Quantum Computing Introduction • The mechanics of light and matter at the atomic and subatomic scale are described by quantum theory • Quantum Theory forms the underlying principles of most of physics. • The developments in the field of information technology has been brought in by quantum mechanics. • Applications of quantum mechanics are not straight forward. • This branch of physics opens up an entirely new world of possibilities in science, technology and information processing. • One of the most promising one is the quantum computer. Moore’s Law and its end • Statement: “The number of transistors on a microchip doubles every year“ • Moore’s Law, predicting the development of more robust computer systems (with more transistors), is coming to an end simply because engineers are unable to develop chips with smaller (and more numerous) transistors. Moore’s Law and its end • Computer chips need new developmental architectures implemented into them in order to be as efficient if more transistors are to be utilized. • While the creation of more powerful computers is regarded as the most important aspect of a computer system, energy efficiency and device lifetime are also as important. • Reduce in the size of the transistor results in causing problems for current microelectronics, which results in electron tunneling. • ∴ Quantum Computation is the option for the further computational studies. Quantum Computing / Computer • Quantum computing is a growing field at the intersection of Physics and Computer Science. • A quantum computer is a device performing quantum computations • It manipulates the quantum states of qubits in a controlled way to perform algorithms. • The development of a quantum computer is currently in its infancy; • Main challenges in further development are to make the quantum computer scalable and to make it fault-tolerant. Quantum Computing / Computer • Quantum computers perform calculations based on the probability of an object's state before it is measured - instead of just 1s or 0s. • This means they have the potential to process exponentially more data compared to classical computers. • Classical computers carry out logical operations using the definite position of a physical state. • These are usually binary, meaning its operations are based on one of two positions. A single state - such as on or off, up or down, 1 or 0 - is called a bit. • In quantum computing, operations use the quantum state of an object to produce what's known as a qubit. Qubit & Representation (Bloch’s sphere) • A qubit (or quantum bit) is the quantum mechanical analogue of a classical bit. • In classical computing the information is encoded in bits, where each bit can have the value zero or one. • In quantum computing the information is encoded in qubits. • A qubit is a two-level quantum system where the two basic qubit states are usually written as ∣0⟩ and ∣1⟩. Qubit • A qubit can be in state ∣0⟩ , ∣1⟩ or (unlike a classical bit) in a linear combination of both states. This phenomenon is called superposition. • ∣ψ⟩ = α ∣0⟩ + β ∣1⟩, where α and β are the amplitude of the states which are complex in nature. Properties of Qubit: • Qubit is a superposition of both ∣0⟩ & ∣1⟩ given by ∣ψ⟩ = α ∣0⟩ + β ∣1⟩ • The total probability of all the states of the quantum system must be 100% • ∣α2∣ + ∣ β 2∣ = 1, is called Normalization rule Bloch’s sphere • In quantum computing, there are state vectors, which point to a specific point in space that corresponds to a particular quantum state. • This can be visualized using a Bloch sphere. • For instance, a vector representing the state of a quantum system could look something like this arrow, enclosed inside the Bloch sphere, which is the so-called "state space" of all possible points to which our state vectors can "point". • This particular state corresponds to an even superposition between |0⟩and |1⟩ (the arrow is halfway between|0⟩ at the top and |1⟩ at the bottom of the sphere). • These vectors are allowed to rotate anywhere on the surface of the sphere, and each of these points represents a different quantum state. Matrix representation of Qubit Quantum Superposition • One of the properties that sets a qubit apart from a classical bit is that it can be in superposition. • Superposition is one of the fundamental principles of quantum mechanics. • In classical physics, a wave describing a musical tone can be seen as several waves with different frequencies that are added together, superposed. • Similarly, a quantum state in superposition can be seen as a linear combination of other distinct quantum states. • This quantum state in superposition forms a new valid quantum state. • Qubits can be in a superposition of both the basis states ∣0⟩, ∣1⟩. • When a qubit is measured it will collapse to one of its eigen states and the measured value will reflect that state. • For example, when a qubit is in a superposition state of equal weights, a measurement will make it collapse to one of its two basis states ∣0⟩ and ∣1⟩ with an equal probability of 50%. • ∣0⟩ is the state that when measured, and therefore collapsed, will always give the result 0. • Similarly, ∣1⟩ will always convert to 1. Quantum Entanglement • Quantum entanglement is a physical phenomenon that occurs when a group of particles share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. • The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics • entanglement is a primary feature of quantum mechanics lacking in classical mechanics. • In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows to implement valuable transformations. Quantum Entanglement • Quantum entanglement is the state where two systems are so strongly correlated that gaining information about one system will give immediate information about the other no matter how far apart these systems are. • In quantum computers, changing the state of an entangled qubit will change the state of the paired qubit immediately. • Entanglement improves the processing speed of quantum computers exponentially over classical computers. Applications of Entanglement • Superdense coding: In simple words, superdense coding is the process of transporting 2 classical bits of information using 1 entangled qubit. • Quantum cryptography: Cryptography is the process of exchanging information between two parties using an encrypted code and a deciphering key to decrypt the message. The key to cryptography is to provide a secure channel between 2 parties. Entanglement enables that. • If two systems are purely entangled that means they are correlated with each other (i.e. when one changes, the other also changes) and no third party shares this correlation. • Additionally, quantum cryptography benefits from the no-cloning theorem which states that: “it is impossible to create an independent and identical copy of an arbitrary unknown quantum state”. Therefore, it is theoretically impossible to copy data encoded in a quantum state. • Quantum teleportation: Quantum teleportation is