# Quantum Error Correction And Fault Tolerance

## Contents |

J Math Phys **43:4452MathSciNetMATHCrossRef21.DiVincenzo DP** (2000) The physical implementation of quantum computation. To view the rest of this content please follow the download PDF link above. Copying quantum information is not possible due to the no-cloning theorem. Mikael Lassen, Metin Sabuncu, Alexander Huck, Julien Niset, Gerd Leuchs, Nicolas J. weblink

Cancel Please Select Your Country -- Select one -- Afghanistan Albania Algeria American Samoa Andorra Anguilla Angola Antarctica Antigua And Barbuda Argentina Armenia Aruba Austria Australia Azerbaijan Bahamas Bahrain Bangladesh Barbados On a stabilizer code, therefore, logical Pauli operations can be performed via a transversal Pauli operation on the physical qubits. What is more, the outcome of this operation (the syndrome) tells us not only which physical qubit was affected, but also, in which of several possible ways it was affected. pt is the threshold for fault-tolerant quantum computation. Continued

## Quantum Error Correction For Beginners

The salient point in these error-correction conditions is that the matrix element Cab does not depend on the encoded basis states i and j, which roughly speaking indicates that neither the However, in order to have a stabilizer code at all, the generators produced by the above procedure must commute. Comments: **46 pages, with large margins.**

Please try the request again. By using this site, you agree to the Terms of Use and Privacy Policy. According to the quantum Hamming bound, encoding a single logical qubit and providing for arbitrary error correction in a single qubit requires a minimum of 5 physical qubits. Quantum Threshold Theorem Schindler, J.

Proc Royal Soc London Series A, 454:365–384. An Introduction To Quantum Error Correction And Fault-tolerant Quantum Computation That's the problem it solves.» P≟NP This theoretical computer science–related article is a stub. ACM Comput Surv 34(3):375–408CrossRef24.Forney GD Jr, Grassl M, Guha S (2007) Convolutional and tail‐biting quantum error‐correcting codes. The free VitalSource Bookshelf® application allows you to access to your eBooks whenever and wherever you choose.

Nature 402:390–393CrossRef31.Grassl M (2002) Algorithmic aspects of quantum error‐correcting codes. Daniel Gottesman AT&T Bell Laboratories. ^ A.R.Calderbank E.M.Rains P.W.Shor and N.J.A.Sloane "Quantum Error Correction Via Codes Over GF(4)"IEEE.Transactions on Information Theory,Vol.44,No.4,July 1998 ^ D. Knill, R. It could be through conference attendance, group discussion or directed reading to name just a few examples.

- Comput.
- Blakestad, J.
- D.
- Unfortunately, it does not appear to be possible to perform universal quantum computations using just transversal gates.
- However, by taking appropriate products, we get an infinite set of gates, one that is dense in the unitary group U(2n), allowing universal quantum computation.
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- Thus, for the 7-qubit code, the full logical Clifford group is accessible via transversal operations.
- The 1st, 4th and 7th qubits are for the sign flip code, while the three group of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code.
- The 7-qubit code is much studied because its properties make it particularly well-suited to fault-tolerant quantum computation.
- This theorem seems to present an obstacle to formulating a theory of quantum error correction.

## An Introduction To Quantum Error Correction And Fault-tolerant Quantum Computation

In: Proceedings of 35th Annual Symposium on Fundamentals of Computer Science (FOCS'96). https://www.crcpress.com/Quantum-Error-Correction-and-Fault-Tolerant-Quantum-Computing/Gaitan/p/book/9780849371998 Russ Math Surv 52(6):1191–1249MathSciNetMATHCrossRef43.Knill E (2005) Quantum computing with realistically noisy devices. Quantum Error Correction For Beginners Int J Found Comput Sci 14(5):757–77538.Hamming RW (1986) Coding and information theory. Stabilizer Codes And Quantum Error Correction. The procedure is transversal, so an error on a single qubit in the initial cat state or in a single gate during the interaction will only produce one error in the

Preprint quant-ph/961202871.Zanardi P, Rasetti M (1997) Noiseless quantum codes. have a peek at these guys Preprint quant-ph/050208617.Chau HF (1998) Quantum convolutional codes. Quantum Inf Comput 4(3):222–228MathSciNetMATH61.Reichardt BW (2006) Error‐detection‐based quantum fault tolerance against discrete Pauli noise. The system returned: (22) Invalid argument The remote host or network may be down. Quantum Error Correction Lidar

Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. With the Shor code, a qubit state | ψ ⟩ = α 0 | 0 ⟩ + α 1 | 1 ⟩ {\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } will Since, assuming the cat state is correct, all of its qubits are either ∣0⟩ or ∣1⟩, the procedure either leaves the data state alone or performs M on it uniformly. http://vealcine.com/quantum-error/quantum-error-correction-usc.php Schoelkopf, "Realization of Three-Qubit Quantum Error Correction with Superconducting Circuits," Nature 482, 382-385 (2012), doi:10.1038/nature10786, arXiv:1109.4948 ^ M.

To further lower the logical error rate, we turn to a family of codes known as concatenated codes. Quantum Code Phys Rev A 58(2):905–909MathSciNetCrossRef18.Cleve R, Gottesman D (1997) Efficient computations of encodings for quantum error correction. Princeton University Press, Princeton, pp 43–9856.Nielsen MA, Chuang IL (2000) Quantum computation and quantum information.Cambridge University Press, CambridgeMATH57.Ollivier H, Tillich JP (2003) Description of a quantum convolutional code.

## Preprint quant-ph/960501166.Steane AM (1996) Error correcting codes in quantum theory.

To build a quantum computer, we face an even more daunting task: If our quantum gates are imperfect, everything we do will add to the error. Phys Rev Lett 79(17):3306–3309CrossRef About this Reference Work Entry Title Quantum Error Correction and Fault Tolerant Quantum Computing Reference Work Title Computational Complexity Reference Work Subtitle Theory, Techniques, and Applications Pages It is assumed that measurements and classical computations can be performed quickly and reliably, and that quantum gates can be performed between arbitrary pairs of qubits in the computer, irrespective of Quantum Computer IEEE Trans Inf Theory 48(7):2122–2124MATHCrossRef55.von Neumann J (1956) Probabilistic logic and the synthesis of reliable organisms from unreliable components.

The disadvantage is that it is not transversal, and thus not fault-tolerant. Classical error correcting codes use a syndrome measurement to diagnose which error corrupts an encoded state. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. this content Nebendahl, D.

w ⋅ v = 0 for all v ∈ C. The first step of the three qubit bit flip code is to entangle the qubit with two other qubits using two CNOT gates with input | 0 ⟩ {\displaystyle |0\rangle } Todd Brun, Igor Devetak, and Min-Hsiu Hsieh also constructed the entanglement-assisted stabilizer formalism as an extension of the standard stabilizer formalism that incorporates quantum entanglement shared between a sender and a For lower physical error rates, overhead requirements are more modest, particularly if we only attempt to optimize for calculations of a given size, but are still larger than one would like.

If the three bit flip group (1,2,3), (4,5,6), and (7,8,9) are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code. CRC Press eBooks are available through VitalSource. Pittman, B. February 7, 2008 by CRC Press Textbook - 312 Pages - 24 B/W Illustrations ISBN 9781420006681 - CAT# E7199 eBooks are subject to VAT, which is applied during the checkout process.

A syndrome measurement can determine whether a qubit has been corrupted, and if so, which one.