It is shown that encoding the state of a quantum computer (or other quantum information-processing device) for error correction has the effect of making its operating states macroscopically indistinguishable; all the more, the more ``stable'' the code is, that is, the more errors it can correct in each pass. It is also shown that for nondegenerate codes the expectation values of macroscopic observables in such encoded states are identical to those that would be obtained for a totally mixed state of maximum entropy, where again the degree of indistinguishability increases with the number of errors corrected by the code. The results can be used to estimate the decoherence rates for systems of qubits under certain forms of interaction with the environment: it is found that when the qubits are encoded all coherent superpositions decay, to lowest order, at the same rate, which scales more slowly with the number of physical qubits than the decoherence rate for macroscopic coherent superpositions (``Schr\"odinger cats'') of the bare qubits.
It is shown that encoding the state of a quantum computer (or other quantum information-processing device) for error correction has the effect of making its operating states macroscopically indistinguishable; all the more, the more stable'' the code is, that is, the more errors it can correct in each pass. It is also shown that for nondegenerate codes the expectation values of macroscopic observables in such encoded states are identical to those that would be obtained for a totally mixed state of maximum entropy, where again the degree of indistinguishability increases with the number of errors corrected by the code. The results can be used to estimate the decoherence rates for systems of qubits under certain forms of interaction with the environment: it is found that when the qubits are encoded all coherent superpositions decay, to lowest order, at the same rate, which scales more slowly with the number of physical qubits than the decoherence rate for macroscopic coherent superpositions (Schr"odinger cats'') of the bare qubits.
A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (two-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n=1-2H2(2t/n) where H2(p) is the binary entropy function -plog2p-(1-p)log2(1-p). Upper bounds on this asymptotic rate are given. \textcopyright{} 1996 The American Physical Society.
The smallest details of living systems are molecular devices that operate between the classical and quantum levels, i.e. between the potential dimension (microscale) and the actual three-dimensional space (macroscale). They realize non-demolition quantum measurements in which time appears as a mesoscale dimension separating contradictory statements in the course of actualization. These smaller devices form larger devices (macromolecular complexes), up to living body. The quantum device possesses its own potential internal quantum state (IQS), which is maintained for prolonged time via error-correction being a reflection over this state. Decoherence-free IQS can exhibit itself by a creative generation of iteration limits in the real world. To avoid a collapse of the quantum information in the process of correcting errors, it is possible to make a partial measurement that extracts only the error-information and leaves the encoded state untouched. In natural quantum computers, which are living systems, the error-correction is internal. It is a result of reflection, given as a sort of a subjective process allotting optimal limits of iteration. The IQS resembles the properties of a quasi-particle, which interacts with the surround, applying decoherence commands to it. In this framework, enzymes are molecular automata of the extremal quantum computer, the set of which maintains stable highly ordered coherent state, and genome represents a concatenation of error-correcting codes into a single reflective set. Biological systems, being autopoietic in physical space, control quantum measurements in the physical universe. The biological evolution is really a functional evolution of measurement constraints in which limits of iteration are established possessing criteria of perfection and having selective values.
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.
It is known that a macroscopic quantum superposition (MQS), when it is exposed to an environment, decoheres at a rate scaling with the separation of its component states in phase space. This is more or less consistent with the well-known proposition that a more macroscopic quantum state is reduced more quickly to a classical state in general. Effects of initial mixedness, however, on the subsequent decoherence of MQSs have been less known. We study the evolution of a highly mixed MQS interacting with an environment and compare it with that of a pure MQS having the same size as the central distance between its component states. Although the decoherence develops more rapidly for the mixed MQS in short times, its rate can be significantly suppressed after a certain time and becomes smaller than the decoherence rate of its corresponding pure MQS. In an optics experiment to generate a MQS, our result has the practical implication that nonclassicality of a MQS can still be observable in moderate times, even though a large amount of noise is added to the initial state.
This work is a study of the influence of the medium on the quantum system of qubits. The article assumes a description of the fundamentals of the quantum information theory, as well as a place in it the concept of quantum entanglement. Quantum error correction is a set of methods for protecting quantum information and quantum state from unwanted interactions of the environment (decoherence) and other forms and types of noise. Today, there are modern quantum technologies that are capable of supporting fundamentally new computational algorithms based on the principles of quantum mechanics. The dependences of data distortion on noise and measures of decoherence on noise in one qubit are demonstrated. This article is devoted to solving the problem of research and development of corrective codes for correcting several types of quantum errors. These errors appear during computational processes in quantum algorithms and models of quantum computing devices. The aim of the work is to study existing methods for correcting various types of quantum errors and to create a corrective code for quantum error correction. The scientific novelty of this research is expressed in the one shortcoming exclusion of the quantum computing process. Currently, in many advanced countries of the world, intensive research is being carried out to develop and create quantum computers and their software, and there has been a rapid increase in interest in quantum computers. A large number of articles and monographs are published.