩⊗|b⟩) = A|a⟩ ⊗B|b⟩,

(|a⟩+|b⟩)⊗|c⟩= |a⟩⊗|c⟩+|b⟩⊗|c⟩,

|a⟩⊗(|b⟩+ |c⟩)= |a⟩⊗|b⟩+|a⟩⊗|c⟩.
We also have the notations $\ket{a}\otimes \ket{b}\equiv \ket{a}\ket{b} \equiv \ket{ab}$. If $\ket{a}$ and $\ket{b}$ are column vectors of two elements each, then $\ket{ab}$ is a column vector of four elements. Corresponding operators $A$ and $B$ would become a four by four matrix $A\otimes B$. It can be generalized to $n$-dimensional Hilbert space. The number of states is increased exponentially with the increase of the number of qubits.

A typical example of a two-qubit gate is the controlled-NOT or CNOT gate shown in Fig. <ref> (a). This gate flips the second qubit when the first qubit state is $\ket{1}$. The first qubit is called control qubit and the second qubit is called target qubit. This gate is a classical analog of exclusive-OR gate based on exclusive-OR logic represented by $\oplus$. The symbol $\oplus$ is defined such that the output state $\ket{x\oplus y}$ will give a value 0 when both inputs are either zero or 1, but the output value will be 1 when one of the inputs is 1.
A two-qubit state
|ψ⟩    = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩
is changed by the operation of CNOT gate as
CNOT|ψ⟩    =
1     0     0     0

0     1     0     0

0     0     0     1

0     0     1     0

δ =

= α|00⟩ + β|01⟩ + δ|10⟩ + γ|11⟩.
The second qubit remains the same when the first qubit is $\ket{0}$, whereas $X$ gate is applied to the target qubit when the control state is $\ket{1}$. But in general, there can be any one-qubit gate at the place of $X$ as shown in Fig. <ref> (b). In that case, we can write the controlled-U (CU) gate as
CU =
1     0     0     0

0     1     0     0

0     0     u_1     u_2

0     0     u_3     u_4

2.250$\ket{a\oplus b}$

[ultra thick,cyan] (0,0)–(0,1);

[ultra thick,cyan] (0,0)–(0,1);

[Controlled-NOT, Controlled-U, SWAP gate, and its physical realization.](a) CNOT gate (b) Controlled-U gate and (c) SWAP gate (d) Physical realization of SWAP gate.

Another two-qubit gate is a SWAP gate that swaps the states of input qubits. The SWAP gate and its physical realization are shown in Fig. <ref> (c) and (d). It can also be written in matrix notation as
1     0     0     0

0     0     1     0

0     1     0     0

0     0     0     1

The computation cannot be performed by a single qubit only. A system should consist of several qubits and should have the capability to entangle these qubits. One qubit is a superposition of two basis states, but when there is a quantum correlation among two systems, then we say that these systems are entangled. The entangled state is non-separable and cannot mathematically be factorized into two separate superposition states. The Bell's state
|Ψ^+⟩ = α_00|00⟩ + α_11|11⟩,
with $\abs{\alpha_{00}}^2 + \abs{\alpha_{11}}^2 =1$, is an example of the entangled state. It is non-local, that is, the information of only one qubit is not accessible locally when two states are far apart. Classically, the first qubit can be in state $\ket{0}$ or $\ket{1}$, so can be the second qubit. Therefore, there are four possibilities of values on measurement. But an entangled state like this Bell's state would give $\ket{00}$ or $\ket{11}$ with the probabilities $\abs{\alpha_{00}}^2$ and $\abs{\alpha_{11}}^2$. On measurement, if the first qubit collapse to the state $\ket{0}$ $(\ket{1})$ then the second qubit is forced to collapse to $\ket{0}$ $(\ket{1})$. The state is maximally entangled when $\abs{\alpha_{00}}^2 = \abs{\alpha_{11}}^2 = \frac{1}{2}$. The entangled state is created in a process that can be shown as in Fig. <ref>.

The entanglement generating gate.

Toffoli and Fredkin gates shown in Fig. <ref> are examples of three-qubit gates. Toffoli gate, also known as controlled-controlled-NOT or CCNOT, consists of two control qubits and one target qubit. When the first and second qubits will be in state $\ket{1}$, then the NOT gate $X$ will be applied to the third gate, but nothing will happen in all other cases. For the Fredkin gate, a SWAP gate is applied on the second and third qubit when the first one is in state $\ket{1}$, nothing will happen otherwise. Therefore, this gate is also known as the controlled-SWAP gate.

Gates are drawn from left to right in a diagram but mathematically they appear in order from right to left. For example, the gate in Fig. <ref> is written as
|Ψ^+⟩ = CNOT(H⊗I)|00⟩.

2.250$\ket{ab\oplus c}$

[ultra thick,cyan] (0,0)–(0,2);

[Toffoli and Fredkin gates.](a) Toffoli and (b) Fredkin gates.

The set of elementary gates used to perform all kinds of computations is called the universal set of gates [42]. There are several universal sets of gates. Hadamard, $\pi/8$, and a CNOT gate can make one of the sets of universal quantum gates [32]. The number of gates used to implement a circuit is called quantum cost of the circuit.
Quantum algorithms are used to solve a particular problem. The most popular quantum algorithms are Shor's factoring algorithm [43] and Grover search algorithm [44]. The former is an exponential speedup over classical factoring algorithm, and the latter is a quadratic speedup.

The gates and circuits in quantum computing are made reversible.
The reversible circuits are the ones with the same number of outputs as the number of inputs, and there is also a one-to-one correspondence between input and output states. Landauer [45] proved that the minimum energy dissipation for the processing of information is  $KT \ln 2$.
Bennett et al. [46] proposed that the energy dissipation can be avoided if the information processing is made reversible.

The conditions necessary for constructing a quantum computer are known as DiVincenzo' criteria [31]. Five requirements for quantum computations are scalability of the physical system, ability to initialize the qubits in a particular basis state, long decoherence time, universal set of gates, and measurement capability. Two more conditions are added for quantum communication, which are the ability to interconvert stationary and flying qubits, and the ability to transmit the qubits between two locations.
Some physical systems used to construct a quantum computer are: ion trap, neutral atoms trapped in an optical lattice, superconductors, quantum dots, nitrogen vacancy centers in diamond, optical, and topological.

§ TERNARY QUANTUM GATES

In quantum technologies, hybrid circuits are sometimes employed, which are a combination of binary and multivalued circuits. Such gates and corresponding circuits may be advantageous in some ways, such as the reduction in inputs and outputs, reduction in the quantum cost, and the complexity of interconnects. Ternary logic is the most popular multi-value logic. The basic unit of information for multivalued logic is called a qudit and that of ternary logic is called a qutrit.
Khan and Perkowski [47] showed that the ternary logic needs fewer gates comparing with its respective binary system. A binary quantum system requires $n_2 = \log_2N$ qubits for a Hilbert space of dimensions $N$. On the other hand, an $m$-valued quantum system requires $n_m = \log_mN$ qudits, we have
n_m = log_mN = log_2N/log_2m = n_2/log_2m.
Therefore, an $m$-valued quantum system requires $1/\log_2m$ times the memory of its binary counterpart. Hence, a logarithmic reduction in the number of qudits for an $m$-valued logic. It is also shown in [47] that $m=3$ is the most favorable choice. Haghparast et al. [29] proved that the ternary is $37\%$ more compact than binary. Therefore, by using ternary logic gates, we can reduce the cost of circuits and make them more efficient.

In ternary quantum logic, the state $\ket{0}$, $\ket{1}$ and $\ket{2}$ are computational bases. A state can be in a superposition of these three basis states, and is written as
ϕ= α|0⟩ + β|1⟩ + γ|2⟩,
with $\alpha$, $\beta$ and $\gamma$ being complex numbers such that $\abs{\alpha}^2 + \abs{\beta}^2 + \abs{\gamma}^2 = 1$. The state vector $\ket{\psi}$ is a three-dimensional column vector
γ =α

The elementary gates for ternary logic are $3\cross 3$ unitary matrices [48, 47, 49, 50, 51]. The qutrit gates and their symbols are shown in Fig. <ref>, where $Z_3(+1)$ shifts the qutrit state by 1 and $Z_3(+2)$ gate shifts the qutrit state by 2. $Z_3(01)$, $Z_3(12)$ and $Z_3(02)$ permute the states $\ket{0}$ and $\ket{1}$, $\ket{1}$ and $\ket{2}$, and $\ket{0}$ and $\ket{2}$ respectively [51, 47].

0 & 0 & 1\\
1 & 0 & 0\\
0 & 1 & 0
\end{pmatrix}$

0 & 1 & 0\\
0 & 0 & 1\\
1 & 0 & 0
\end{pmatrix}$

0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 1
\end{pmatrix}$

1 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{pmatrix}$

0 & 0 & 1\\
0 & 1 & 0\\
1 & 0 & 0
\end{pmatrix}$

Quantum one-qutrit gates.

Analogous to the Hadamard in binary, there is a Chrestenson transform that creates a superposition state from the bases states and is written as [52, 53]
1    1     1

1     ω    ω^*

1     ω^*     ω,
where $\omega= \exp(2\pi i/3)$. The $\omega$ is a cube root of unity, that means that it is equal to unity if we raise it to the cubic power.

A two-qutrit state is written as
|ψ⟩ =    |ϕ_1⟩ ⊗|ϕ_2⟩
= (α_1 |0⟩ + β_1 |1⟩ + γ_1|2⟩ ) ⊗(α_2 |0⟩ + β_2 |1⟩ + γ_2|2⟩)

=    α_1 α_2 |00⟩ + α_1 β_2 |01⟩ + α_1 γ_2 |02⟩ + β_1 α_2 |10⟩ + β_1 β_2|11⟩ + β_1 γ_2|12⟩ +

γ_1 α_2|20⟩ + γ_1β_2|21⟩ + γ_1γ_2|22⟩.
The two-qutrit gates analogous to CNOT gate in Fig. <ref> is such that the $U$ is applied when the controlling qutrit is at $\ket{2}$ otherwise the second qutrit does not change. Here, $U$ is one of the five one-qutrit gates in Fig. <ref>. The Toffoli gate is implemented in such a way that $U$ is applied on the third qutrit when both the controlling qutrit at $\ket{2}$. The one-qutrit gates are called shift gates and the two-qutrit gates are referred to as Muthukrishnan-Stroud (MS) gates [54]. The shift gates and MS gates have quantum cost unity [55, 56]. We will further discuss ternary gates and circuits in Chapter <ref>.

The physical realization of ternary logic was suggested by Ref. [54] for an ion-trap quantum computer, by Ref. [57, 58] for a Josephson junction, by Ref. [59] for cold atoms, and by Ref. [60] for entangled photons. Some circuit architectures are better described by the multi-valued logic. In certain systems containing the non-Abelian anyons, called metaplectic anyons, qutrits naturally appear. We will discuss ternary gates with metaplectic anyons in Chapter <ref>.

The main factors in designing reversible circuits are as follows. The total quantum cost refers to the number of ternary shift gates required to realize the circuit. The total hardware complexity is the complexity of a circuit in which $\varepsilon$ denotes a ternary one-qutrit shift gate, and $\gamma$ denotes a two-qutrit MS gate. The number of constant inputs and the number of unutilized garbage outputs is also one of the factors taken into consideration. The delay time indicated by $\Delta$ is the logical depth of a circuit. It is 1 for the shift gates and MS gates.

§ COMPUTATIONAL COMPLEXITY

Is a quantum computer more powerful than a classical computer? The answer is in the computational complexity theory. To discuss the computational complexity, we will first talk about the questions such as: what is computable? And which problems are solvable and how efficiently?
In 1936, Alan Turing proposed a thought machine to answer these questions [61]. A classical computer is based on a Turing machine model, so we will use these two interchangeably. Turing showed that there exists a universal Turing machine that can simulate any problem that is solvable on any other hardware. This machine can truly capture the algorithmic process. Alonzo Church simultaneously arrived at a similar result [62].

The classical Church-Turing thesis can be stated as, Any computation process or algorithm that can be devised by a mathematician can be effectively implemented on a Turing machine.

After the invention of the transistor, several other models were suggested. So for the comparison of the models, efficiency became an important factor. The Church-Turing thesis was modified and strengthened with the word efficient. According to the strong Church-Turing thesis: Any computation that can be performed by any physical machine can be simulated efficiently by a Turing machine.

