Computations with a future quantum computer will be implemented through the operations by elementary quantum gates.It is now well known that the collection of 1-bit and 2-bit quantum gates are universal for quantum computation, i.e., any n-bit unitary operation can be carried out by concatenations of 1-bit and 2-bit elementary quantum gates.Three contemporary quantum devices-cavity QED, ion traps and quantum dots-have been widely regarded as perhaps the most promising candidates for the construction of elementary quantum gates.In this paper, we describe the physical properties of these devices, and show the mathematical derivations based on the interaction of the laser field as control with atoms, ions or electron spins, leading to the following:(i) the 1-bit unitary rotation gates; and(ii) the 2-bit quantum phase gates and the controlled-not gate.This paper is aimed at providing a sufficiently self-contained survey account of analytical nature for mathematicians, physicists and computer scientists to aid interdisciplinary understanding in the research of quantum computation.
Computations with a future quantum computer will be implemented through the operations by elementary quantum gates.It is now well known that the collection of 1-bit and 2-bit quantum gates are universal for quantum computation, i.e., any n-bit unitary operation can be carried out by concatenations of 1-bit and 2-bit elementary quantum gates.Three contemporary quantum devices-cavity QED, ion traps and quantum dots-have been widely regarded as perhaps the most promising candidates for the construction of elementary quantum gates.In this paper, we describe the physical properties of these devices, and show the mathematical derivations based on the interaction of the laser field as control with atoms, ions or electron spins, leading to the following:(i) the 1-bit unitary rotation gates; and(ii) the 2-bit quantum phase gates and the controlled-not gate.This paper is aimed at providing a sufficiently self-contained survey account of analytical nature for mathematicians, physicists and computer scientists to aid interdisciplinary understanding in the research of quantum computation.
Present-day computers are essentially non-quantum mechanical in operation. The logic gates of a classical computer are irreversible in nature. For easy understanding, elementary quantum mechanics is reviewed including the Schrödinger’s equation, the particle-in-a-box problem, state superposition and collapsing. Bloch sphere representation of a quantum bit is explained. The qubits display interesting properties of quantum superposition, entanglement and interference, characteristic of atomic and subatomic particles. Measurements carried out on qubits cause their wave functions to collapse forcing them into a definite 0 or 1 state. Qubit decoherence on interaction with the environment is a major problem in quantum computing Three types of quantum computing devices are discussed including silicon, superconducting and trapped qubit devices. Among silicon devices, gate-controlled silicon quantum dots and phosphorous donor implanted Si substrates are considered. Superconducting qubit archetypes and their realization techniques are explained. The transmon qubit design is preferred for charge noise mitigation. Under the trapped ion qubit devices, construction and operation of Paul ion traps is discussed, and the types of trapped ion qubits are elucidated.
Semiconductor quantum dots are a promising candidate for future quantum computer devices. Presently, there are three major proposals for designing quantum computing gates based on quantum dot technology: (i) electrons trapped in microcavity; (ii) spintronics; (iii) biexcitons. We survey these designs and show mathematically how, in principle, they will generate 1-bit rotation gates as well as 2-bit entanglement and, thus, provide a class of universal quantum gates. Some physical attributes and issues related to their limitations, decoherence and measurement are also discussed.
Today's computers work on bits that exist as either 0 or 1. Quantum computers aren't limited to two states; they encode information as quantum bits, or qubits, which can exist in superposition. Qubits represent atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor. Because a quantum computer can contain these multiple states simultaneously, it has the potential to be millions of times more powerful than today's most powerful supercomputers. A processor that can use registers of qubits will be able to perform calculations using all the possible values of the input registers simultaneously. This superposition causes a phenomenon called quantum parallelism, and is the motivating force behind the research being carried out in quantum computing. Due to technical obstacles, till date, a quantum computer has not yet been realized. But the concepts and ideas of quantum computing has been demonstrated using various methods like NMR, Ion Trap, Quantum Dot, Optical Methods, etc. A quantum computer manipulates qubits by executing a series of quantum gates, each a unitary transformation acting on a single qubit or pair of qubits. In applying these gates in succession, a quantum computer can perform a complicated unitary transformation to a set of qubits in some initial state. The qubits can then be measured, with this measurement serving as the final computational result. Research must devise a way to maintain decoherence and other potential sources of error at an acceptable level. Probably the most important idea in this field is the application of error correction in phase coherence as a means to extract information and reduce error in a quantum system without actually measuring that system. Thereby, quantum computers will emerge as the superior computational devices and perhaps one day make today's modern computer obsolete.
