Overhead and noise threshold of fault-tolerant quantum error correction

Fault tolerant quantum error correction (QEC) networks are studied by a combination of numerical and approximate analytical treatments. The probability of failure of the recovery operation is calculated for a variety of CSS codes, including large block codes and concatenated codes. Recent insights into the syndrome extraction process, which render the whole process more efficient and more noise-tolerant, are incorporated. The average number of recoveries which can be completed without failure is thus estimated as a function of various parameters. The main parameters are the gate (γ) and memory (ǫ) failure rates, the physical scale-up of the computer size, and the time tm required for measurements and classical processing. The achievable computation size is given as a surface in parameter space. This indicates the noise threshold as well as other information. It is found that concatenated codes based on the [[23, 1, 7]] Golay code give higher thresholds than those based on the [[7, 1, 3]] Hamming code under most conditions. The threshold gate noise γ0 is a function of ǫ/γ and tm; example values are {ǫ/γ, tm, γ0} = {1, 1, 10 }, {0.01, 1, 3×10}, {1, 100, 10}, {0.01, 100, 2×10}, assuming zero cost for information transport. This represents an order of magnitude increase in tolerated memory noise, compared with previous calculations, which is made possible by recent insights into the fault-tolerant QEC process. The possibility of robust storage and manipulation of quantum information has profound practical and theoretical implications. It would allow highly complex quantum interference and entanglement phenomena, including quantum computing, to be realized in the laboratory, and it also underlies a new and as yet little understood area of physics concerning the thermodynamics of complex entangled quantum systems. The challenge of achieving precise manipulation of quantum information has inspired much ingenuity, and many established methods of experimental physics, such as adiabatic passage, geometric phases, spin echo and their generalizations can be useful. These provide an improvement in the precision of some driven evolution by a given factor at a cost in speed, for example a slow-down of the evolution by the same factor. Such methods may play a useful role in a quantum computer, but they cannot provide all the stability required, for two reasons. First the slow-down is unacceptable when large quantum algorithms are contemplated, and secondly it is doubtful whether they will in practice achieve the relative precision of order 1/KQ which is needed to allow a successful computation involving Q elementary steps on K logical qubits, when KQ reaches values ≫ 10 which are needed for computations large enough that a quantum computer could out-perform the best available classical computer. Quantum error correction (QEC) [1, 2, 3, 4] may allow a precision ≪ 10 per logical operation to be attained in quantum computers. In order for this to be possible, QEC must be applied in a fault-tolerant manner, that is, the QEC process is constructed so that it removes more noise than it generates when it is itself imperfect. The main concepts of fault-tolerance were introduced by Shor [5], and further insights have been discovered by several authors [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Most of these studies have