The Church-Turing thesis and the strong Church-Turing thesis were stated for the deterministic Turing machine. Later, some algorithms were devised for which a solution could be found with some bounded probability [63]. One of the examples is the Solovay-Strassen algorithm, which appeared in 1977 [64] for finding whether an integer is a prime or not. It did not answer with certainty but with some probability. These kinds of algorithms could not be efficiently implemented on a deterministic Turing machine. A probabilistic Turing machine can make a random choice at each step. The modified strong Church-Turing thesis states that: Any computation that can be performed by any physical machine can be simulated efficiently by a probabilistic Turing machine.

Since nature is quantum mechanical, another challenge to the strong Church-Turing thesis came from quantum theory. Quantum theory is not based on probabilities, but on probability amplitudes, so complex numbers are used. And in addition, there is a measurement problem in quantum theory. The quantum Turing machine should have circuits that preserve the inner product. The quantum circuit model is equivalent to the quantum Turing machine [63].

In 1985, David Deutsch [2] proposed a model of the Turing machine that is based on the laws of quantum theory. Now the strong Church-Turing thesis can be stated as:
A quantum Turing machine can efficiently simulate any realistic model of computation.

Now the question of what is computable is answered regarding what is computable on a Turing machine. There is a problem of uncomputability or undecidability called the halting problem.
On a computer, an algorithm runs for several steps and gives out an answer, yes or no. For simple programs, we know that a computer will halt after some number of steps or run forever. But for some complex algorithms, it is not easy to know if it will halt. Alan Turing proved that it is impossible to have an algorithm that can tell if an arbitrary algorithm will eventually halt. It is related to Gödel's two incompleteness theorems, according to which a formal axiomatic system cannot be complete and consistent at the same time. In every formal system, some statements cannot be proved or disproved by using the same system. The undecidability brought great philosophical consequences to the field of computing [65, 66].

Apart from the halting problem, to classify the difficulty of solving different problems on a Turing machine, the idea of complexity classes is introduced. These classes share some common features when one talks about the resources of time and space needed to solve a problem. The resource of time is related to the number of steps needed for an algorithm to solve a problem.

The complexity class P is the set of problems that can be solved on a classical computer in polynomial time, whereas when the solution of a problem can be checked on the classical computer, then the problem is in the complexity class NP. For example, there is no fast way to solve the problem of finding factors of a large number, so it is not in the class P. But if someone gave the factors, then it can easily be checked if it is the right solution. Therefore, the factor finding problem is in the class NP. The P is a subset of the NP. Whether $\textbf{P} = \textbf{NP}$ is an open problem. It is one of the million-dollar problems of Clay Mathematics Institute.

When a problem needs small space resources but no limit on time, then it falls into the complexity class of PSPACE. Another class is BPP when the problem can be solved in polynomial time if a bounded probability of error is allowed. The BQP is defined as the set of problems that can be solved on quantum computers with some bounded probability of error is allowed [67]. This complexity class lies somewhere between the classes P and PSPACE [32].
It implies [68] that
P⊆BPP⊆BQP⊆PSPACE,     and     P⊆NP ⊆PSPACE.
This is shown in the Fig. <ref>.
There are many more complexity classes, see [67, 38]. So far, there is no proof that the containments in Eq. <ref> are strict, but it is widely believed that P$\ne$NP, NP$\ne$PSPACE, and BPP$\ne$BQP [68]. It is a non-trivial problem to prove that the quantum computer is more powerful than the classical computer, despite evidences in the favor of this proposition [32].

[ultra thick,blue] (-0.2,0.2) ellipse (3 and 2.5);
[ultra thick,blue] (1,-1.1) ellipse (0.5 and 0.7);
[ultra thick,blue] (0,-1) ellipse (1.6 and 1.2);
[ultra thick,red] (-0.2,-0.5) ellipse (2.2 and 1.7);
[ultra thick,blue] (0.7,0.2) ellipse (1.4 and 2.1);
at (1,-1.1) P;
at (-1,-1.2) BPP;
at (-1.5,0) BQP;
at (1,1.4) NP;
at (-1.5,1.5) PSPACE;

[Complexity classes.]Complexity classes [68].

To analyze the cost of an algorithm with the increase of the input size, three notations are used in computer science. When two functions $f(n)$ and $g(n)$ are defined on non-negative integers, we say $f(n)$ is in the class of functions $O(g(n))$ or $f(n)$ in $O(g(n))$ if there are positive integers $c$ and $N$ such that for all values of $n\ge N$, $f(n)\le cg(n)$. That is $g(n)$ is an upper bound on $f(n)$ up to a constant factor for large $n$. $O$ is useful for worse case scenario or upper bound of the resources consumed by an algorithm.
$\Omega$ notation is used for a lower bound on resources. A function $f(n)$ is $\Omega(g(n))$ if there are positive integers $c$ and $N$ such that for all values of $n\ge N$, $f(n) \ge cg(n)$.
We say that $f(n)$ is $\Theta(g(n))$ if it is both $O(g(n))$ and $\Omega(g(n))$.
As an example, let $24n + 4[\log n] +8$ be gates required to perform a task. For large $n$, only the first term is important. The function $2n$ is in the class $O(2n^2)$ since $2n\le 2n^2$ for large $n$.

The calculations show that there is a significant speedup in solving problems on a quantum computer. Peter Shor [69] invented an algorithm to find prime factors of a number on the quantum computer in $O((logN)^3)$ steps. The same problem on a classical computer takes is $O(N^{1/3})$ steps. Therefore, it is an exponential speed-up solving the prime factors finding a problem on a quantum computer.
Simon's period finding problem falls into BQP class [63] and the Grover search algorithm has a quadratic speedup comparing with the classical search. See [32, 68] for a detailed analysis of the complexity of different quantum computing algorithms.

CHAPTER: TOPOLOGY AND KNOT THEORY

The knot theory is of fundamental importance in topological quantum computing. Topological quantum gates are made up of knots, links, and braids. The integral along the worldlines of quasiparticles in topological materials gives knot invariants. The geometric phases are associated with such invariants. The knot theory is studied as a branch of topology.

§ TOPOLOGY

The equivalence of two spaces in Euclidean geometry is shown by comparing their lengths and angles, but angles and lengths are irrelevant in topology.
Instead, imagine that the spaces are made up of a stretchable and moldable material so that we can continuously deform one space to the other without tearing. For example, a sphere cannot be turned into a torus without tearing a hole, so they are topologically different. A hole or a handle in a topological space is called a genus. From this point of view, a disk is equivalent to a rectangle or a square but different from an annulus. A curve and a straight line are equivalent shapes, but both are different from a closed curve. A closed curve is equivalent to a circle. A torus is equivalent to a coffee cup as each has one hole in it, and we can smoothly deform one into the other. Some examples of topological spaces are shown in Fig. <ref>. For further knowledge on topology, see Appendix <ref>.

[blue,ultra thick] (0,0)–(1.5,0);$\ =$
[ultra thick,blue] (0,-0.5) arc (90:-180:0.5);$=$
[decorate,ultra thick,blue, decoration=snake] (0,0) arc (-180:0:1);$\ne$
[blue,ultra thick] (0,0) circle[radius=0.6];$=$
[ultra thick,blue] (0,0) rectangle (1,1);

[scale=0.7][teal,ultra thick,fill = cyan!20] (1,0) arc (0:90:1)–(0,2) arc (90:0:2)–cycle; $=$
[teal,ultra thick,fill=cyan!20,opacity=0.8] (0,0) ellipse (0.7 and 0.7);$=$
[teal,ultra thick,fill = cyan!20] (0,0) rectangle (1.5,1);$\ \ne$
[scale=0.7][teal,ultra thick,fill=cyan!20,opacity=0.8] (0,0) ellipse (1 and 1);
[teal,ultra thick,fill=white,opacity=0.8] (0,0) ellipse (0.4 and 0.4);

Equivalence of topological spaces.

Topology is the study of properties that are preserved under continuous deformation in such a way that the dimension of the diagram should not change. The continuous deformation is called homeomorphism. The topological properties that characterize the equivalence of two shapes under homeomorphism are called topological invariants. These invariants can be numbers, or certain properties of the topological spaces like connectedness, compactness, homotopy group, homology group, or cohomology group [70, 71, 72, 73, 74]. A genus is a topological invariant, but in some cases, it is not a very useful one.

The first homotopy group provides an intuition for anyonic statistics and braids.
Consider two regions $X_1$ and $X_2$ in Euclidean space as shown in Fig. <ref>. Imagine that any loop in $X_2$ can be shrunk to a point, but when there is a hole as in $X_1$, a loop cannot be shrunk to a point. From the Fig. <ref>, $\alpha _1$ can be deformed to $\beta _1$, and $\gamma_1$ can be deformed to $\delta_1$, but the loops $\alpha_1$ and $\beta_1$ cannot be deformed to $\gamma_1$ and $\beta_1$.
When two loops can be continuously deformed to each other then they are in the same equivalence class or a homotopic class [72].

[Homotopy group.]In the region $X_1$, the loops $\beta_1$ and $\alpha_1$ can be deformed to each other but they cannot be deformed to $\delta_1$ or $\gamma_1$. In the region $X_2$, both the loops can be shrunk to identity.

Spaces are distinguished by working with equivalent classes rather than loops. This suggests that the holes are determined by using these equivalence classes. The group structure on these equivalence classes is called the first homotopy group or a fundamental group represented by $\pi _1(X)$. See Appendix <ref> for the formal definition of a group. The group axioms of a homotopy group are described below.
The composition of two group elements corresponds to two loops that start at the same point and are combined to make the third one. In Fig. <ref>, $\gamma_1$ and $\beta_1$ loops are combined to make $\tilde {\gamma}$ that can be written as $\tilde {\gamma} = \gamma_1 \ \beta_1$. This loop first goes along the $\beta_1$ then along the $\gamma_1$. The inverse $\beta_1^{-1}$ is given by a loop that goes in the opposite direction. The identity loop is the one that stays at some point all the time. The loop $\varepsilon = \beta_1 \ \beta_1^{-1}$ is not an identity but homotopic to the identity [72]. When these loops are physically made by the motion of particles on a two-dimensional space then they make braids in the third dimension which is time.

[ultra thick,blue,fill=orange] (2,4) ellipse (0.3 and 0.3);
[ultra thick,blue] (0.5,0.5) rectangle (3.8,5.6);
[ultra thick,blue] (1.5,3) .. controls(1,6.5) and (4.5,3.5)..(1.5,3);
[ultra thick,blue] (1.5,3) .. controls(0,8) and (6.5,3.5)..(1.5,3);
[ultra thick,blue] (1.5,3) .. controls(0,1) and (3,1)..(1.5,3);
[ultra thick,blue] (1.5,3) .. controls(-1,0) and (4,0)..(1.5,3);
[ultra thick,->,red] (3.3,3.7)–(3.4,3.9);
[ultra thick,->,red] (2.7,3.7)–(2.72,3.85);
[ultra thick,->,red] (1.5,1.5)–(1.6,1.5);
[ultra thick,->,red] (1.5,0.75)–(1.6,0.75);
[ultra thick,blue,fill=orange] (1.5,3) ellipse (0.057 and 0.057);
[] at (3,1)$X_1$;
[] at (3,4.1)$\alpha_1$;
[] at (3.3,5)$\beta_1$;
[] at (1.8,1.35)$\gamma_1$;
[] at (1.25,1)$\delta_1$;
[ultra thick,blue] (4,0.5) rectangle (7.3,5.6);
[ultra thick,blue] (4.5,2) .. controls(4.55,5.5) and (8.5,3.5)..(4.5,2);
[ultra thick,blue] (4.5,2) .. controls(3,7) and (10.5,3.5)..(4.5,2);
[ultra thick,->,red] (6.6,2.97)–(6.71,3.13);
[ultra thick,->,red] (6.13,3)–(6.22,3.15);

[] at (6.5,1)$X_2$;
[] at (5.5,4.1)$\alpha_2$;
[] at (6,4.8)$\beta_2$;
[below] at (4.5,1.9)$\alpha_2$;

§ TANGLE

A tangle is a collection of strands embedded in three-dimensional space in such a way that two sides are fixed and strands can have over crossings and under crossings, turning back as shown in Fig. <ref>.
A tangle is an isotopy class of a collection of non-interacting smooth curves $R^2 \cross [0,1]$ have ends fixed at $R^2 \cross {0} \cross {0}$ and $R^2 \cross {0} \cross {1}$.
In general, the strands of a tangle could be oriented or framed. The double backs are the loops that correspond to the creation and annihilation of particles. The closure of a tangle by identifying the top of the tangle to the bottom is called a trace. A closed tangle is a knot or a link.

