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 (gamma) and memory (epsilon) failure rates, the physical scale-up of the computer size, and the time t_m 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 gamma_0 is a function of epsilon/gamma and t_m; example values are {epsilon/gamma, t_m, gamma_0} = {1, 1, 0.001}, {0.01, 1, 0.003}, {1, 100, 0.0001}, {0.01, 100, 0.002}, 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.Comment: 21 pages, 12 figures, minor mistakes corrected and layout improved, ref added; v4: clarification of assumption re logic gate

Efficient fault-tolerant quantum computing

Fault tolerant quantum computing methods which work with efficient quantum error correcting codes are discussed. Several new techniques are introduced to restrict accumulation of errors before or during the recovery. Classes of eligible quantum codes are obtained, and good candidates exhibited. This permits a new analysis of the permissible error rates and minimum overheads for robust quantum computing. It is found that, under the standard noise model of ubiquitous stochastic, uncorrelated errors, a quantum computer need be only an order of magnitude larger than the logical machine contained within it in order to be reliable. For example, a scale-up by a factor of 22, with gate error rate of order $10^{-5}$, is sufficient to permit large quantum algorithms such as factorization of thousand-digit numbers.Comment: 21 pages plus 5 figures. Replaced with figures in new format to avoid problem

Local Fault-tolerant Quantum Computation

We analyze and study the effects of locality on the fault-tolerance threshold for quantum computation. We analytically estimate how the threshold will depend on a scale parameter r which estimates the scale-up in the size of the circuit due to encoding. We carry out a detailed semi-numerical threshold analysis for concatenated coding using the 7-qubit CSS code in the local and `nonlocal' setting. First, we find that the threshold in the local model for the [[7,1,3]] code has a 1/r dependence, which is in correspondence with our analytical estimate. Second, the threshold, beyond the 1/r dependence, does not depend too strongly on the noise levels for transporting qubits. Beyond these results, we find that it is important to look at more than one level of concatenation in order to estimate the threshold and that it may be beneficial in certain places, like in the transportation of qubits, to do error correction only infrequently.Comment: REVTeX, 44 pages, 19 figures, to appear in Physical Review

Numerical simulation of information recovery in quantum computers

Decoherence is the main problem to be solved before quantum computers can be built. To control decoherence, it is possible to use error correction methods, but these methods are themselves noisy quantum computation processes. In this work we study the ability of Steane's and Shor's fault-tolerant recovering methods, as well a modification of Steane's ancilla network, to correct errors in qubits. We test a way to measure correctly ancilla's fidelity for these methods, and state the possibility of carrying out an effective error correction through a noisy quantum channel, even using noisy error correction methods.Comment: 38 pages, Figures included. Accepted in Phys. Rev. A, 200

Quantum Computing with Very Noisy Devices

In theory, quantum computers can efficiently simulate quantum physics, factor large numbers and estimate integrals, thus solving otherwise intractable computational problems. In practice, quantum computers must operate with noisy devices called ``gates'' that tend to destroy the fragile quantum states needed for computation. The goal of fault-tolerant quantum computing is to compute accurately even when gates have a high probability of error each time they are used. Here we give evidence that accurate quantum computing is possible with error probabilities above 3% per gate, which is significantly higher than what was previously thought possible. However, the resources required for computing at such high error probabilities are excessive. Fortunately, they decrease rapidly with decreasing error probabilities. If we had quantum resources comparable to the considerable resources available in today's digital computers, we could implement non-trivial quantum computations at error probabilities as high as 1% per gate.Comment: 47 page

Quantum Teleportation is a Universal Computational Primitive

We present a method to create a variety of interesting gates by teleporting quantum bits through special entangled states. This allows, for instance, the construction of a quantum computer based on just single qubit operations, Bell measurements, and GHZ states. We also present straightforward constructions of a wide variety of fault-tolerant quantum gates.Comment: 6 pages, REVTeX, 6 epsf figure

Active stabilisation, quantum computation and quantum state synthesis

Active stabilisation of a quantum system is the active suppression of noise (such as decoherence) in the system, without disrupting its unitary evolution. Quantum error correction suggests the possibility of achieving this, but only if the recovery network can suppress more noise than it introduces. A general method of constructing such networks is proposed, which gives a substantial improvement over previous fault tolerant designs. The construction permits quantum error correction to be understood as essentially quantum state synthesis. An approximate analysis implies that algorithms involving very many computational steps on a quantum computer can thus be made possible.Comment: 8 pages LaTeX plus 4 figures. Submitted to Phys. Rev. Let

Optimal correction of concatenated fault-tolerant quantum codes

We present a method of concatenated quantum error correction in which improved classical processing is used with existing quantum codes and fault-tolerant circuits to more reliably correct errors. Rather than correcting each level of a concatenated code independently, our method uses information about the likelihood of errors having occurred at lower levels to maximize the probability of correctly interpreting error syndromes. Results of simulations of our method applied to the [[4,1,2]] subsystem code indicate that it can correct a number of discrete errors up to half of the distance of the concatenated code, which is optimal.Comment: 7 pages, 2 figures, published versio

Quantum key distribution with 2-bit quantum codes

We propose a prepare-and-measure scheme for quantum key distribution with 2-bit quantum codes. The protocol is unconditionally secure under whatever type of intercept-and-resend attack. Given the symmetric and independent errors to the transmitted qubits, our scheme can tolerate a bit error rate up to 26% in 4-state protocol and 30% in 6-state protocol, respectively. These values are higher than all currently known threshold values for prepare-and-measure protocols. A specific realization with linear optics is given.Comment: Approved for publication in Physical Review Letter