# Quantum Error Correction Code

## Contents |

Found. Fault-Tolerant Measurement and Error Correction Since all our gates are unreliable, including those used to correct errors, we will need some sort of fault-tolerant quantum error correction procedure. Despite being efficiently simulable, most stabilizer states on a large number of qubits exhibit maximal bipartite entanglement[Dahlsten and Plenio, QIC 2006]. Through the transmission in a channel the relative sign between | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } can become inverted. weblink

Niset, **G. **Unfortunately, it does not appear to be possible to perform universal quantum computations using just transversal gates. If the classical distance d = 2t + 1, the quantum code can correct t bit flip (X) errors, just as could the classical code. So we’re hoping that will be the case for ours, too.” Stephen Bartlett, a physics professor at the University of Sydney who studies quantum computing, doesn’t find the additional qubits required https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

However, in order to have a stabilizer code at all, the generators produced by the above procedure must commute. Cambridge University Press. ^ W.Shor, Peter (1995). "Scheme for reducing decoherence in quantum computer memory". Shor (AT&T Research) (Submitted on 30 Dec 1995 (v1), last revised 16 Apr 1996 (this version, v2)) Abstract: A quantum error-correcting code is defined to be a unitary mapping (encoding) of 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.

- Cerf and U.
- Contents 1 The bit flip code 2 The sign flip code 3 The Shor code 4 General codes 5 Models 6 Experimental realization 7 See also 8 References 9 Bibliography 10
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- In most codes, the effect is either a bit flip, or a sign (of the phase) flip, or both (corresponding to the Pauli matrices X, Z, and Y).
- Note that this gives us a finite generating set of gates.
- We give a simple circuit which takes the initial state with four extra qubits in the state |0> to the encoded state.
- Because of the linearity of quantum mechanics, we can always take the set of errors E to be a linear space: If a QECC corrects Ea and Eb, it will also
- We use the notation [[n, k, d]] to a refer to such a stabilizer code.

This theorem seems to present an obstacle to formulating a theory of quantum error correction. Cornell University Library We gratefully acknowledge support fromthe Simons Foundation and member institutions arXiv.org > quant-ph > arXiv:0904.2557 Search or Article-id (Help | Advanced search) All papers Titles Authors Abstracts G. Quantum Error Correction Book A 71, 052332 (2005), doi:10.1103/PhysRevA.71.052332 ^ J.

A distance d stabilizer code which has nontrivial P ∈ S with wt(P) < d is called degenerate, whereas one which does not is non-degenerate. Conversely, we are also interested in the problem of setting upper bounds on achievable values of (logK)/n and d/n. Lett. 81, 2152–2155 (1998), doi:10.1103/PhysRevLett.81.2152 ^ T. http://arxiv.org/abs/quant-ph/9602019 Sun, L.

And that’s what’s really got people excited. Quantum Code 7 Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Jost, E. Now these qubits will be sent through a channel E bit {\displaystyle E_{\text{bit}}} where we assume that at most one bit flip may occur.

## Stabilizer Codes And Quantum Error Correction.

Itano, J. https://arxiv.org/abs/0904.2557 Of these, only the assumption of independent errors is at all necessary, and that can be considerably relaxed to allow short-range correlations and certain kinds of non-Markovian environments. Quantum Error Correction For Beginners Unfortunately, the practical requirements for this result are not nearly so good. Steane Code Found.

The simplest method, due to Shor, is very general but also requires the most overhead and is frequently the most susceptible to noise. have a peek at these guys **Math. **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. Category:Introductory Tutorials Category:Quantum Error Correction Category:Handbook of Quantum InformationLast modified:Monday, October 26, 2015 - 17:56 ERROR The requested URL could not be retrieved The following error was encountered while trying to 5 Qubit Code

In addition, CSS codes have some very useful properties which make them excellent choices for fault-tolerant quantum computation. Maas, E. Comments: 4 pages (including figures), latex file using RevTex Subjects: Quantum Physics (quant-ph) Citeas: arXiv:quant-ph/9602019 (or arXiv:quant-ph/9602019v1 for this version) Submission history From: Raymond LaFlamme [view email] [v1] Tue, 27 check over here The theory of fault-tolerant quantum computation tells us how to perform operations on states encoded in a quantum error-correcting code without compromising the code's ability to protect against errors.

We use the notation ((n, K, d)) to refer to an ((n, K)) QECC with distance d. 5-qubit Quantum Error Correction The best rigorous proofs of the threshold to date show that the threshold is at least 2 × 10 − 5 (meaning one error per 50, 000 operations). The latter is counter-intuitive at first sight: Since noise is arbitrary, how can the effect of noise be one of only few distinct possibilities?

## So a single qubit can not be repeated three times as in the previous example, as any measurement of the qubit will change its wave function.

H. Rev. A 5-qubit code is the smallest possible code which protects a single logical qubit against single-qubit errors. Bit Flip Memory Error Zurek, T.

B. **D. **Shor is also responsible for the theoretical result that put quantum computing on the map, an algorithm that would enable a quantum computer to factor large numbers exponentially faster than a http://vealcine.com/quantum-error/quantum-error-correction-ppt.php If distinct of the set of correctable errors produce orthogonal results, the code is considered pure.[3] Models[edit] Over time, researchers have come up with several codes: Peter Shor's 9-qubit-code, a.k.a.

We convert H1 into stabilizer generators as above, replacing each 0 with I and each 1 with Z. The researchers’ protocol performs one of those agreement measurements on all three qubits, modifying the state of any qubit that’s out of alignment with the other two. Calderbank, Peter Shor and Andrew Steane. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty

Cornell University Library We gratefully acknowledge support fromthe Simons Foundation and member institutions arXiv.org > quant-ph > arXiv:quant-ph/9512032 Search or Article-id (Help | Advanced search) All papers Titles Authors Abstracts To determine whether a given subspace is able to correct a given set of errors, we can apply the quantum error-correction conditions: Theorem 1 A QECC C corrects the set of Revised April 1996 to give more intuition and an example. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Quantum error correction From Wikipedia, the free encyclopedia Jump to: navigation, search Quantum error correction is used in quantum

Sabuncu, A. Sloane ([2], [3]); these are also called additive codes. In the case of certain codes, such as the 7-qubit code, a number of different gates can be performed transversally. The Clifford group on n qubits is defined as the set of unitary operations which conjugate the Pauli group Pn into itself; Cn can be generated by the Hadamard transform, the

Ser., Chapman & Hall/CRC, Boca Raton, FL, 2002. Andersen, "Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code," Nature Photonics 4, 700 (2010), doi:10.1038/nphoton.2010.168 Bibliography[edit] Daniel Lidar and Todd Brun, ed. (2013). "Quantum Error Correction". Please try the request again. We encode a 0 as 000 and a 1 as 111.

This is a big problem for a quantum computer.