[ultra thick,knot=blue,dashed] (-0.3,3.5)–(-0.3,0.5)–(5,0.5);
[ultra thick,knot=olive](0,0) .. controls(0,1) and (2,1)..(2,3);
[ultra thick,knot=cyan](0,3) .. controls(0,2) and (1,2)..(1,3);
[ultra thick,knot=teal](2,0) .. controls(0,1) and (0,1.5)..(1,1.3);
[ultra thick,knot=violet](1,0) .. controls(1,1) and (4,2)..(3,3);
[ultra thick,knot=red](4,0) .. controls(4,1) and (4,2)..(4,3);
[ultra thick,knot=teal](3,0) .. controls(2,0.8) ..(1.3,1.2);
[ultra thick,knot=green](2.5,1.5) .. controls(2,2) and (2,2.5)..(2.6,2);
[ultra thick,knot=green](2.9,1.7) .. controls(3.5,1.3) and (3.5,0.6)..(2.5,1.5);
[ultra thick, knot=blue] (-1,-0.5) rectangle (4.5,2.5);
[ultra thick,blue] (-0.3,3.5)–(5,3.5)–(5,0.5);
[ultra thick,blue,dashed] (-1,-0.5)–(-0.3,0.5);
[ultra thick,blue] (4.5,-0.5)–(5,0.5);
[ultra thick,blue] (-1,2.5)–(-0.3,3.5);
[ultra thick,blue] (4.5,2.5)–(5,3.5);
[orange] (0,0) circle(3pt);
[orange] (1,0) circle(3pt);
[orange] (2,0) circle(3pt);
[orange] (3,0) circle(3pt);
[orange] (4,0) circle(3pt);
[orange] (0,3) circle(3pt);
[orange] (1,3) circle(3pt);
[orange] (2,3) circle(3pt);
[orange] (3,3) circle(3pt);
[orange] (4,3) circle(3pt);

A tangle diagram

§ KNOT INVARIANTS

A knot is a closed loop embedded in the three-dimensional space. A link is a disjoint union of more than one loop.
A knot diagram is a projection of a knot into the plane $\mathbb{R}^2$ such that the points are segments and double points are under-crossings and over-crossings.  A circle is an unknot or a trivial link. The simplest non-trivial link is a Hopf link. For example, the trefoil, figure 8 knot, and Hopf link are shown in Fig. <ref>. For further study on knots, see [75, 76, 77].

[use Hobby shortcut,every path/.style=line width=1mm,white,double=red, double distance=.5mm,scale=0.8]
foreach įn 1,...,
.. ([blank=soft]90+360*/̨-180/:-.5) .. (90+360*/̨:2)
[use previous Hobby path=invert soft blanks,disjoint,double=blue];
[use Hobby shortcut]
[consider self intersections=true,ignore endpoint intersections=false,flip crossing=3,only when rendering/.style=ultra thick,blue]
([closed,ultra thick,blue]0,0) .. (1.5,1) .. (.5,2) .. (-.5,1) .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);

[flip crossing=2]
[red,ultra thick](2,0) circle[radius=1cm];
[blue,ultra thick,->] (1,0) circle[radius=1cm];

Trefoil knot, figure 8 knot, and the Hopf link.

Two knot diagrams are equivalent if we can bend, stretch and smoothly deform one to the other without cutting. In knot theory, the equivalence of two knots is called ambient isotopy. Other than stretching and bending, a simple way of showing that two knots are isotopic to each other is by a finite number of the Reidemeister moves shown in Fig. <ref>. These moves are always permitted but not always sufficient to show the isotopy of two knots. When a knot is modified by applying these moves on a small portion of the diagram while keeping the rest of the diagram fixed, we get the formulas called Skein Relations.

[scale = 0.5]
[blue,knot,ultra thick] (1,4) .. controls +(0,-2) .. (0,1.7);
[knot=blue,ultra thick] (1,0) .. controls +(0,2) .. (0,2.3);
[blue,ultra thick] (0,1.7) to [curve through=(-0.4,2)](0,2.3);
[left] at (2.5,2) =;
[red,knot,ultra thick] (3,0) – (3,4);
[] at (4,2) =;
[blue,knot,ultra thick] (5,4) .. controls +(0,-2) .. (6,1.7);
[knot=blue,ultra thick] (5,0) .. controls +(0,2) .. (6,2.3);
[blue,ultra thick] (6,1.7) to [curve through=(6.4,2)](6,2.3);

[scale = 0.5]
[knot=blue,ultra thick] (1,0) .. controls +(0,1) and +(0,-1) .. (0,2) .. controls +(0,1) and +(0,-1) .. (1,4);
[red,knot,ultra thick] (0,0) .. controls +(0,1) and +(0,-1) .. (1,2) .. controls +(0,1) and +(0,-1) .. (0,4);
[left] at (2.5,2) =;

[scale = 0.5]
[blue,ultra thick] (1,0) – (1,4);
[red,knot,ultra thick] (0,0) – (0,4);
[left] at (2,2) =;

[scale = 0.5]
[red,knot,ultra thick] (0,0) .. controls +(0,1) and +(0,-1) .. (1,2) .. controls +(0,1) and +(0,-1) .. (0,4);
[knot=blue,ultra thick] (1,0) .. controls +(0,1) and +(0,-1) .. (0,2) .. controls +(0,1) and +(0,-1) .. (1,4);

[scale = 0.5]
[blue,knot,ultra thick] (1,0) – (1,4);
[red,knot,ultra thick] (0,0) – (3,4);
[teal,knot,ultra thick] (3,0) – (0,4);
[left] at (4,2) =;

[scale = 0.5]
[blue,knot,ultra thick] (2,0) – (2,4);
[red,knot,ultra thick] (0,0) – (3,4);
[teal,knot,ultra thick] (3,0) – (0,4);

[Reidemeister moves.]Reidemeister moves: (a) The move I is undoing a twist in the strand, (b) the move II separates two unbraided strands, and (c) the move III slides a strand under a crossing.

The knot invariants are a set of rules that give the same output for two equivalent knots. That is, these invariants should not change under an ambient isotopy. Therefore, different knots are distinguished by their knot invariants.
The knot invariants have their merits and limitations. The knot polynomials are among several knot invariants assigned to knots and relatively easy to calculate.
The Jones polynomial [78] is of particular interest to us because of its connection to physics. This connection was first explored by Edward Witten in 1989 in his seminal paper [79]. Physically, the trajectories of anyons in spacetime make knots. The knot invariants of trajectories are calculated by the path integral approach to the Chern-Simons theory. The present form of Jones polynomial is due to Kauffman who formulated it in a simpler way [75].

§.§ Kauffman Bracket

The Kauffman bracket is a polynomial invariant of unoriented link. The normalized version of Kauffman bracket yields the Jones polynomial when framing of a knot or a link is also considered. A Kaufman bracket $\langle L \rangle$ of a knot or a link $L$ assigns to each crossing a number that is either $A$ or $B$ as in diagrammatic Eq. <ref>. The values of variables $A$ and $B$ are to be computed in present section. This Kauffman skein relation is used recursively until we get a resulting link diagram to have no crossings. Thus it consists of a finite set of unlinks or circles [80, 75]. The Kauffman skein relation is written as

[scale = 0.5]
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.9,-0.8) – (1.1,0.8);
[red,knot,ultra thick] (1.1,-0.8) – (1.9,0.8);
[left] at (5,0) $ = A $;

[scale = 0.5]
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (1.1,-0.8) to [curve through=(1.3,0)](1.1,0.8);
[blue,ultra thick] (1.9,-0.8) to [curve through=(1.7,0)](1.9,0.8);
[left] at (5,0) $ + B $;

[scale = 0.5]
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (0.8,0.5) to [curve through=(1.4,0.2)](2.2,0.5);
[blue,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.2)](2.2,-0.5);

Here the variables $A$ and $B$ are assigned according to the convention such that when the first strand goes over the second while going in upward direction, we call it a positive crossing or an overcrossing,. In contrast, when the first strand goes below the second while going in upward direction, we call it a negative crossing or an undercrossing. The variables $A$ and $B$ on the right hand side will exchange their places comparing with the case of overcrossing. Now we will look for the invariance of Kauffman bracket under the Reidemeister moves. The Kauffman bracket is not an invariant under the Reidemeister move I. We will see later that it is an invariant when one considers a strand not as a string but as a ribbon or frame. First, we will discuss the invariance of Kauffman bracket under the moves II and III. For the move II, we can write an equation as

[scale = 0.5]
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,knot,ultra thick] (1.1,-0.8) to [curve through=(1.7,0)](1.1,0.8);
[blue,knot,ultra thick] (1.9,-0.8) to [curve through=(1.3,0)](1.9,0.8);
[scale = 0.5]
[left] at (0,0) $ = A $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,knot,ultra thick] (1.1,-0.8) to [curve through=(1.7,0)](1.9,0.8);
[blue,knot,ultra thick] (1.9,-0.8) to [curve through=(1.3,0)](1.1,0.8);

[scale = 0.5]
[left] at (0,0) $ + B $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,knot,ultra thick] (0.9,0.7) to [curve through=(1.4,0.5)](2.1,0.7);
[blue,knot,ultra thick] (1.1,-0.8) to [curve through=(1.3,-0.8)](1.7,0);
[blue,knot,ultra thick] (1.9,-0.8) to [curve through=(1.7,-0.8)](1.3,0);
[blue,ultra thick] (1.3,0) to [curve through=(1.5,0.1)](1.7,0);

[scale = 0.5]
[left] at (0,0) $ = A \Bigg[B$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (1.1,-0.8) to [curve through=(1.3,0)](1.1,0.8);
[blue,ultra thick] (1.9,-0.8) to [curve through=(1.7,0)](1.9,0.8);

[scale = 0.5]
[left] at (0,0) $ + A $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (0.8,0.5) to [curve through=(1.4,0.2)](2.2,0.5);
[blue,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.2)](2.2,-0.5);
[left] at (4,0) $\Bigg]$;

[scale = 0.5]
[left] at (0,0) $ + B \Bigg[A$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue, ultra thick] (1.5,0) circle (0.25);
[red,ultra thick] (0.8,0.6) to [curve through=(1.4,0.4)](2.2,0.6);
[blue,ultra thick] (0.8,-0.6) to [curve through=(1.4,-0.4)](2.2,-0.6);

[scale = 0.5]
[left] at (0,0) $ + B $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (0.8,0.5) to [curve through=(1.4,0.2)](2.2,0.5);
[blue,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.2)](2.2,-0.5);
[left] at (4,0) $\Bigg]$;

[scale = 0.5]
[left] at (0,0) $ = AB$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (1.1,-0.8) to [curve through=(1.3,0)](1.1,0.8);
[blue,ultra thick] (1.9,-0.8) to [curve through=(1.7,0)](1.9,0.8);

[scale = 0.5]
[left] at (0,0) $+A^2 $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (0.8,0.5) to [curve through=(1.4,0.2)](2.2,0.5);
[blue,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.2)](2.2,-0.5);

[scale = 0.5]
[left] at (0,0) $+ BAd$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (0.8,0.5) to [curve through=(1.4,0.2)](2.2,0.5);
[blue,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.2)](2.2,-0.5);

[scale = 0.5]
[left] at (0,0) $ + B^2$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (0.8,0.5) to [curve through=(1.4,0.2)](2.2,0.5);
[blue,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.2)](2.2,-0.5);

[scale = 0.5]
[left] at (0,0) $ = AB$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (1.1,-0.8) to [curve through=(1.3,0)](1.1,0.8);
[blue,ultra thick] (1.9,-0.8) to [curve through=(1.7,0)](1.9,0.8);

[scale = 0.5]
[left] at (0,0) $ + (A^2 + B^2 + BAd)$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red,ultra thick] (0.8,0.5) to [curve through=(1.4,0.2)](2.2,0.5);
[blue,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.2)](2.2,-0.5);

where the $d$ is a real number assigned to an unknot or a circle. For the Reidemeister move II, only the first term in Eq. <ref> must survive. Therefore, we must have the following conditions
AB =1,     A^2 + B(Ad+B) = 0.
From the first condition in Eq. <ref>, we have $B= A^{-1}$. The second condition would become

A^2 + B(Ad+B)    = A^2 + A^-1(Ad + A^-1) = A^2 +d + A^-2,

⇒d    = -(A^2 + A^-2).