Preface FOUNDATIONS OF QUANTUM INFORMATICS Spins: The Stern-Gerlach experiment and spin filter EPR, Bell's inequalities, and hidden variables The Landauer principle QUANTUM COMPUTATION AND QUANTUM SYSTEMS Turing machines and binary logic gates Quantum mechanical systems Hilbert spaces Complex finite dimensional Hilbert Spaces Quantum Turing machines Universality of elementary quantum gates Quantum algorithms Quantum adder and multiplier Quantum error correction codes Lasers: a heuristic introduction Quantum computing devices and requirements TWO-LEVEL ATOMS AND CAVITY QED Two-level atoms Quantization of the electromagnetic field Cavity QED Cavity QED for the quantum phase gate Quantum eraser Quantum disentanglement eraser IMPERFECT QUANTUM OPERATIONS Fidelity Density matrices Time evolution of density matrices Examples of master equations Fidelity calculations ION TRAPS Introduction Ion qubits Summary of ion preparation Coherence Quantum gates Large scale confined-ion quantum computer Trap architecture and performance Teleportation of coherent information Experimental DFS logic gates Quantum error correction by ion traps Summary of ion quantum computation QUANTUM LOGIC USING COLD, CONFINED ATOMS Introduction Atom trapping and detection Atom interactions with external fields Atom trapping Qubits and gates Controlled two-qubit gates Coherence properties of atom gates Assessment QUANTUM DOTS QUANTUM COMPUTING GATES Introduction Electrons in quantum dots microcavity Coupled electron spins Biexciton in a single quantum dot Conclusions LINEAR OPTICS COMPUTERS Classical electrodynamics - Classical computers Quantum electrodynamics - Quantum computers Teleportation Summary and outlook SUPERCONDUCTING QUANTUM COMPUTING DEVICES Introduction Superconductivity More on Cooper pairs and Josephson junctions Superconducting circuits: classical Superconducting circuits: quantum Quantum gates Measurement NMR QUANTUM COMPUTING Nuclear magnetic resonance Basic technology with NMR Solid state NMR Shor's algorithm and its experimental realization Quantum algorithm for lattice-gas systems Conclusion Appendix A: The Fock-Darwin States Appendix B: Evaluation of the exchange energy Appendix C: Transformation of quantum states: SU(2) and SO(3) Appendix D: The Homeomorphism from SU(2) to SO(3)
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit , which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices , which are also called operators . I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere . The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible , because they are characterized by unitary matrices ; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I , X , Y , Z , H , CNOT, CCNOT, CROSSOVER or SWAP, controlled- U , and controlled-controlled- U . These gates can be used to form quantum circuits , involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product , as progressively introduced through the above description, is eventually formalized.
We are still at the very beginnings of physical implementations of quantum computers. The devices listed below have been used successfully to entangle two qubits (at most!), except for NMR which has gone up to 7 qubits. It is still premature to try to predict which device will prove most effective for building a quantum computer capable of dealing with several hundred qubits (if such a computer someday exists); perhaps it will be something new, not on this list at all. In any case, it would be as foolish to predict that such a computer will not be available by 2050 as to predict the contrary. The storage and processing of quantum information requires physical systems possessing the following properties (di Vincenzo criteria): (i) they must be scalable, that is, capable of being extended to a sufficient number of qubits, with well defined qubits; (ii) they must have qubits which can be initialized in the state |0〉; (iii) they must have qubits which are carried by physical states of sufficiently long lifetime, so as to ensure that the quantum states remain coherent throughout the calculation; (iv) they must possess a set of universal quantum gates: unitary transformations on individual qubits and a cNOT gate, which are obtained by controlled manipulations; (v) there must be an efficient procedure for measuring the state of the qubits at the end of the calculation ( readout of the results).