This thesis is about the quantum error-correcting codes. Although there are many methods to construct them, finding a good quantum code is still a complicated and difficult task, for we do not know which methods to use. To solve this problem, we did a thorough research on a special class of quantum codes--additive cyclic quantum codes. A new search algorithm has been designed and a lot of good additive cyclic quantum codes found by this algorithm are presented in this thesis. By showing the success of this algorithm and great value of additive cyclic quantum codes, we have greatly reduced the complexity of finding good quantum codes. Quantum error-correction theory is key part of quantum information theory. As quantum information theory is a quite new field which has been developing fast during the last decade, we spent much time explaining the primary concepts of it. We introduced linear algebra, quantum mechanics, quantum operations, quantum error-correction theory and quantum error-correcting codes. In a word, this thesis serves two functions: (1) an exploration of quantum error-correcting codes and (2) an introduction to quantum information theory.
This thesis is about the quantum error-correcting codes. Although there are many methods to construct them, finding a good quantum code is still a complicated and difficult task, for we do not know which methods to use. To solve this problem, we did a thorough research on a special class of quantum codes--additive cyclic quantum codes. A new search algorithm has been designed and a lot of good additive cyclic quantum codes found by this algorithm are presented in this thesis. By showing the success of this algorithm and great value of additive cyclic quantum codes, we have greatly reduced the complexity of finding good quantum codes. Quantum error-correction theory is key part of quantum information theory. As quantum information theory is a quite new field which has been developing fast during the last decade, we spent much time explaining the primary concepts of it. We introduced linear algebra, quantum mechanics, quantum operations, quantum error-correction theory and quantum error-correcting codes. In a word, this thesis serves two functions: (1) an exploration of quantum error-correcting codes and (2) an introduction to quantum information theory.
The theory of quantum error correction was established more than a decade ago as the primary tool for fighting decoherence in quantum information processing. Although great progress has already been made in this field, limited methods are available in constructing new quantum error correction codes from old codes. Here we exhibit a simple and general method to construct new quantum error correction codes by nesting certain quantum codes together. The problem of finding long quantum error correction codes is reduced to that of searching several short length quantum codes with certain properties. Our method works for all length and all distance codes, and is quite efficient to construct optimal or near optimal codes. Two main known methods in constructing new codes from old codes in quantum error-correction theory, the concatenating and pasting, can be understood in the framework of nested quantum error correction codes.
Research in the field of quantum algorithms and quantum error correction is progressing at an astounding rate. There are many good papers on both subjects, but reading even a few of these may seem a daunting task to the newcomer. The aim of this paper is to give a leisurely introduction to the basic theory of quantum error correcting codes without appealing to even the most basic notions in physics. Thus the article is not a substitute for important papers such as [12] or [7] but rather an advertisement for them. I would be pleased if, in addition, some readers view this as a useful companion article if and when they go on to read more substantial literature on the subject of quantum error correction. I present nothing new here. Rather, I give an elementary account of the important theorems and proofs which appear in these fundamental works using only undergraduate algebra and a bit of classical coding theory. In particular, I give a full proof of the Knill/Laflamme theorem as well as an elementary treatment of stabilizer codes. The goal is to make the literature dealing with this exciting new area more accessible to discrete mathematicians.
. We introduce a new class of quantum error--correcting codes derived from (classical) Reed--Solomon codes over finite fields of characteristic two. Quantum circuits for encoding and decoding based on the discrete cyclic Fourier transform over finite fields are presented. 1 Introduction During the last years it has been shown that computers taking advantage of quantum mechanical phenomena outperform currently used computers. The striking examples are integer factoring in polynomial time (see [18]) and finding pre--images of an n--ary Boolean function ("searching") in time O( p 2 n ) (see [12]). Quantum computers are not only of theoretical nature---there are several suggestions how to physically realize them (see, e. g., [6, 7]). On the way towards building a quantum computer, one very important problem is to stabilize quantum mechanical systems since they are very vulnerable. A theory of quantum error--correcting codes has already been established (see [15]). Nevertheless, the pr...
We introduce joint difference sets as a generalization of cyclic difference sets, and we construct a new class of quantum error-correcting codes (QECCs) from these joint difference sets. The main benefits of our method are as follows. First, we can construct quantum codes that are both high rate and with large block length, while maintaining good performance. Second, the density of constructed quantum parity check matrix can approach zero when the code length is very large. This allows us to use a simple iterative decoding algorithm. Interestingly, our method yields the well-known [5,1,3] QECC.
Quantum computation has been a rich field of study for decades because it promises possible spectacular advances, some of which may run counter to our classically rooted intuitions. At the same time, quantum computation is still in its infancy in both theoretical and practical areas. Efficient quantum algorithms are very limited in number and scope; no real breakthrough has yet been achieved in physical implementations. Grover's search algorithm can be applied to a wide range of problems; even problems not generally regarded as searching problems can be reformulated to take advantage of quantum parallelism and entanglement leading to algorithms which show a square root speedup over their classical counterparts. This dissertation discusses a systematic way to formulate such problems and gives as an example a quantum scheduling algorithm for an R||C_max problem. This thesis shows that quantum solution to such problems is not only feasible but in some cases advantageous. The complexity of the error correction circuitry forces us to design quantum error correction codes capable of correcting only a single error per error correction cycle. Yet, time-correlated errors are common for physical implementations of quantum systems; an error corrected during a certain cycle may reoccur in a later cycle due to physical processes specific to each physical implementation of the qubits. This dissertation discusses quantum error correction for a restricted class of time-correlated errors in a spin-boson model. The algorithm proposed allows the correction of two errors per error correction cycle, provided that one of them is time-correlated. The algorithm can be applied to any stabilizer code, perfect or non-perfect, and simplified the circuit complexity significantly comparing to the classic quantum error correction codes.
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.