From this last relation we can get another relation which will be useful later
Ad+B = -A(A^2 + A^-2) + A^-1 = -A^3.
The Kauffman bracket is invariant under the Reidemeister move III as shown in Eq. <ref>,

[scale = 0.5]
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.09,-0.8) – (1.09,0.8);
[red,knot,ultra thick] (2.2,-0.6) – (0.8,0.6);
[green,knot,ultra thick] (0.8,-0.6) – (2.2,0.6);

[scale = 0.5]
[left] at (0,0) $ = A $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.15,-0.9) – (1.15,0.9);
[red,knot,ultra thick] (0.9,-0.5) to [curve through=(1.5,0)](0.9,0.5);
[green,knot,ultra thick] (2.2,-0.5) to [curve through=(1.7,0)](2.2,0.5);

[scale = 0.5]
[left] at (0,0) $ + B $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.2,-0.8) – (1.2,0.8);
[red,knot,ultra thick] (0.8,0.5) to [curve through=(1.4,0.3)](2.2,0.5);
[green,knot,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.3)](2.2,-0.5);

[scale = 0.5]
[left] at (0,0) $ = A $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.8,-0.9) – (1.8,0.9);
[red,knot,ultra thick] (0.8,-0.5) to [curve through=(1.3,0)](0.8,0.5);
[green,knot,ultra thick] (2.2,-0.5) to [curve through=(1.5,0)](2.2,0.5);

[scale = 0.5]
[left] at (0,0) $ + B $;
[blue, ultra thick] (0,0) – (1,1);
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.9,-0.8) – (1.9,0.8);
[red,knot,ultra thick] (0.8,0.5) to [curve through=(1.4,0.3)](2.2,0.5);
[green,knot,ultra thick] (0.8,-0.5) to [curve through=(1.4,-0.3)](2.2,-0.5);

[scale = 0.5]
[left] at (0,0) $ = $;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.9,-0.8) – (1.9,0.8);
[red,knot,ultra thick] (2.2,-0.6) – (0.8,0.6);
[green,knot,ultra thick] (0.8,-0.6) – (2.2,0.6);
The invariance of Kauffman bracket under the Reidemesister move I is shown in Eq. <ref>,

[scale = 0.5]
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.1,0.8) .. controls (1.2,-0.5) and (2.3,-0.5) .. (2.3,0);
[blue,knot,ultra thick] (1.1,-0.8) .. controls (1.2,0.5) and (2.3,0.5) .. (2.3,0);

[scale = 0.5]
[left] at (0,0) $ = A$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[red, ultra thick] (2.1,0) circle (0.4);
[blue,knot,ultra thick] (1,0.8) to [curve through=(1.5,0)](1,-0.8);

[scale = 0.5]
[left] at (0,0) $ + B$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.1,0.8) .. controls (1.2,-0.8) and (2.3,1) .. (2.3,0);
[blue,knot,ultra thick] (1.1,-0.8) .. controls (1.2,0.8) and (2.3,-1) .. (2.3,0);

[scale = 0.5]
[left] at (0,0) $ = Ad$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.5,-0.8) – (1.5,0.8);

[scale = 0.5]
[left] at (0,0) $ + B$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.5,-0.8) – (1.5,0.8);

[scale = 0.5]
[left] at (0,0) $ = (Ad+B)$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.5,-0.8) – (1.5,0.8);

[scale = 0.5]
[left] at (0,0) $ = -A^{3}$;
[teal, ultra thick] (1,-1)–(0,0)–(1,1);
[teal, ultra thick] (2,-1)–(3,0)–(2,1);
[blue,knot,ultra thick] (1.5,-0.8) – (1.5,0.8);
In the last line of Eq. <ref>, Eq. <ref> is used.
The Kauffman bracket polynomial is based on the skein relations which are written as
⟨L ⟩= A ⟨L_A ⟩+ A^-1⟨L_B ⟩

⟨L ∪O ⟩= d ⟨L ⟩= -(A^2+A^-2) ⟨L ⟩

⟨O ⟩= 1
where $L$ is a link and $O$ refers to an unknot or a trivial link. The first relation is the same as <ref>, the second relation tells that if a knot or a link is a union of a knot and an unknot then the resultant knot would be $d$ times that knot. The third relation implies that an isolated unknot is assigned a value of 1.

As we can see, the Kauffman bracket is not an invariant under the Reidemeister move I. When two knot diagrams are the same under only Reidemeister moves II, and III then the diagrams are called regular isotopic but not ambient isotopic. The smoothing out of the twist gives a factor of $-A^3$. This factor will be compensated for by taking a twist in a framed or ribbon strand as in Fig. <ref>. A ribbon strand has some width rather than a line.
Hence a ribbon is related to a string with a factor $-A^{3}$ multiplied. In quantum theory, it is related to the phase accumulated by a particle with a spin when it does a $2\pi$ rotation.

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[ultra thick,blue](0.34,0.9)..(0.5,0.8).. (0.7,0.9) .. (0.7,1.05) .. (0.33,1)..(0.2,0.6);
[ultra thick,blue](0.27,0.7).. (0.45,0.57).. (0.95,0.95) .. (0.5,1.37) .. (0.25,1.26)..(0,0.75);
[ultra thick,blue] (0,1) .. (0,1.8);
[ultra thick,blue] (0.2,1.3) .. (0.2,1.8);
[ultra thick,blue] (0,0) .. (0,0.8);
[ultra thick,blue] (0.2,0) .. (0.2,0.6);
[ultra thick,blue] (0,1.8) – (0.2,1.8);
[ultra thick,blue] (0,0) – (0.2,0);

[] at (-0.2,1) $ \rightarrow $;
[ultra thick,blue] (0,0) – (0,0.2);
[ultra thick,blue] (0.2,0) – (0.2,0.2);
[ultra thick,knot=blue] (0.2,0.2) – (0,0.6);
[ultra thick,knot=blue] (0,0.2) – (0.2,0.6);
[ultra thick,blue] (0,0.6) – (0,1.2);
[ultra thick,blue] (0.2,0.6) – (0.2,1.2);
[ultra thick,knot=blue] (0.2,1.2) – (0,1.6);
[ultra thick,knot=blue] (0,1.2) – (0.2,1.6);
[ultra thick,blue] (0,1.6) – (0,1.8);
[ultra thick,blue] (0.2,1.6) – (0.2,1.8);
[ultra thick,blue] (0,0) – (0.2,0);
[ultra thick,blue] (0,1.8) – (0.2,1.8);

[] at (-0.4,1) $ =-A^3 $;
[ultra thick,blue] (0,0) – (0,1.8)–(0.2,1.8)–(0.2,0)–cycle;

Straightening a loop in a ribbon gives a twist factor.

§.§ Jones Polynomial

The Kauffman bracket is not an invariant under all the Reidemeister moves. We need to account for the twist factor or the self linking. This twist is also called a writhe. Now we will also assign an orientation to the knot diagrams. If $w_+$ is an overcrossing and $w_-$ is an undercrossing for an oriented knot or link, then the writhe is given by $w(L) = w_+ - w_-$. We can construct a quantity as
V_L(A) = (-A^3)^-w (L) ⟨L ⟩.
This is called Jones polynomial. It is an invariant under all three Reidemeister moves. Two knots are equivalent if they have the same value of the Jones polynomial. A twist that contributes a factor $(-A^{-3})$ would get canceled with $(-A^3)^{w(L)}$. With the change of variable $t^{1/2} = A^{-2}$, the Jones polynomial agrees with the original form in Jones' paper [78]. The Jones polynomial is an invariant under orientation change.
The skein relations for Jones polynomial can be written as
-t^-1 V(L_+) + (t^1/2 - t^1/2)V(L_0) + t V(L_-) = 0,
where $L_0,L_-,L_+$ are shown in Fig. <ref>.

[scale = 0.5]
[left] at (0,2) $L_+ = $;
[blue,knot,ultra thick] (3,0) – (0,4);
[red,knot,ultra thick] (0,0) – (3,4);

[scale = 0.5]
[left] at (0,2) $, \ L_0 = $;
[blue,ultra thick] (1,0) – (1,4);
[red,knot,ultra thick] (0,0) – (0,4);

[scale = 0.5]
[left] at (0,2) $, \ L_- = $;
[red,knot,ultra thick] (0,0) – (3,4);
[blue,knot,ultra thick] (3,0) – (0,4);

Skein relations for Jones polynomial

The writhe is $+1$ for $L_+$, $-1$ for $L_-$, and $0$ for $L_0$. The Jones polynomial is an invariant for a knot's mirror image when $t$ is replaced by $t^{-1}$. The Jones polynomial for a Hopf link is $V(\text{Hopf})= -t^{5/2}-t^{1/2}$. The polynomial invariant for the knot in Fig. <ref> is given as follows. Since $L_+$ and $L_-$ are isotopic to an unknot, we have $V(L_+)= V(L_-)=1$, that is the unknot is assigned the value of the Jones polynomial as 1. Therefore,
-t^-1 V(L_+) +    (t^1/2 - t^1/2)V(L_0) + t V(L_-) = 0

V(L_0)= t^-1 -t/t^1/2 - t^-1/2.
The knot invariants are calculated in terms of topological quantum field theory (TQFT) and quantum group in Appendix <ref>. The TQFT is discussed in Chapter <ref> and the quantum group is defined in Appendix <ref>.

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[blue,ultra thick] ([closed]0,0) .. (1,-1) .. (2,0) .. (3,1) .. (4,0) .. (3,-1) .. (2,0)..(1,1)..(0,0);

[flip crossing=2]
[teal, ultra thick](1,0) circle[radius=1];
[teal,ultra thick] (-1.2,0) circle[radius=1];

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[red,ultra thick] ([closed]0,0) .. (1,1) .. (2,0) .. (3,-1) .. (4,0) .. (3,1) .. (2,0)..(1,-1)..(0,0);

Skein relations for unlinked circles.

§ BRAID GROUP

The trajectories of $N$ particles from their initial position at a time $t_i$ to the final position at a time $t_f$ are in one-to-one correspondence with the elements of the braid group $\cal{B}_N$.
The time direction is taken vertically upward. The trajectories of particles are equivalence classes of all those trajectories which can be continuously deformed into each other.
Assume that the particle number is fixed, which is the same as saying that there are no loops inside the braids. We will discuss in Chapter <ref> that the loops correspond to the creation and annihilation of particles.

Let the braiding of the first and the second strand be represented by  $\sigma_1$ and braiding of the second and the third strands be represented by $\sigma_2$ and so on.
The braid group generators are shown in Fig. <ref>. (a) is the identity element of the braid group. It consists of all straight strands. As shown in (b), the clockwise exchange of strands $i$ and $i+1$ is represented by the generator $\sigma_i$, whereas counterclockwise exchange is represented by the inverse $\sigma_i^{-1}$.
The group composition of two braids is given by stacking the strands on top of each other. For a non-Abelian group, the multiplication is noncommutative, and the order of stacking matters in this case. This is because of the degeneracy of the ground state, as discussed in Chapter <ref>. The braid group generators also satisfy two conditions shown in Fig. <ref> (c) and (d).

σ_iσ_j    = σ_jσ_i  for  i-j > 1,

σ_iσ_i+1σ_i    = σ_i+1σ_iσ_i+1.
The second relation is the famous Yang-Baxter equation.

