Bits as well as qubits are affected by noise in transmission, the process of computation or even storage. We will refer to the error-prone area as a channel. The error is introduced by unwanted but unavoidable interaction with the rest of the world and needs to be corrected before the data can be reliable. Classically this is important in communication systems and there is a whole thriving field of study of classical error correction. In quantum systems, states are intrinsically so fragile that unless they are impossibly isolated from the rest of the world, errors or noise would make computation or communication using them infeasible. Fortunately, it was discovered early in the history of this subject that it is possible to encode qubits in special ways so as to make the information resilient to errors. The qubits are decoded at the end of the computation/communication (schematic of Figure 10.1).
Bits as well as qubits are affected by noise in transmission, the process of computation or even storage. We will refer to the error-prone area as a channel. The error is introduced by unwanted but unavoidable interaction with the rest of the world and needs to be corrected before the data can be reliable. Classically this is important in communication systems and there is a whole thriving field of study of classical error correction. In quantum systems, states are intrinsically so fragile that unless they are impossibly isolated from the rest of the world, errors or noise would make computation or communication using them infeasible. Fortunately, it was discovered early in the history of this subject that it is possible to encode qubits in special ways so as to make the information resilient to errors. The qubits are decoded at the end of the computation/communication (schematic of Figure 10.1).
Today's communications networks are filled with Web pages, e-mails, telephone conversations, video clips, financial transactions, and more, all encoded digitally as bursts of light flashing through fiber-optic cables. All of these messages can be boiled down to sequences of bits, 0's and 1's—usually long sequences, measured in kilobits, megabits, or sometimes even gigabits. But light, like everything else, behaves according to the laws of quantum mechanics, and a sequence of bits is not a good way to describe the properties of a quantum object. Instead, we should turn to quantum bits, or “qubits,” and someday, perhaps, today's network traffic will be supplemented by streams of such quantum information. Quantum transmissions would enable cryptographic systems to detect attempted eavesdropping, and quantum computers capable of manipulating quantum information could solve some problems far more rapidly than any conceivable classical computer. Just as classical information is encoded today in error-correcting codes to protect it against errors in transmission, quantum information will need to be encoded in quantum error-correcting codes in order to reliably reach its destination. Indeed, quantum information is more sensitive to errors than is classical information, so error correction will be even more important. On page 436 of this issue, Brun et al. (1) show how it is possible to improve the performance of quantum error-correcting codes, using a quantum state previously shared between the sender and receiver.
Introduction Quantum error-correcting codes (QECCs) have turned out to have many applications; but their primary purpose, of course, is to protect quantum information from noise. This need manifests itself in two rather different situations. The first is in quantum computing. The qubits in a quantum computer are subject to noise due both to imprecision in the operations (or gates) and to interactions with the external environment. They undergo errors when they are acted upon, and when they are just being stored (though hopefully not at the same rate). All the qubits in the computer must be kept error-free long enough for the computation to be completed. This is the principle of fault tolerance. A rather different situation occurs in quantum communication . Here, the sender and receiver (Alice and Bob) are assumed to be physically separated, and the qubits travel from the sender to the receiver over a (presumably noisy) channel. This channel is assumed to be the dominant source of errors. While the qubits may undergo processing before and after transmission, errors during this processing are considered negligible compared to the channel errors. This picture of transmission through a channel is similar in spirit to the paradigm underlying much of classical information theory.
Communication is the prototypical application of error-correction methods. To communicate, a sender needs to convey information to a receiver over a noisy "communication channel." Such a channel can be thought of as a means of transmitting an information-carrying physical system from one place to another. During transmission, the physical system is subject to disturbances (noise) that can adversely affect the information carried. To use a communication channel, the sender needs to encode the information to be transmitted in the physical system. After transmission, the receiver decodes the information. Quantum error correction is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal with both noise on stored quantum information, and also with faulty quantum gates, faulty quantum preparation,and faulty measurements. In this dissertation, we look at how additional information about the structure of the quantum circuit and noise can improve or alter the performance of techniques in quantum error correction. Chapter 1 and 2, are an introduction to the quantum computation, quantum error correction codes and fault-tolerant quantum computing. These chapters are written to be a useful for students at the graduate and advanced undergraduate level. Also. The first two chapters of this dissertation will be useful to researchers in other fields who would like to understand how quantum error correction and fault-tolerant quantum computing are possible. In chapter 3, we present numerical simulation results comparing the logical error rates for the fault-tolerant [[7, 1, 3]] 's 7 code using the technique of ancilla verification vs. the newer method of ancilla decoding as described in [1]. In chapter 4, we determine how often one should apply error correction. Therefore, we provide a relationship between the logical error rate and the physical error rate for a sequence of logical gates, sometimes followed by noisy quantum error correction
Realization of long distance quantum cryptography requires quantum error correction (QEC) of the errors introduced by noise on communication channels. This chapter begins with presenting common sources of errors in QEC. For quantum information processing qubits using photons, trapped ions, superconducting circuits and spins in semiconductors have been considered. Coherent quantum errors arise due to the errors associated with system controls and/or characterization where imprecise operation on the qubit introduces inaccurate Hamiltonian dynamics. The basic principle involved in classical error correction is to increase the number of bits more than is required for a given amount of information. Digitization of quantum errors makes it possible to correct any quantum error using the ability to correct a finite set of errors. The use of stabilizer codes to describe QEC codes is highly useful. It allows for easy synthesis of correction circuits and also shows how logical operations can be performed directly on encoded data.
The algorithms discussed in Chapter 21 rely on having isolated systems to store information. In practical applications this is not possible, and the environment interacts with the systems causing decoherence. Suppose the data is contained in the state |x〉, and the environment is described by |E〉. The initial state of the entire system is described by the tensor product of the states |X〉⊗|E〉, which evolves according to some unitary operation U. The state |x〉 evolves according to the unitary operation U x which describes the algorithm. In classical error correction codes, all that needs to be corrected are bit flips (see Chapter 10). In the quantum case errors such as bit flips, phase changes and rotations complicate the error correction techniques. Since arbitrary errors in an encoding of information cannot be corrected, only certain types of errors are assumed to occur (this was also the case in classical error correction, where an upper bound in the number of bits that could be flipped in a transmission was assumed). The types of errors depend on the implementation. For example, suppose the types of errors (which we assume are distinguishable due to an encoding) are described by the unitary basis E 1,...E n , so that all errors are a linear combination [138] E = al.E 1 + ... + a n E n ,...E † E = I.
One of the applications of quantum error correction is protecting quantum computers from noise. Certainly, there is nothing particularly quantum mechanical in the idea of protecting information by encoding it. Even ordinary digital computers use various fault tolerance methods at the software level to correct errors during the storage or the transmission of information; e.g., the integrity of the bits stored in hard disks is verified by using parity checks (checksums). In addition, for critical computing systems such as those inside airplanes or nuclear reactors, software fault tolerance methods are also applied during the processing of information; e.g., airplane control computers compare the results from multiple parallel processors to detect faults. In general, however, the hardware of modern computers is remarkably robust to noise, so that for most applications the processing of information can be executed with high reliability without using software error correction. In contrast to the ease and robustness with which classical information can be processed, the processing of quantum information appears at present to be much more challenging. Although constructing reliable quantum computing hardware is certainly a daunting task, we have nevertheless strong hopes that large-scale quantum computers, able to implement useful long computations, can in fact be realized. This optimism is founded on methods of quantum fault tolerance , which show that scalable quantum computation is, in principle, possible against a variety of noise processes.