Information carried by a quantum system has notoriously weird properties. Physicists and engineers are now learning how to put that weirdness to work. Quantum computers, which manipulate quantum states rather than classical bits, may someday be able to perform tasks that would be inconceivable with conventional digital technology. (See the article by Charles H. Bennett, PHYSICS Today, October 1995, page 24, and the “Search and Discovery” report in PHYSICS Today, March 1996, page 21.)b
Information carried by a quantum system has notoriously weird properties. Physicists and engineers are now learning how to put that weirdness to work. Quantum computers, which manipulate quantum states rather than classical bits, may someday be able to perform tasks that would be inconceivable with conventional digital technology. (See the article by Charles H. Bennett, PHYSICS Today, October 1995, page 24, and the “Search and Discovery” report in PHYSICS Today, March 1996, page 21.)b
In a ground breaking discovery in 1994, Shor has shown that quantum computers, if built, can factor numbers efficiently. Since then quantum computing has become a burgeoning field of research, attracting theoreticians and experimentalists alike, and regrouping researchers from fields like computer science, physics, mathematics and engineering. Quantum information is very fragile and prone to decoherence. Yet by the middle of 1996 it has been shown that fault-tolerant quantum computation is possible. We give a simple description of the elements of quantum error-correction and quantum fault-tolerance. After characterizing quantum errors we present several error correction schemes and outline the elements of a full fledged fault-tolerant computation, which works error-free even though all of its components can be faulty. We also mention alternative approaches to error-correction, so called error-avoiding or decoherence-free schemes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.
Behold your computer. Your computer represents the culmination of years of technological advancements beginning with the early ideas of Charles Babbage (1791-1871) and eventual creation of the first computer by German engineer Konrad Zuse in 1941. Surprisingly however, the high speed modern computer sitting in front of you is fundamentally no different from its gargantuan 30 ton ancestors, which were equipped with some 18000 vacuum tubes and 500 miles of wiring! Although computers have become more compact and considerably faster in performing their task, the task remains the same: to manipulate and interpret an encoding of binary bits into a useful computational result. A bit is a fundamental unit of information, classically represented as a 0 or 1 in your digital computer. Each classical bit is physically realized through a macroscopic physical system, such as the magnetization on a hard disk or the charge on a capacitor. A document, for example, comprised of n-characters stored on the hard drive of a typical computer is accordingly described by a string of 8n zeros and ones. Herein lies a key difference between your classical computer and a quantum computer. Where a classical computer obeys the well understood laws of classical physics, a quantum computer is a device that harnesses physical phenomenon unique to quantum mechanics (especially quantum interference) to realize a fundamentally new mode of information processing. In a quantum computer, the fundamental unit of information (called a quantum bit or qubit), is not binary but rather more quaternary in nature. This qubit property arises as a direct consequence of its adherence to the laws of quantum mechanics which differ radically from the laws of classical physics. A qubit can exist not only in a state corresponding to the logical state 0 or 1 as in a classical bit, but also in states corresponding to a blend or superposition of these classical states. In other words, a qubit can exist as a zero, a one, or simultaneously as both 0 and 1, with a numerical coefficient representing the probability for each state. This may seem counterintuitive because everyday phenomenon are governed by classical physics, not quantum mechanics – which takes over at the atomic level. This rather difficult concept is perhaps best explained through an experiment. Consider Figure 1 below:
Introduction Almost a century after its birth, quantum theory remains odd and counterintuitive. As Richard Feynman wrote, it seems that nobody really understands it. This is indeed true if by understanding we mean being able to explain it using our common sense and everyday experience. The development of some kind of “quantum common sense” has been very slow even though our everyday life is being continuously influenced by technologies whose roots lie in quantum laws. In recent years the growing field of quantum information and quantum computation became a fruitful playground for physicists, mathematicians, computer scientists, and researchers from other fields who developed new interesting ways of storing, transmitting, and processing information using quantum mechanics at its best. Thus, both theoretical and experimental research on multiparticle entanglement, on the manipulation of individual quantum systems, on decoherence, and on the transition from quantum to classical are subjects of interest not only for their basic relevance but also for their potential practical significance as they might be of help for the development of a real quantum computer. Even though this technology may be far in the future, it is interesting to speculate about what lessons on quantum reality could be learned from quantum computation (eg., what would happen if one could operate a quantum computer).
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.