[ultra thick,red] (1,0) – (1,-2.5);
[ultra thick,blue] (2,0) – (2,-2.5);
[left] at (1.2,-1.3) $...$;

[ultra thick, style strands=1blue,style strands=2red,style strands=3green]
[ultra thick,blue] (1,0) – (1,0.5);
[ultra thick,red] (2,0) – (2,0.5);
[ultra thick,red] (1,-1.5) – (1,-2);
[ultra thick,blue] (2,-1.5) – (2,-2);

[ultra thick, style strands=1red,style strands=2blue,style strands=3green]
[ultra thick,red] (1,0) – (1,0.5);
[ultra thick,blue] (2,0) – (2,0.5);
[ultra thick,blue] (1,-1.5) – (1,-2);
[ultra thick,red] (2,-1.5) – (2,-2);

$\sigma_i$ and $\sigma_i^{-1}$

[ultra thick, style strands=1red,style strands=2blue,style strands=3green]
[ultra thick,red] (1,0) – (1,1);
[ultra thick,blue] (2,0) – (2,1);

[ultra thick, style strands=1red,style strands=2blue,style strands=3green]
[ultra thick,blue] (1,-2.5) – (1,-1.5);
[ultra thick,red] (2,-2.5) – (2,-1.5);
[left] at (3,-1.5) $=$;

[ultra thick, style strands=1red,style strands=2blue,style strands=3green]
[ultra thick,blue] (1,-2.5) – (1,-1.5);
[ultra thick,red] (2,-2.5) – (2,-1.5);

[ultra thick, style strands=1red,style strands=2blue,style strands=3green]
[ultra thick,red] (1,0) – (1,1);
[ultra thick,blue] (2,0) – (2,1);

$\sigma_i\sigma_j =
\sigma_j \sigma_i$ for $\abs{i-j}>1$

[ultra thick, style strands=1red,style strands=2blue,style strands=3teal]
s_1^-1 s_2^- s_1^-1;
[left] at (4,-2) $=$;

[ultra thick, style strands=1red,style strands=2blue,style strands=3teal]
S_2^-1 s_1^-1 s_2^-1;

The braid group; generators and their properties.

[green,ultra thick,fill=cyan!20] (1.5,-2.8) ellipse (2 and 0.5);
[ultra thick, style strands=1red,style strands=2blue]
s_1 s_1 s;
[ultra thick,red] (1,-2.5) .. controls(0,-4.5) and (0,2)..(1,0);
[ultra thick,blue] (2,-2.5) .. controls(3,-4.5) and (3,2)..(2,0);

The creation, braiding, and annihilation of two pairs of particles in $(2+1)$-dimensional spacetime

§.§ Matrix Representation of the Braid Group

The generators of a braid group above can be written in the matrix representation as follows. Suppose, $\sigma_i$ represents a positive crossing when two stands are going upward in such a way that the first strand is at the top and the second strand is at the bottom. The $\sigma_i^{-1}$ represents a negative crossing when the second strand is at the top and the first is at the bottom while going upward. See [81] for further details.

To find the braid matrices, first consider the case of a two-strand braid group $\mathcal{B}_2$. For positive crossing, that is $\sigma$, we assigned a value $\psi_{12}= t$ and $\psi_{21}= b$. Similarly, for negative crossing, that is $\sigma^{-1}$, we assign $\psi_{21}^{-1} = t^{-1}$ and $\psi_{12}^{-1} = b^{-1}$. When there is no strand connecting the upper and lower points, the corresponding values are zero. Therefore, $\psi_{11}= \psi_{22}^{-1} = 0$ and $\psi_{11}^{-1}= \psi_{22}^{-1} = 0$. We can write the $\sigma$ matrices as
σ_1 =
0   t

b   0
,     σ_1^-1 =
0   b^-1

t^-1   0
For a braid in $\mathcal{B}_n$, we would have $n\cross n$ matrix representation. $\mathcal{B}_2$ and its inverse are shown in Fig. <ref> (a).
To check whether these are the inverse of each other, compose $\sigma$ with $\sigma^{-1}$. Since we get $\sigma_1\sigma_1^{-1}= I_2$, the braid gives an identity which is also the Reidemeister move I as shown in Fig. <ref> (b).

[ultra thick, style strands=1blue,style strands=2red,style strands=3green]

[ultra thick, style strands=1red,style strands=2blue,style strands=3green]

[ultra thick, style strands=1red,style strands=2blue,style strands=3green]
[ultra thick,red] (3,-2.5) – (3,0);
[ultra thick,blue] (4,-2.5) – (4,0);
[] at (2.5,-1.5) $=$;

[Generators of the braid group and their composition.](a) The generator $\sigma_1$ and $\sigma^{-1}$ of the braid $\mathcal{B}_2$, (b) the composition of $\sigma$ and $\sigma^{-1}$.

We can extend the procedure for writing the braid matrices to the case of $\mathcal{B}_3$. On similar lines, we can write the $\sigma$ and $\sigma^{-1}$ matrices for three strands shown in Fig. <ref>.

σ_1 =
0   t   0

b   0   0

0   0   1
, 	σ_2 =
1   0   0

0   0   t

0   b   0

σ_1^-1 =
0   b^-1   0

t^-1   0   0

0   0   1
, σ_2^-1 =
1   0   0

0   0   b^-1

0   t^-1   0
It is easy to check that
$\sigma_1 \sigma_1^{-1} = \sigma_2\sigma_2^{-1} = I_3$.
The Yang-Baxter equation from the Fig. <ref> is given by
σ_1 σ_2 σ_1 = σ_2σ_1 σ_2 =
0   0   t^2

0   bt   0

b^2   0   0

[ultra thick, style strands=1blue,style strands=2red,style strands=3teal]
[ultra thick,teal] (3,-1.5) – (3,0);

[ultra thick, style strands=1red,style strands=2blue,style strands=3teal]
[ultra thick,teal] (3,-1.5) – (3,0);

[ultra thick, style strands=1red,style strands=2blue,style strands=3teal]

[ultra thick, style strands=1red,style strands=2teal,style strands=3blue]

The generators $\sigma_1,\sigma_1^{-1},\sigma_2,\sigma_2^{-1}$ in  $\mathcal{B}_3$

§ KNOT AND PHYSICS

A knot or a link can be obtained from a braid by the closure. There are two types of closures. The trace closure is identifying the top ends of the strands to their corresponding bottom ends by keeping track of each strand. So that each colored strand at the top is connected to the same color at the bottom. Whereas the plat closure is to identify each strand at the top to the adjacent end at the bottom without knowing the paths in the middle. That is, to identify the rightmost strand at the top to the rightmost at the bottom without bothering about its color. These two types of closure are shown in Fig. <ref> (a) and (b). Physically, the strands are paths of moving particles in a time direction that is taken upward. A closure corresponds to taking the trace of multi-particle input and output states. The braiding causes the change in the phase of the wave function. These phases are geometric phases discussed in Chapter <ref>. If we get the same phase for the motion of an anyon clockwise and anticlockwise around the other anyon then the braid group is an Abelian or commutative, otherwise it is a non-Abelian group.

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[ultra thick,red](1,-2.5)..(0.9,-2.7)..(0.2,-1.25)..(0.9,0.2)..(1,0);
[ultra thick,blue](2,-2.5)..(2.1,-2.7)..(2.8,-1.25)..(2.1,0.2)..(2,0);

[ultra thick, style strands=1red,style strands=2blue]
s_1 s_1 s;

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[ultra thick,olive](3,0)..(3.2,0)..(4.5,-1.8).. (3.2,-3.5)..(3,-3.5);

[ultra thick,olive](2,0)..(3,0.5)..(5,-1.8).. (3,-4)..(2,-3.5);
[ultra thick,olive](1,0)..(3,1)..(5.5,-1.8).. (3,-4.5)..(1,-3.5);

[ultra thick, style strands=1red,style strands=2blue,style strands=3teal]
S_2^1 s_1^-1 s_2^1;

The trace closure and plat closure.

The amplitude of the process in quantum physics is related to the Kauffman bracket. For example, a Hopf link in Fig. <ref> (a) is corresponding to the creation of two pairs of particles and antiparticles in a $(2+1)$-dimensional spacetime, their braiding, and then fusion. The particles may or may not be fused to the vacuum. It will have some probability amplitude associated with the fusion. This probability amplitude corresponds to taking the trace of a braid.
The quantum amplitude for these processes does not depend on the geometry, speed of evolution, or the distance between particles, but depends only on the topology of the worldlines. The same quantum amplitude occurs when two paths have the same topology. That is, the quantum amplitude is a knot invariant, we will see that in Chapter <ref>. Edward Witten made the connection between the knot theory and quantum physics [79]. He won the field medal for this work with Vaughn Jones in 1990.

CHAPTER: GEOMETRIC PHASES IN QUANTUM PHYSICS

Anyons are charge-flux composites that arise in a highly correlated system, such as the quantum Hall state. The interaction between these charges and fluxes is through braiding. Topological quantum gates are implemented by braiding anyons on a two-dimensional manifold. The quantum gates are the unitary operators that change a system from one state to the other, as explained in Chapter <ref>. The state-space of a topological quantum gate is the ground state degeneracy of the system. This braiding and the ground state degeneracy can be understood through geometric phases [82], such as Aharonov-Bohm phase [83] and Berry phase [27].

§ A CHARGED PARTICLE IN A MAGNETIC FIELD

Classically, the motion of a nonrelativistic charge particle in a magnetic field is described by the Lorentz equation given as
mẍ_i = q (E + (vB)).
The electric field $\bm{E}$ and the magnetic field $\bm{B}$ can be written in terms of vector potential $\textbf{A} = (A_x,A_y,A_z)$ and scalar potential $\phi(x)$ as
E= - ∂/∂tA - ∇ϕ,   B= ∇A.
In quantum mechanics, the momentum $p$ is used instead of $v$ and its status is raised to an operator $\bm{p}= -i\hbar \nabla$.
The Schrödinger's equation for a charged particle moving in an electromagnetic field can be written as
iħ∂ψ(x)/∂t = [1/2m(p - q A)^2 + q ϕ]ψ(x).
Let $\psi_0(x)$ be an eigenstate of the Hamiltonian when there is no vector potential. The wave function of the particle in the presence of vector potential is related to $\psi_0$ as
ψ(x) = exp(iq/ħ ∫A·dx) ψ_0(x).
The wave function in a magnetic field will get a phase $\phi=\frac{q}{\hbar} \int \bm{A}\cdot d\bm{x}$ other than the dynamical phase. A dynamical phase is the one that a wave function gets during the time evolution. The momentum operator $(\bm{p}-q\bm{A})$ appears as a combination of $\bm{p} = -i\hbar \nabla$ and the vector potential $\bm{A}$.

§.§ Gauge Transformation

Let us transform the vector potential as
A →A^' = A + ∇Λ,  ϕ→ϕ' = ϕ- ∂Λ/∂t,
where $\Lambda$ is a scalar function. The transformation in Eq. <ref> is called gauge transformation. Now the Schrödinger equation will be written in terms of $\bm{A}',\psi'$, and $\phi'$. The wave function in Eq. <ref> can be written as
ψ(x)→ψ'(x)    = exp(iq/ħ∫A'·dx)ψ_0(x)

= exp(iq/ħΛ(x))ψ(x).
As a result of the gauge transformation, the wave function gets an additional phase of $\exp(i\frac{q}{\hbar} \Lambda(x))$. The fields $\bm{E}$ and $\bm{B}$ will remain invariant under this transformation. The physical quantities are modulus squared, so the complex phases do not appear.
This gauge transformation is local as $\Lambda (x)$ is a function of $x$. The global gauge transformation is not as significant. It is independent of the position and is corresponding to the transformation of the whole system.
The gauge transformation should be thought of as a phase transformation rather than some scale transformation as the name implies.

Quantum field theory of the electromagnetic field is the Abelian gauge field. Yang and Mills in 1954 pointed out that this phase function $\Lambda (x)$ can be a matrix instead of just a number. Since matrices are non-commutative in general, this phase transformation is non-Abelian. The Yang-Mills field is a non-Abelian gauge field. The Chern-Simons theory, which is the effective field theory for the topological phases, also used to compute the knot polynomial, is a type of non-Abelian gauge field in two dimensions. The gauge theory is discussed in Appendix <ref>.

