Basics of Quantum Error Correction

Chiara Macchiavello

Rodriguez, Derek

The introduction of quantum computing has presented algorithmic solutions to computationally difficult challenges that are far more efficient than those of classical computers. These algorithms leverage the properties of quantum mechanics to manipulate the quantum properties of subatomic particles, requiring immense precision and stability. Current quantum hardware, however, is too noisy and introduces too many errors for these algorithms to be useful in practice, necessitating the use of error correction algorithms. This field survey seeks to introduce various principles of quantum mechanics relevant to quantum computing and quantum error correction (QEC), detail the implementation and motivations of a basic QEC algorithm, and provide insight into the current research direction.

Simon J. Devitt, William J. Munro, Kae Nemoto

Quantum error correction (QEC) and fault-tolerant quantum computation represent one of the most vital theoretical aspects of quantum information processing. It was well known from the early developments of this exciting field that the fragility of coherent quantum systems would be a catastrophic obstacle to the development of large-scale quantum computers. The introduction of quantum error correction in 1995 showed that active techniques could be employed to mitigate this fatal problem. However, quantum error correction and fault-tolerant computation is now a much larger field and many new codes, techniques, and methodologies have been developed to implement error correction for large-scale quantum algorithms. In response, we have attempted to summarize the basic aspects of quantum error correction and fault-tolerance, not as a detailed guide, but rather as a basic introduction. The development in this area has been so pronounced that many in the field of quantum information, specifically researchers who are new to quantum information or people focused on the many other important issues in quantum computation, have found it difficult to keep up with the general formalisms and methodologies employed in this area. Rather than introducing these concepts from a rigorous mathematical and computer science framework, we instead examine error correction and fault-tolerance largely through detailed examples, which are more relevant to experimentalists today and in the near future.

Victor V. Albert

S. Rajasekar, R. Velusamy

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.

Ivan B. Djordjević