I make a rough estimate of the accuracy threshold for fault tolerant quantum computing with concatenated codes. First I consider only gate errors and use the depolarizing channel error model. I will follow P.Shor (quant-ph/9505011) for fault tolerant error correction (FTEC) and the fault tolerant implementation of elementary operations on states encoded by the 7-qubit code. A simple computer simulation suggests a threshold for gate errors of the order ε\approx 10^{-3} or better. I also give a simple argument that the threshold for memory errors is about 10 times smaller, thus ε\approx 10^{-4}.
The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially encoding and decoding, implementing gates and quantum error correction will be considered error free. Finally we will relax this non realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarizing channel will be able to provide a simple estimation of the storage or memory computation error threshold: < 5.2 10-5. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code and the Shor's fault-tolerant method to measure the stabilizer's generators is used.
This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when
η, the probability for an error in a qubit or a gate, is smaller than a constant threshold
ηc. This result improves on Shor's result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56–65], which shows how to perform fault-tolerant quantum computation when the error rate
η decays polylogarithmically with the size of the computation, an assumption which is physically unreasonable. The cost of making the quantum circuit fault-tolerant in our construction is polylogarithmic in time and space. Our result holds for a very general local noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault-tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with
p>2 states, namely, p-qudits, and is also generalized to one-dimensional quantum computers with only nearest-neighbor interactions. No measurements, or classical operations, are required during the quantum computation. We estimate the threshold of our construction to be
ηc≃10−6, in the best case. By this we show that local noise is in principle not an obstacle for scalable quantum computation. The main ingredient of our proof is the computation on states encoded by a quantum error correcting code (QECC). To this end we introduce a special class of Calderbank–Shor–Steane (CSS) codes, called polynomial codes (the quantum analogue of Reed–Solomon codes). Their nice algebraic structure allows all of the encoded gates to be transversal. We also provide another version of the proof which uses more general CSS codes, but its encoded gates are slightly less elegant. To achieve fault tolerance, we encode the quantum circuit by another circuit by using one of these QECCs. This step is repeated polyloglog many times, each step slightly improving the effective error rate, to achieve the desired reliability. The resulting circuit exhibits a hierarchical structure, and for the analysis of its robustness we borrow terminology from Khalfin and Tsirelson [Found. Phys., 22 (1992), pp. 879–948] and Gács [Advances in Computing Research: A Research Annual: Randomness and Computation, JAI Press, Greenwich, CT, 1989]. The paper is to a large extent self-contained. In particular, we provide simpler proofs for many of the known results we use, such as the fact that it suffices to correct for bit-flips and phase-flips, the correctness of CSS codes, and the fact that two-qubit gates are universal, together with their extensions to higher-dimensional particles. We also provide full proofs of the universality of the sets of gates we use (the proof of universality was missing in Shor's paper). This paper thus provides a self-contained and complete proof of universal fault-tolerant quantum computation in the presence of local noise.
"The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially, encoding and decoding, implementing gates and quantum error correction will be considered error-free. Finally, we shall relax this non-realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits us to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarising channel will be able to provide a simple estimate of the storage or memory computation error threshold: 5.2 10. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code, and Shor´s fault-tolerant method is used to measure the stabiliser´s generators."
The error correcting capabilities of the Calderbank-Shor-Steane [[7,1,3]] quantum code, together with a fault-tolerant syndrome extraction by means of several ancilla states, have been numerically studied. A simple probability expression to characterize the code ability for correcting an encoded qubit has been considered. This probability, as a correction quality criterion, permits the error correction capabilities among different recovery schemes to be compared. The memory error threshold is calculated by means of the best method of those considered.
The error correcting capabilities of the Calderbank-Shor-Steane [[7,1,3]] quantum code, together with a fault-tolerant syndrome extraction by means of several ancilla states, have been numerically studied. A simple probability expression to characterize the code ability for correcting an encoded qubit has been considered. This probability, as a correction quality criterion, permits the error correction capabilities among different recovery schemes to be compared. The memory error threshold is calculated by means of the best method of those considered.