[fill=red!10] (-0.3,1.1) – (4.4,1.1) – (3.3,-0.2) – (-1.5,-0.2)– cycle;
[teal,ultra thick,fill=cyan!20,opacity=0.8] (0.5,0.5) ellipse (0.8 and 0.4);
[teal,ultra thick,fill=cyan!20,opacity=0.8] (2.5,0.5) ellipse (0.8 and 0.4);
[teal,bend right,ultra thick] (0.5,0.5) to (0,2);
[teal,bend right,ultra thick] (2.5,0.5) to (2,2);
[blue] (0.5,0.5) circle (2pt);
[blue] (2.5,0.5) circle (2pt);
[blue] (0.25,1.7) circle (3pt);
[blue] (2.45,1.4) circle (3pt);
[red] (0.1,0.85) circle (3pt);
[red] (3,0.8) circle (3pt);
[above] at (-0.7,-0.2)$x^\mu$;
[above] at (0.7,1.7)$A^\mu$;
[above] at (3.7,1.2)$phase$;

Gauge transformation of $A^\mu$ causes the change of phase of the wave function.

§.§ Aharonov-Bohm Effect

In 1959, Yakir Aharonov and David Bohm [83] suggested that in quantum mechanics, the vector potential is not just a mathematical artifact, but it leads to detectable results. The effect of vector potential can be observed in a region where $\bm{B}= 0$ but $\bm{A} \ne 0$. They proposed an experiment shown in Fig. <ref>.
Suppose an infinitely long solenoid having a current through it produces a magnetic field along the z-axis. Since according to the right-hand rule the magnetic field is along the axis of the solenoid and zero outside, it can be taken as a tube of magnetic flux. The electron beam from the source $S$ is separated into two parts as shown in Fig. <ref>. The two parts of the beam combined at the screen make an interference pattern.
The phase we get with the wave function of a charged particle in a magnetic field is given in the Eq. <ref>.
The phase acquired by the evolution of the wave function around a loop $C$ can be derived as
ϕ= q/ħ ∮A·dr = q/ħ∫_S ∇A ·ds = q/ħ ∫_S B·ds = q/ħΦ,
where $d\bm{r}$ is a segment of the loop $C$ and $S$ is the surface enclosed by $C$, $d\bm{s}$ is the surface area element, and $\Phi$ is the total flux through $S$. This phase is gauge invariant, i.e. it is independent of the choice of $\bm{A}$ provided that it gives the same $\bm{B}$.
This phase is topological, as it does not depend on the shape of the path around the flux. Also, it remains invariant under the deformation of the surface that makes $\Phi$ fixed.
The two paths in Fig. <ref> are facing different relative vector potentials, hence interference fringes are modulated by the magnetic flux in the coil which is affected by the change of the electric current through the coil. Therefore, the choice of potentials instead of fields is not merely a convenience but a necessity. The electromagnetic field needs to be described in terms of an abstract four-dimensional vector $A_\mu = (\bm{A}, \phi)$.

[ultra thick,blue,->] (-3,-1) .. controls(0,2) and (1.5,1.5)..(3,1.1);
[cylinder,draw=black,thick,aspect=1.5,minimum height=3cm,minimum width=1cm,shape border rotate=90,cylinder uses custom fill, cylinder body fill=green!30,cylinder end fill=red!5,opacity=0.8] at (0.5,0);
[ultra thick,->,red] (0.5,-1.5)–(0.5,2);
[ultra thick,blue,->] (-3,-1) .. controls(-1,-1) and (1.5,-1.5)..(3,0.9);
[snake=coil, segment amplitude=12pt, segment length=4pt](0.5,-1.2)–(0.5,1.3);
[snake=coil,segment aspect=0,segment amplitude=15pt,ultra thick,red] (3.2,1) – (4.8,1);
[above] at (0.5,2)$\Phi$;

[Aharonov-Bohm Effect]Aharonov-Bohm Effect: switching on and off the flux in the flux tube, causes a shift in the interference fringes.

§.§ Anyon and Aharonov-Bohm Effect

An anyon is a quasiparticle having fractional charge and fractional statistics. We can think of these particles as a composite of charge $q$ and flux $\Phi$. These composites arise in two-dimensional physical systems [16, 15]. We will explain the formation of these composites in Chapters <ref> and <ref>.
Let us exchange two anyons in a 2-dimensional space. The movement of anyons around each other in the $(2+1)$-dimensional space, is described by the braid group, see Chapter <ref>. The charge 1 going around the flux of 2 gets the Aharanov-Bohm phase $e^{iq \Phi}$. At the same time, the flux of 1 going around the charge of 2 and gets the phase $e^{iq \Phi}$, as in Fig. <ref>. Therefore, the system gets a total phase $e^{2iq \Phi}$. This phase depends on the number of times one charge circulates the other, but it does not depend on the shape of the path, provided that the adiabaticity condition is satisfied. The adiabaticity condition dictates that the charges must be moved slowly enough so that the system is not perturbed drastically from the ground state. The number of times a charge circulates another charge is called the winding number. The phase will be written as $e^{imq\Phi}$ when the winding number is $m$. The statistical angle $\phi = q \Phi$ on an exchange corresponds to the phase shift of their wave function.
The $2\pi$ rotation of an anyon around itself gives a phase of $e^{iq \Phi}$ due to the charge ring around it. The spin-statistics theorem [84] says that if $s$ is the effective spin of an anyon taken counterclockwise, we get a phase $e^{i2\pi s}$. Thus, we have a non-trivial spin $s= \frac{q \Phi}{2\pi}$ [28].
The topological spin or a twist of an anyon is the rotation of the charge around its own flux. The phase due to the topological spin and the phase due to the braiding are related to each other. We will discuss that again later in Chapters <ref>, <ref> and Appendix <ref>.

[fill=red!10] (-0.5,1.3) – (6.5,1.3) – (5.2,-1.3) – (-2.1,-1.3)– cycle;
[ultra thick,fill=cyan] (0,0) ellipse (0.8 and 0.4);
[ultra thick,blue] (2.5,0) ellipse (2.6 and 1);
[ultra thick,fill=cyan] (4,0) ellipse (0.8 and 0.4);
[cylinder,draw=black,thick,aspect=1.5,minimum height=2cm,minimum width=1cm,shape border rotate=90,cylinder uses custom fill, cylinder body fill=green!30,cylinder end fill=red!5] at (0,0.6);
[ultra thick,->,red] (0,1)–(0,2);
[ultra thick,->,red] (0.2,1)–(0.2,2);
[ultra thick,->,red] (-0.2,1)–(-0.2,2);
[cylinder,draw=black,thick,aspect=1.5,minimum height=2cm,minimum width=1cm,shape border rotate=90,cylinder uses custom fill, cylinder body fill=green!30,cylinder end fill=red!5,opacity=0.8] at (4,0.6);
[ultra thick,->,red] (4,1)–(4,2);
[ultra thick,->,red] (3.8,1)–(3.8,2);
[ultra thick,->,red] (4.2,1)–(4.2,2);
[->,bend left,ultra thick] (4.5,2) to (4.8,1);
[above] at (0,2)$\Phi$;
[above] at (4,2)$\Phi$;
[] at (-0.8,-0.5)$q_1$;
[] at (3.2,-0.5)$q_2$;
[] at (4.2,-1)$C$;
Anyons moving around each other acquire Aharanov-Bohm phase.

§ BERRY PHASE

The Aharanov-Bohm phase is a special case of the geometric phases when the system has time reversal symmetry [85, 86]. It appears when the underlying geometry is changed by the magnetic field in an abstract way. A more general geometric phase, acquired by a wave function during the evolution in parametric space, is called the Berry phase. As an example, consider a spin-1/2 particle in a magnetic field that is oriented in a particular direction. By slowly varying the magnetic field orientation and bringing it back to the initial value, the system will come back to the initial state up to an overall phase with the wave function of the particle [28].

Let us compute the geometric phase in quantum mechanics for a general situation described by two variables $\bm{r}$ and $\bm{R}(t)$. Let $\bm{r}$ describe a fast motion and $\bm{R}(t)$ be a variable that describes a slow motion. The slow variable describes a parameter that varies slowly with time and modulates the fast variable. For example, the motion of electrons of the atoms in a diatomic molecule is described by the fast variable $r$ and the vibratory motion of atoms is described by a slower variable $\bm{R}(t)$.
We suppose that the system returns to the original state after completing a loop in parametric space.
The adiabaticity condition should be satisfied, which means that the motion of the system in the parametric space should be slow enough so that the system does not go to the excited state.
The Schrödinger equation with state vector $\psi(t)$ can be written as
iħ∂/∂t |ψ(t)⟩ = H(R(t)) |ψ(t)⟩.
Let $\ket{n, \bm{R}}$ be an eigenstate of the Hamiltonian that has energy eigenvalue as $E_n(\bm{R})$.
As the $\bm{R}(t)$ is slowly varying, at an instant of time $t$ we can take $\ket{n, \bm{R}(t)}$ as a basis vector, therefore we can write
H(R(t)) |n, R(t)⟩ = E_n(R(t))|n, R(t)⟩.
The solution of Schrödinger equation is given by
|ψ(t)⟩ = e^iγ_n (t) exp[-i/ħ ∫_0^tdt^' E_n(R(t^'))] |n,R(t)⟩.
The phase in brackets is the dynamical phase that depends on time, whereas $\gamma_n(t)$ is the Berry phase that depends on the geometry of parametric space.
In 1984, Berry pointed out that $\gamma_n(t)$ has deep physical meaning and cannot be ignored [27]. Consider a situation when the slow variable $\bm{R}(t)$ returns to the starting point $(\bm{R}(t)=\bm{R}(T))$ at a time $t=T$ after completing a turn in the parametric space in a closed path $C$. If we put the solution <ref> in the Schrödinger equation, take derivative, and cancel then exponentials on both sides, and then apply $\bra{n,\bm{R(t')}}$, we get
γ_n(C) = i ∫_0^t dt'⟨n, R(t')| d/dt' |n, R(t')⟩ = i∮_C dR·⟨n, R| ∇_R |n, R⟩ .
One of the examples of the Berry phase is the evolution of a system from one ground state to the other in topological materials. The degeneracy corresponds to the parametric space $\bm{R}(t)$. This idea is used in topological quantum computation in Chapter <ref>. In addition to the quantum computation, the Berry phase is also used to classify the topological materials, as discussed in Chapter <ref>.
The quantum analog of the geometric phase in polarization optics was discussed by Pancharatnam in 1956 [87]. The non-adiabatic generalization of the Berry phase was proposed by Aharonov-Anandan [88].

The topological aspect of the Berry phase is holonomy on a fiber bundle. The holonomy is a failure of parallel transport around some manifold [77, 82] as shown in Fig. <ref> (a). A conic manifold is drawn because a cylinder is a trivial bundle, so the geometric phase may not appear.
A fiber bundle is a topological space that is a product space locally, but globally it may have a different structure than the product space. See Appendix <ref> for the concept of the fiber bundle and parallel transport.

The parallel transport is moving a vector on a manifold such that the direction of the vector keeps pointing in the same direction. In a plane or a flat space, the initial direction and the final direction would coincide. But in a curved space, the initial and the final directions are different in general. This can be visualized if we bring the vector back after making a complete loop and match it with the initial direction, as shown in Fig. <ref> (b). If we start from the North Pole, parallel transporting the blue and red arrows while keeping the direction pointing in one direction, when we come back to the North Pole, the final direction of the vector is different when comparing with the starting one.

Two vectors at two different positions at the manifold belong to different tangent spaces, so they are not related. To compare the two vectors, a covariant derivative is taken. It involves an extra term with the flat space derivative. The failure of the covariant derivatives along two different paths gives the curvature. In our case, we are not moving a vector in a physical space but in a space where vector potential and the magnetic field strength play the role of connection and curvature respectively. This is explained in terms of the gauge theory on fiber bundle in Appendix <ref>.

[Parallel transport and the holonomy on a fiber bundle.](a) Parallel transport on curved space. (b) Geometric phase is the holonomy on a fiber bundle. This diagram is taken from [86].

