There are an infinite number of ways to represent a classical bit with a quantum state. By taking advantage of this fact, one can develop methods for reducing error rates in computation, communication and sensing that do not require the use of quantum codes. But how well do they work? In this paper, we give results that can be used to quantify the improvement offered by such methods.
There are an infinite number of ways to represent a classical bit with a quantum state. By taking advantage of this fact, one can develop methods for reducing error rates in computation, communication and sensing that do not require the use of quantum codes. But how well do they work? In this paper, we give results that can be used to quantify the improvement offered by such methods.
All free-space quantum communications require the use of a quantum channel, which transports quantum bits in such a way that the quantum mechanical states of the qubits remain preserved from one end of the channel to the other one. In quantum computing the classical error coding methods could not be used, however we can construct a classical channel with zero redundancy error correction for any unitary channel. In our basically new quantum error correction approach, the classical states are coded into the eigenvectors and unitary transformations. In this paper, we show that with our new algorithm it’s possible to create redundancy-free quantum error correction. We also consider the redundancy-free implementation of a unitary error correcting operator. Our protocol achieves the redundancy-free quantum communication using local unitary operations and unitary matrices. These solutions could be useful for the free-space quantum communication.
With the rapid improvement of quantum computing technology, quantum key distribution(QKD) is a hot technology. Information reconciliation is a key step of QKD which is useful for correcting key error. Classical message interaction is necessary in a practical information reconciliation scheme, which makes the efficiency of these protocols decreased. Therefore, some one-way information reconciliation schemes based on low-density parity-check(LDPC) codes and polar codes are proposed. Here we propose a concatenated method of IR schemes which can achieve any given error rate level without the need of interactions. Compared with the one-way IR schems based on LDPC codes and polar codes, the IR schemes based on the proposed concatenated method can get lower bit error rates after error correction, which can also reduce the communication delay and system complexity of QKD, improve the final key generation rate and enhance the practicability of QKD system.
The limits of classical information theory are stretched at several points by the consideration of quantum mechanics. Entanglement in itself is a unique phenomenon, and its application has enabled solutions such as super-dense compression, teleportation or super-activation of quantum channels. Noisy classical channels can only be made error-free by adding redundancy, and even this can only be done for certain channels. However, in this article, we examine an equally unique possibility, namely, what properties can make the quantum channels error-free without adding redundancy just by using appropriate quantum transformations. We will show the quantum coding required for this and illustrate its efficiency through examples.
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.
Quantum Key Distribution is meant to be an ultimate computer security system that will not need upgrade/overhaul from time to time. QKD allows the generation of long key length on demand. QKD, coupled with the unconditional secure "one-time pad" encryption system, will be unbreakable for eavesdropper with infinite resources. Numerous experimental QKD prototypes have demonstrated that QKD is likely to be a reality before quantum computer. One of the steps in a Quantum Key Distribution protocol is to remove the transmission errors from quantum communication, which typically has a high error rate (one percent or higher). An interactive error control method is utilized in the reconciliation of QKD. This procedure divides transmitted bits into blocks. The size of the block is chosen so that the chance of having multiple error bits in one block is small. By checking parity and doing interactive BINARY search when a parity error is found, error bits can be located and removed. Some error bits escape the detection from the first pass. By repeating this procedure several times, each time randomly dividing bits into blocks, most error bits can be detected and removed. Each parity check means loss of one bit. The goal is to minimize the number of parity checks to locate all (or most) errors and to have a high reliability that the remaining bits have very small residue error rate. Brassard and Salvail devised a better error control procedure in 1993. By keeping track of block parity information from pass to pass, which they called CASCADE, the number of parity checks per error bit is reduced. They observed no errors left after four passes from 10 simulation runs. However, they offered upper bound of residue error rate that was much higher than observed from simulation. Therefore, there is a need to find the optimal block size and to improve the residue error rate analysis of this procedure. In this work, improvement in reconciliation procedure is obtained in throughput (i.e. the number of input bits minus the number of parity checks) and in lower residue error rate. Throughput is doubled at high input error rate and less throughput gain at low input error rate. Three most important factors that enhance the system throughput are: 1) Using larger block size to fully utilize the power of CASCADE method. 2) Avoiding any pair of bits to stay in the same block of any later pass, which allows the use of three passes only (instead of four passes in the BS procedure). 3) Some bits are already known to be lost from the current pass, so I avoid carrying these bits to the later pass. The residue error rate at the end of the second pass (and the third pass) for small initial block size is derived. The residue error rate at the end of the third pass is found much lower than the upper bound given by the previous workers. This residue error rate is found to be a function of input size and the block size. This error control method has many similar features and difficulties to the recent development of "turbo-code" in the error control-coding field.
Methods of finding good quantum error-correcting codes are discussed, and many example codes are presented. The recipe ${\mathit{C}}_{2}^{\mathrm{\ensuremath{\perp}}}$\ensuremath{\subset}C1, where C1 and C2 are classical codes, is used to obtain codes for up to 16 information quantum bits (qubits) with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C1 to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error-correcting codes for 3, 4, and 5 information qubits with block lengths of 8, 10, and 11 qubits, respectively. \textcopyright{} 1996 The American Physical Society.