also the process of exchanging quantum information such as photons, atoms, electrons, and superconducting circuits between two parties. Quantum Gates • Classical computers manipulate bits using classical logic gates, such as OR, AND, NOT, NAND etc. • Similarly, quantum computers manipulate qubits using quantum gates. • The gates are applied to qubits and the state of the qubits changes depending on which gate is applied. • Quantum gates are reversible and are represented by unitary matrices • Quantum NOT gate, Pauli X,Y, Z gates, Hadamard gate, Phase (S) gate are few gates explained here NOT gate Superconductivity Introduction: • Super conductivity is the phenomenon observed in some metals and materials. • Kammerlingh Onnes in 1911 observed that the electrical resistivity of pure mercury drops abruptly to zero at about 4.2K . • This state is called super conducting state. • The material is called superconductor . • The temperature at which they attain superconductivity is called critical temperature Tc. Temperature dependence of resistivity of a superconductor: • One of the most interesting properties of solid at low temperature is that electrical resistivity of metals and alloys vanish entirely below a certain temperature. • This zero resistivity or infinite conductivity is known as superconductivity. • Temperature at which transition takes place is known as transition temperature or critical temperature (Tc). • Above the transition temperature, the substance is in the normal state and below it will be in superconducting state. • Tc value is different for different materials “The resistance offered by certain materials to the flow of electric current abruptly drop to zero below a threshold temperature. This phenomenon is called superconductivity and threshold temperature is called “critical temperature.” Critical Temperature : Temperature at which the resistivity of the material drops to zero is called as Critical Temperature(Tc)or Transitiontemperature Ex : Hg = 4.2 K , Pb = 7.2 K, Nb = 4.5 K , Yitrium Barium Copper Oxide = 92 K etc Meissner effect: • A superconducting material kept in a magnetic field expels the magnetic flux out of its body when it is cooled below the critical temperature and thus becomes perfect diamagnet. This effect is called Meissner effect. • When the temperature is lowered to Tc, the flux is suddenly and completely expelled, as the specimen becomes superconducting. • The Meissner effect is reversible. • When the temperature is raised the flux penetrates the material, after it reaches Tc. Then the substance will be in the normal state. • The magnetic induction inside the specimen B = µo (H + M) Where 'H' is the intensity of the magnetizing field and ‘M’ is the magnetization produced within the material. For T < Tc, B=0 µ0 (H + M) = 0 M = -H M/H = -1= χ Susceptibility is -1 i.e. it is perfect diamagnetism. Hence superconducting material do not allow the magnetic flux to exist inside the material. Effect of magnetic field: • Superconductivity can be destroyed by applying magnetic field. The strength of the magnetic field required to destroy the superconductivity below the Tc is called critical field. It is denoted by Hc(T). • If ‘T’ is the temperature of the superconducting material, ‘Tc’ is the critical temperature, ‘Hc’ is the critical field and ‘Ho’ is the critical field at 0K. They are related by Hc = Ho[1-(T/Tc)2] CRITICAL CURRENT • The maximum current that a super conducting material can carry and remain in its superconducting state is known as its critical current Ic. • The maximum current which can flow through a superconductor without destroying its superconducting nature is called critical current. • A current greater than Ic will cause the wire to revert to its normal state. • This critical current is proportional to the radius of the wire. Types of Superconductors • There are two types of superconductors. • They are type-I superconductors and type-II superconductors. • Type-I superconductors exhibit complete Meissner effect. Below the critical field it behaves as perfect diamagnet. • If the external magnetic field increases beyond Hc the superconducting specimen gets converted to normal state. The magnetic flux penetrates and resistance increase from zero to some value. As the critical field is very low for type-I superconductors, they are not used in construction of solenoids and superconducting magnets. • Type-II superconductors • Type-II superconductors are hard superconductors. They exist in three states • Superconducting state • Mixed state • Normal state • They are having two critical fields Hc1 and Hc2. • For the field less then Hc1, it expels the magnetic field completely and becomes a perfect diamagnetic. • Between Hc1 and Hc2 the flux starts penetrating throughout the specimen. • This state is called vortex state. • Hc2 is 100 times higher than Hc1. • At Hc2 the flux penetrates completely and becomes normal conductor. • Type-II superconductors are used in the manufacturing of the superconducting magnets of high magnetic fields above 10 Tesla. BCS Theory • Bardeen, Cooper and Schrieffer (BCS) in 1957 explained the phenomenon of superconductivity based on the formation of cooper pairs, which is a quantum mechanical concept. • When a current flow in a superconductor, electrons come near a positive ion core of lattice, due to attractive force. • The ion core also gets displaced from its position, which is called lattice distortion. • The lattice vibrations are quantized in a term called Phonons. • Now an electron which comes near that place will interact with the distorted lattice. • This tends to reduce the energy of the electron. • It is equivalent to interaction between the two electrons through the lattice. • This leads to the formation of cooper pairs. BCS Theory • “Cooper pairs are a bound pair of electrons formed by the interaction between the electrons with opposite spin and momenta in a phonon field”. • According to quantum mechanics a cooper pair is treated as single entity. • A wave function is associated with each cooper pair. This holds good over a large volume with finite value for its amplitude. • The wave function of similar cooper pairs overlaps. For one cooper pair overlapping may extend over 106 other pairs. Thus it covers entire volume of the superconductor. It leads to union of large number of cooper pairs. The resistance encountered by any single cooper pair is overcome by combined action of other pairs in the union. • When the electrons flow in the form of cooper pairs in materials, they do not encounter any scattering and the resistance factor vanishes or in other words conductivity becomes infinity which is called as superconductivity. • In superconducting state electron-phonon interaction is stronger than the coulomb force of attraction of electrons. Cooper pairs are not scattered by the lattice points. They travel freely without slow down as their energy is not transferred. Due to this they do not posses any electrical resistivity.