§.§ Anyons on a Torus

The system may have multiple types of anyons categorized according to their topological charge. As we discussed in <ref>, anyons are charge-flux composite found in topological materials. When two anyons are brought close to each other, their fusion may result in another anyon or a superposition of several anyons. These two anyons may annihilate to vacuum if they are antiparticles to each other. The ground state somehow knows what types of anyons can be created. Let there be two paths $C_1$ and $C_2$ on a torus along meridian and longitude, as shown in Fig. <ref>.
Let $T_1$ and $T_2$ be operators correspond to the creation of anyon-antianyon pair from the vacuum and carrying around meridian and longitude respectively.
$T_2^{-1}T_1^{-1}T_2T_1$ is two particles created, braided around each other and then re-annihilated. Since the operators $T_1$ and $T_2$ are implemented with some time-dependent Hamiltonian [11], they are unitary. These two operators do not commute with each other, we have
T_2T_1 = e^-2iθ T_1T_2.
Therefore, the system has ground state degeneracy. As $T_1$ is unitary, its eigenvalues must have a unit modulus, that is, they are just complex numbers. The operation of $T_1$ on a state $\alpha$ can be written as
T_1|α⟩ = e^iα|α⟩.
where $\alpha$ is the space of possible ground states. $T_2\ket{\alpha}$ must also be a ground state since $T_2$ commutes with $H$. Therefore, we can write
T_1(T_2 |α⟩) = e^2iθe^iα (T_2 |α⟩).
Let us call this new ground state $\ket{\alpha+2\theta} = T_2\ket{\alpha}$.
On similar lines, we can generate more ground states. Consider a system where anyons have a statistical phase $\theta = \pi p/m$, where $p$ and $m$ are relatively prime so that $p/m$ is an irreducible fraction. The ground states can be written as
|α⟩,|α+2πp/m⟩, |α+4πp/m⟩,...,|α+2π(m-1)/m⟩.
The phase $\alpha+2\pi = \alpha$ so that we are back to the original state. Now we have $m$ independent ground states. Since anyons have the statistical angle $\theta = \pi p/m$, the charge-flux composite will get $(q,\Phi) = (\pi p/m,1)$. When there is a fusion of $n$ elementary anyons then we have $\ket{n} = (q=n\pi p/m,\Phi =n) = (n\pi p/m,n)$. When there are $m$ anyons, we have $\ket{m} = (\pi p,m)$.
Now if we braid $\ket{n} = (n \pi p/m ,n)$ around one of these $\ket{m} = (\pi p/m)$, we obtain a net phase of $2\pi p$ which is equivalent to no phase at all.
Hence, the cluster of $m$ elementary anyons is equivalent to a vacuum. In this way, we have $m$ species of anyon and $m$ different ground states on torus [11]. The subspace used to implement the topological gates depends on a particular model of anyons and also on the number of anyons present.

In the case of an annulus instead of a torus, the $T_1$ operator corresponds to a particle moving along a circular loop and $T_2$ to the particle moving from the inside edge to the outside edge. The degeneracy is $2$. On similar bases, the degeneracy for the higher genus space is $m^g$, where $g$ is genus. The genus is a handle in a topological space. The integral along the path followed by a particle gives a phase accumulated by the particle. It is an example of the Wilson line operator. We can also explain the degeneracy using punctures on a topological surface, as discussed in Chapter <ref>. The above discussion is related to the gauge invariance and the coupling constant in Chern-Simons field theory [11] and the number of anyons in the fractional quantum Hall effect as discussed in Chapter <ref>,.

[scale=0.8][bend left,ultra thick,teal] (-1,-0) to (1,-0);
[bend right,ultra thick,teal] (-1.2,0.1) to (1.2,0.1);
[rotate=0,ultra thick,blue] (0,0) ellipse (2.5 and 1.3);
[densely dashed,red] (0,0) ellipse (2 and 1);
[densely dashed,red] (0,-1.3) arc (270:90:.2 and 0.550);
[thick,red] (0,-1.3) arc (-90:90:.2 and .550);
[] at (0.5,-0.6) $C_1$;
[] at (-1.5,-0.3) $C_2$;
Two non-trivial paths on a torus.

The total charge and flux of a fusion of two particles must be zero: that is, we should get the vacuum. Therefore, the antianyon must have charge $-q$ and phase $\Phi$. The phase of an anyon moving clockwise around another anyon is the same as an antianyon moving clockwise around another antianyon. However, the phase of an anyon around an antianyon is $-2\phi$. The fusion of a particle with its antiparticle gives the vacuum, but when two particles are pushed together, we get a charge $2q$ and flux $2\Phi$. Now, the phase of exchanging these two particles is $\phi= 4q\Phi/\hbar$.

Integration is taken along the curve $C$ in the parametric space of $\bm{R}$ with time $t$ as a parameter. Berry introduced vector potential as
A(R) = i ⟨m, R| ∇_R |m, R⟩
γ_m(C)    =  ∮_C dR·A(R)

= ∫_S dS·(∇A)

= ∫_S dS·V ,   V = ∇A
The problem is familiar from electromagnetic theory. When vector potential can be written as the gradient of scalar potential, that is $\bm{A} = \nabla \phi$ the Berry phase $\gamma_m(C)$ is zero. This corresponding to 'pure gauge' which is trivial case. For non-trivial case, when Berry phase is non-zero, magnetic field show a singularity.

which is a magnetic monopole. No magnetic monopole is discovered until now.

In three-dimensional $\bm{R}$space
(∇_R A)_i    = i (∇_R ⟨m, R(t)| ∇_R |m, R(t))⟩

= i ε_ijk ∂_j (⟨m, R(t)| ∂_k |m, R(t))⟩)

= i ε_ijk (∂_j (⟨m, R(t)|)( ∂_k |m, R(t))⟩)

= i (∇_R ⟨m, R(t)|) (∇_R |m, R(t))⟩)
Where $\partial_i \equiv \frac{\partial}{\partial R_i}$. Berry phase then is written as
γ_m(C)    = i ∫_S dS ·[(∇_R ⟨m, R(t)|) (∇_R |m, R(t))⟩)]

= ∑_n i ∫_S dS ·[(∇_R ⟨m, R(t)|) |n, R⟩ ⟨n, R|(∇_R |m, R(t))⟩)
Where $\ket{n, \bm{R}} \bra{n, \bm{R}} = I$ by completeness theorem.

H(R) |ψ(R)⟩ = E(R) |ψ(R)⟩

Time-dependent wave function
|Ψ(t)⟩ = e^iγ(t)|Ψ(R(t))⟩
allowing additional phase at front. Time dependent Schrödinger equation is
iħ∂/∂t |Ψ(t)⟩ = H(R(t)) |Ψ(t)⟩

[-ħγ̇ + iħ∂/∂t ] |Ψ(R(t))⟩ = E(R(t)) |ψ(R(t))⟩

projecting with $\bra{\psi(\textbf{R})}$
γ̇=-E(R(t))/ħ- i ⟨ψ(R(t))|∂/∂t|ψ(R(t))⟩

Integrating over some path $\textbf{R}(t)$ from the initial time to the final time.

[ultra thick,fill=cyan!30,opacity=0.8] (0,0) ellipse (2 and 1);
[blue,ultra thick] (0,-0.3) – (0,3.2);
[blue,ultra thick] (0.3,-0.3) – (0.5,3.3);
[blue,ultra thick] (0.5,-0.2) – (1,3.5);
[blue,ultra thick] (1.5,-0.3) – (1.5,3.7);
[blue,ultra thick] (-0.5,-0.3) – (-0.5,3.3);
[blue,ultra thick] (-1,-0.2) – (-1,3.5);
[blue,ultra thick] (-1.5,-0.1) – (-1.5,3.7);
[blue,ultra thick] (1.7,0.1) – (1.7,4.2);
[blue,ultra thick] (1.2,0.2) – (1.2,4.5);
[blue,ultra thick] (0.7,0.3) – (0.7,4.7);
[blue,ultra thick] (0.2,0.3) – (0.2,4.8);
[blue,ultra thick] (-1.7,0.1) – (-1.7,4.2);
[blue,ultra thick] (-1.2,0.2) – (-1.2,4.5);
[blue,ultra thick] (-0.7,0.3) – (-0.7,4.7);
[blue,ultra thick] (-0.2,0.3) – (-0.2,4.8);
[ultra thick,fill=green!20,opacity=0.8] (0,4) ellipse (2 and 1);
[ultra thick,densely dashed,red,rotate=-90] (-2,-2) arc (270:90:0.6 and 2);
[ultra thick,red,rotate=-100] (-2.4,-2.4) arc (-90:90:0.6 and 2);

CHAPTER: QUANTUM HALL EFFECT

The physical systems for topological quantum computation are particles called non-Abelian anyons [11, 26]. The non-Abelian anyons are found as quasiparticles in fractional quantum Hall effect [16], and as Majorana fermions in topological superconductors [89]. The fractional quantum Hall effect and topological superconductors are examples of the non-Abelian states of matter that are explained by topology. In this chapter, we will discuss the quantum Hall effect. The topological aspect of the quantum Hall effect and Majorana fermions in topological superconductors will be discussed in the next Chapter <ref>.

§ CLASSICAL HALL EFFECT

The Hall effect was discovered in 1879 by Edwin Hall, about eighteen years before the discovery of the electron. Let a magnetic field $\bm{B}$ be applied along the $z$-direction to a metal conductor as shown in Fig. <ref> (a). The conventional current under the influence of the electric field is along the $x$-direction whereas the electrons move in the -$x$-direction. Under the influence of the magnetic field $\bm{B}$, electrons move in the $y$-direction due to the Lorentz force $\bm{F} = q(\bm{E} + \bm{v} \cross \bm{B})$. As a result, the positive and negative charges are accumulated on the opposite sides of the metal sheet. There is a net electric field between two edges and a potential difference called Hall voltage. In a steady-state, the Hall voltage and the Hall conductance are given by
V_H= IB/ned,     ρ= B/en,
where $I,e,n,d$ respectively are the current, electron charge, electron density, and the thickness of the material.
In classical Hall effect, the Hall voltage increases linearly with the magnetic field whereas the longitudinal voltage remains constant as shown in diagram <ref> (b).

[ultra thick,red] (0,0)–(2.5,2.5);
[ultra thick,blue] (0,0.5)–(2.5,0.5);
[ultra thick,->] (-0.5,0)–(3,0);
[ultra thick,->] (0,-0.5)–(0,3);
[left] at (2.5,2.5) $\rho_{xy}$;
[above] at (2.5,0.5) $\rho_{xx}$;
[below] at (2.5,0) $B$;

[Classical Hall effect.](a) The experimental configuration for the Hall effect [90] (b) Resistivity as a function of magnetic field.

[Resistivity as a function of magnetic field in the quantum Hall effect.]Resistivity as a function of magnetic field in the quantum Hall effect [91].

§ INTEGER QUANTUM HALL EFFECT

Almost a century later, the quantum Hall effect was discovered in 1980 by Klaus von Klitzing et al. [92]. Electrons are bound to move on a two-dimensional surface of the 50 nm thin layer metal-oxide-semiconductor-field-effect transistor (MOSFET). After lowering the temperature to $\sim 4 K$, the transversal resistivity is measured in a very strong magnetic field $(\sim 10 T)$ along the $z$-direction [93].
The plateaus appear in the transverse Hall conductivity with simultaneous disappearance of the longitudinal conductivity with the magnetic field as shown in Fig. <ref>. The Nobel Prize was awarded to Klaus von Klitzing in 1985 for the discovery of the quantum Hall effect.

§.§ Two-Dimensional Electrons in a Magnetic Field

The motion of an electron in a magnetic field was first studied by Landau [94]. The quantum mechanical wave function for a two-dimensional electron in the presence of a magnetic field can be derived as follows [95, 16]. The spin gets polarized by the magnetic field, so we can consider a spinless electron in the $x,y$ plane. Let the magnetic field be applied perpendicular to the plane in the $z$-direction.
The wave function is periodic in the $y$-direction with a length $L_y$ and extended in the $x$-direction with a length $L_x$. We will explain the effect of impurities later, but for the moment, let us ignore the disorders. It will simplify the discussion without the loss of generality. The Hamiltonian for a charge particle in a magnetic field is
H= (p-eA)^2/2m,
where $\bm{A}$ is the vector potential. For the Schrödinger equation for an electron in a two-dimensional surface, we can choose the Landau gauge that is written as
$A_x = -B\hat{x}\hat{y}, \ A_y = 0, \ \bm{B}= \nabla \cross \bm{A} = B\hat{z}$,  so that we have
1/2m(p_x^2 + (p_y+eBx)^2 )ψ= Eψ,
where $p_x = -i\hbar\frac{\partial}{\partial x}$ and $p_y= -i\hbar\frac{\partial}{\partial y}$.
Let us introduce the magnetic length $l \equiv (\hbar /eB)^{1/2}$,
that is the minimum radius allowed by the uncertainty principle for an electron moving in a magnetic field. We also choose $\psi(x,y) = e^{ik_yx}\phi_{k_y}(x)$ because the Hamiltonian is translationally invariant in the $y$-direction. Therefore, we can write
{ p_x^2/2m+ 1/2mω_c^2(k_y l^2 + x)^2}ϕ_k_y(x) = Eϕ_k_y(x),
where $\omega_c = eB/m$ is the cyclotron frequency. $\phi$ satisfies the harmonic oscillator equation with the potential shifted by $x= -k_yl^2$, that is a Gaussian wave packet located at $x= -k_yl^2$. The states are extended in the $x$ direction but confined in the $y$-direction. The eigenvalue of energy is given by
E_n = (n +1/2)ħω_c.
The energy $E_n$ corresponds to the quantized eigenstates known as the Landau levels. This situation is shown in Fig. <ref>.

