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# Quantum Error Correction Tutorial

## Contents

Your cache administrator is webmaster. By using this site, you agree to the Terms of Use and Privacy Policy. If U = i σ y {\displaystyle U=i\sigma _{y}} then both a bit flip error and a sign flip error occur. Publisher conditions are provided by RoMEO. news

A newer idea is Alexei Kitaev's topological quantum codes and the more general idea of a topological quantum computer. It is possible that a double-bit error occurs and the transmitted message is equal to three zeros, but this outcome is less likely than the above outcome. Then the Pauli operators of weight t or less form a basis for the set of all errors acting on t or fewer qubits, so a QECC which corrects these Pauli Cambridge University Press. ^ W.Shor, Peter (1995). "Scheme for reducing decoherence in quantum computer memory". https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

Sundar RajanRead full-textData provided are for informational purposes only. Arxiv pre-print, quant-ph/. 3 lectures on quantum error correction. Stabilizer codes have a special relationship to a finite subgroup Cn of the unitary group U(2n) frequently called the Clifford group. D.

1. A.
2. If we are instead interested in erasure errors, when the location of the error is known but not its precise nature, a distance d code corrects d − 1 erasure errors.
3. M.
4. Zoller, eds., pp. 1--32 (IOS Press, Amsterdam 2006).
5. 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

The product is in the Clifford group, and is only performed if the measurement outcome is 1. A syndrome measurement can determine whether a qubit has been corrupted, and if so, which one. Frank Gaitan (2008). "Quantum Error Correction and Fault Tolerant Quantum Computing". Steane Code Ozeri and D.

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 Daniel Lidar and Todd Brun, ed. (2013). "Quantum Error Correction". Quantum Error Correction Codes Zurek, T. H. That these codes allow indeed for quantum computations of arbitrary length is the content of the threshold theorem, found by Michael Ben-Or and Dorit Aharonov, which asserts that you can correct

Jost, E. Quantum Error Correction Book Specifically: Definition 1 Let Hn be a 2n-dimensional Hilbert space (n qubits), and let C be a K-dimensional subspace of Hn. CSS Codes CSS codes are a very useful class of stabilizer codes invented by Calderbank and Shor, and by Steane. The Pauli group Pk, however, can be performed transversally on any stabilizer code.

## Quantum Error Correction Codes

Full-text · Article · Sep 2013 Vladimir P. https://www.researchgate.net/publication/260401433_A_Tutorial_on_Quantum_Error_Correction_A_Tutorial_on_Quantum_Error_Correction_2 Note that the error syndrome does not tell us anything about the encoded state, only about the error that has occurred. Quantum Error Correction For Beginners J. Stabilizer Codes And Quantum Error Correction. pt is the threshold for fault-tolerant quantum computation.

Then C is an ((n,K)) (binary) quantum error-correcting code (QECC) correcting the set of errors E = {Ea} iff ∃R s.t. http://caribtechsxm.com/quantum-error/quantum-error-correction-usc.php In addition, if we write any superoperator S in terms of its operator-sum representation S(ρ) ↦ ∑AkρAk † , a QECC that corrects the set of errors {Ak} automatically corrects S as well. One of the central problems in the theory of quantum error correction is to find codes which maximize the ratios (logK)/n and d/n, so they can encode as many qubits as The system returned: (22) Invalid argument The remote host or network may be down. 5-qubit Quantum Error Correction

First we create special encoded ancilla states in a non-fault-tolerant way, but perform some sort of check on them (in addition to error correction) to make sure they are not too Furthermore, these calculations make a number of assumptions about the physical properties of the computer. D. More about the author Barreiro, T.

More sophisticated techniques for fault-tolerant error correction involve less interaction with the data but at the cost of more complicated ancilla states. 5 Qubit Code To diagnose bit flips in any of the three possible qubits, syndrome diagnosis is needed, which includes four projection operators: P 0 = | 000 ⟩ ⟨ 000 | + | These are encoding, extraction of syndromes, error operators, and code construction.

## Given a codeword of a particular [[n, 1]] QECC, we can take each physical qubit and again encode it using the same code, producing an [[n2, 1]] QECC.

Note that the Gilbert-Varshamov bound simply states that codes at least this good exist; it does not suggest that better codes cannot exist. Y. M. Fault-tolerant Quantum Computation 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

Here R {\displaystyle {\mathcal {R}}} is known as the correction operation. Knill, R. Fault-Tolerant Gates We will focus on stabilizer codes. click site The fault-tolerant procedures concatenate as well, and after L levels of concatenation, the effective logical error rate is pt(p/pt)2L (for a base code correcting 1 error).

The 7-qubit code is much studied because its properties make it particularly well-suited to fault-tolerant quantum computation. 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. Steane, Proceedings of the International School of Physics Enrico Fermi", course CLXII, Quantum Computers, Algorithms and Chaos'', G. Each quantum error correcting code allows a subset of these errors to be corrected.

T. The errors are assumed to be independent and uncorrelated between qubits except when a gate connects them.