For the periodic wave function in the $y$-direction, we have $k_y = \frac{2\pi}{L_y}$. Since the $x$ ranges over $L_x$, so that $k_x$ ranges over $L_x/l^2$.
It implies that the number of states in the lowest Landau level is
N_ϕ=L_xL_y/2πl^2= ABe/h= AB/Φ_0,
where $A$ is the area of the sample $A=L_xL_y$ and $\Phi_0 = h/e$ is the magnetic flux quantum. The number of states in a Landau level is equal to the number of flux quanta incidents on the plane. So we have a massive degeneracy, as an electron can move in a circular path around magnetic flux at many places on the two-dimensional sheet. The number of filled Landau levels is given by
$n= \nu eB/\hbar $ [95, 11].
This $\nu$ is called filling fraction. It is a ratio of the number of electrons to the number of flux quanta present in the system, that is, $N_e = \nu N_\phi$, where $N_\phi = \Phi/\Phi_0$.

In the case that the filled Landau levels are an integer number, then the chemical potential lies somewhere between filled levels and empty levels. The chemical potential is the energy required to insert one more electron into the system. Now we need energy for the excitation of an electron from an occupied level to an empty level, we say that the system is incompressible and the excitations are gapped. At this value of the integer filling fraction, the Hall resistivity and conductivity are quantized as
σ_xy = νe^2/h,     ρ_xy= 1/ν h/e^2 ,
whereas the longitudinal resistance is zero. When there are disorders in the sample, the degeneracy is lifted, that is the degenerate levels split in energy, and we have bands. The value of $\nu$ in these experiments is highly accurate (up to one part in six million [96]) that makes the QHE one of the most accurate methods for the resistance measurement [97].

[fill=orange!20,thick] (-0.3,1.1) – (4.4,1.1) – (3.3,-0.2) – (-1.5,-0.2)– cycle;
[ultra thick,opacity=0.8,teal,fill=cyan!50] (0.5,0.5) ellipse (0.8 and 0.4);
[ultra thick,opacity=0.8,teal,fill=cyan!40] (2.5,0.5) ellipse (0.8 and 0.4);
[red,ultra thick,->] (0.5,0.5) to (0.5,1.7);
[red,ultra thick,->] (2.5,0.5) to (2.5,1.7);
[blue] (0.5,0.5) circle (3pt);
[blue] (2.5,0.5) circle (3pt);

Two-dimensional electrons move in circular paths around the magnetic flux tubes make the Landau levels.

§.§ Explanation of Integer Quantum Hall Effect

To explain the quantum Hall effect, Laughlin proposed a thought experiment [98]. The Laughlin's argument was later refined by Halperin [99].
Consider the system where a disorder is only in the bulk (material other than the boundary) of the annulus, but free of disorder in the outside regions. The eigenstate in the lowest Landau level is characterized by angular momentum $m$
ϕ_m ∼z^m e^-z/4l^2
where $z=x+iy$ is the representation of the position in complex coordinates. The wave function is localized in the region of disorders (impurities). When the flux is changed, the wave function moves outwards in a circle. When the flux is changed by an integral number of quantum flux $\phi_0$, angular momentum is changed by an integer from $m$ to the next eigenstate with $m+1$. While the wave function goes outward radially, it can get pushed into the disordered region. But one electron will be taken out from the disordered region, so the flux only affects the extended states wrapped around the annulus, as in Fig. <ref> (a). The presence of disorder only affects the number of extended states in the way that there are fewer states than could be without the disorder. By the gauge invariance, the system is back to itself up to a geometric phase [97]. The gauge transformation is hinted at the topological nature of the quantum Hall effect [97]. The deformation of the sample does not change the Hall conductivity, so it is a topological invariant [16]. This is discussed in the next Chapter <ref>.

Halperin further explained the integer quantum Hall effect using the semiclassical approach and the band diagram [96].
In the Landau levels, there are many independent states with the same energy, and each of these states has a corresponding wave function. The electrons move in the cyclotron motion at various places in the sample. The number of these states depends on the strength of the magnetic field. The stronger the magnetic field, the tighter the orbits are and more orbits can fit without overlapping. The quantum Hall effect is observed only in two dimensions, because the motion in the third dimension, in the direction of the magnetic field, can add any amount of energy to the energy of Landau levels, and hence gaps are filled up.

The impurities are responsible for the formation of plateaus and vanishing of the longitudinal resistance at the plateaus. In the presence of an impurity, independent states in a given Landau level are no longer equal in energy, that is, the degeneracy is lifted. They form bands. The electrons moving in circular orbits around disorders are in the localized states. Whereas the extended states are responsible for the current flowing through the sample.
When the magnetic field is increased, more states get pushed into the localized regions. Therefore, the longitudinal resistance keeps on increasing. When vacancies in the localized states become exhausted, then the longitudinal resistance becomes zero, until the magnetic field is at a value that the next Landau level starts filling. At this stage, the Hall resistance stays constant. With increasing the strength of the magnetic field, the Fermi energy level keeps on moving from band to band, as shown in Fig. <ref> (b). Electrons scatter to a new Landau level when the magnetic field reaches a value such that the Fermi level overlaps with the Landau level. The ratio of Hall resistances at two plateaus is a ratio of integers because for any given Hall voltage the current is proportional to the number of occupied sub-bands of the extended states. An integral number of sub-bands is filled at the plateau.

As electrons try to move in a circular path in the magnetic field and the applied electric field pushes them towards the right, they follow the cycloid motion. The drift velocity towards the right is directly proportional to the strength of the electric field and inversely proportional to the strength of the magnetic field. As the magnetic field is increased, more electrons deflect towards edges, hence the Hall voltage must be increased. There is a movement of charges along the edges, but the bulk remains an insulator. The bulk of the quantum Hall system is gapped, but the energy gap is reduced at the edges, hence there are excitations at the edges as shown in Fig. <ref> (a) and (b). The edge states are topologically protected, which means when we deform the sample, the resistance remains the same. This happens unless the topology of the sample remains the same. The current at the edges of the QHE has chiral symmetry, that is, the edge modes are moving in one direction on one side and the other on the other side. The chiral symmetry is broken in topological insulators. The quantum Hall states are called the topologically ordered phases, whereas the topological insulators are the symmetry protected topological phases [100, 101]. For a discussion on how topology can explain the topological materials, see Chapter <ref>.

[ultra thick,red,double distance = 5] (0,-1.5) – (0,-1);
[ultra thick,blue] (0,0) ellipse (1 and 0.5);
[ultra thick,blue] (0,0) ellipse (2 and 1);
[ultra thick,red,->,double distance = 5] (0,-0.45) – (0,2);
[ultra thick,teal,->] (-1.3,0.5) .. controls(-1.8,0.2) and (-1.8,-0.2)..(-1.3,-0.5);
[ultra thick,->](-0.5,-0.5)–(-0.8,-0.9);
[right] at (-1.6,-0.1) $E$;
[right] at (0,2) $\Phi$;
[right] at (-0.7,-0.7) $J$;

[ultra thick,fill=cyan] (15,2) ellipse (4 and 1.2);
[ultra thick,fill=cyan] (15,5.7) ellipse (4 and 1.2);
[ultra thick] (15,9.5) ellipse (4 and 1.2);
[ultra thick,->,white] (10,0) rectangle (15,15);
[ultra thick,fill=cyan] (10,2) ellipse (4 and 0.8);
[ultra thick,fill=cyan] (10,5) ellipse (4 and 0.8);
[ultra thick,fill=cyan] (10,8) ellipse (4 and 0.8);
[ultra thick,->,white] (5,0) rectangle (10,15);
[ultra thick,fill=cyan] (5,2) ellipse (4 and 0.5);
[ultra thick,fill=cyan] (5,4.3) ellipse (4 and 0.5);
[ultra thick,fill=cyan] (5,6.6) ellipse (4 and 0.5);
[ultra thick] (5,9.3) ellipse (4 and 0.5);
[ultra thick,->,white] (0,0) rectangle (5,15);
[ultra thick,->,orange] (0,0) rectangle (4,8);
[ultra thick,->] (0,0) –(22,0);
[ultra thick,->] (0,0) –(0,14);
[ultra thick] (15,0) –(15,12);
[ultra thick] (10,0) –(10,10);
[ultra thick] (5,0) –(5,11);
[ultra thick,dashed] (0,8) –(20,8);
[left] at (0,4) $E$;
[above] at (2,8) $B=0$;
[above] at (8,11.5) Increasing $B$;
[below] at (10,0) Density of states;
[right] at (20,8) $E_F$;
[right] at (20.7,2.6) Landau levels;
[thick,->] (2,11.5) –(14,11.5);
[ultra thick] (19,5) –(21,3);
[ultra thick] (19,3) –(21,3);

[Laughlin's argument](a) Laughlin's argument: the adiabatic increase of the flux creates electric field $\bm{E}$ and causes current flow $J$ in the radial direction. (b) Oscillations of density of states at the Fermi level.

[ultra thick,blue,fill=cyan!20] (-0.5,0) rectangle (3,4.8);
[ultra thick,red,->] (1,2.5) – (1,2.6);
[ultra thick,red,->] (1.5,2.5) ellipse (0.5 and 0.5);
[ultra thick,red,->] (3,0) arc (270:90:0.5);
[ultra thick,red,->] (3,1.2) arc (270:90:0.5);
[ultra thick,red,->] (3,2.4) arc (270:90:0.5);
[ultra thick,red,->] (3,3.6) arc (270:90:0.5);
[ultra thick,red,->] (-0.5,1.2) arc (90:-90:0.5);
[ultra thick,red,->] (-0.5,2.4) arc (90:-90:0.5);
[ultra thick,red,->] (-0.5,3.6) arc (90:-90:0.5);
[ultra thick,red,->] (-0.5,4.8) arc (90:-90:0.5);

[ultra thick,blue] plot [smooth] coordinates (1,5)(2,3)(4.5,2.7)(7,3)(8,5);
[ultra thick,blue] plot [smooth] coordinates (1,5)(2,2)(4.5,1.5)(7,2)(8,5);
[ultra thick,blue] plot [smooth] coordinates (1,5)(2,1)(4.5,0.3)(7,1)(8,5);
[ultra thick] (0,0) –(0,5);
[ultra thick] (0,0) –(10,0);
[ultra thick,dashed,red] (0,2) –(10,2);
[red] (1.6,2) circle (3pt);
[red] (2,2) circle (3pt);
[red] (7,2) circle (3pt);
[red] (7.4,2) circle (3pt);
[right] at (10,2) $E_F$;
[right] at (10,0) $x$;
[left] at (0,3) $E$;
[thick] (7.6,1.9) –(7.9,1.5);
[] at (9.2,1.1) Edge states;
[above] at (4.5,0.3) $\nu=0$;
[] at (4.5,1.75) $\nu=1$;
[above] at (4.5,2.7) $\nu=2$;

[The edge modes on quantum Hall bar.]Edge modes: (a) the material remains an insulator in the bulk but there is a flow of current on the edges, (b) the Fermi level in the bulk is lower than that on the edges.

§ FRACTIONAL QUANTUM HALL EFFECT

The integer quantum Hall effect is observed in an impure sample, and no interaction among electrons is considered.
In 1982 Dan Tsui and Horst Stormer [102] discovered plateaus in quantum Hall conductivity at the filling fraction $\nu= p/q$. Hence, the Hall resistance is
R_H = h/e^2p/q,
where $p$ and $q$ are integers.