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# Quantum Error Correction Wikipedia

## Contents

This theorem seems to present an obstacle to formulating a theory of quantum error correction. Adding one half of an ebit to the receiver's side gives the canonical stabilizer whose simultaneous +1-eigenstate is the above state. So even if the error due to the noise was arbitrary, it can be expressed as a superposition of basis operations—the error basis (which is here given by the Pauli matrices Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. have a peek at this web-site

Rains, P. The quantum Singleton bound (or Knill-Laflamme bound) states that any ((n, K, d)) QECC must satisfy n − logK ≥ 2d − 2 We can set a lower bound on the existence of QECCs using the quantum Gilbert-Varshamov bound, An anticommuting error E {\displaystyle E} is detectable by measuring each element g {\displaystyle g} in S {\displaystyle {\mathcal {S}}} and computing a syndrome r {\displaystyle \mathbf {r} } identifying E We perform a multi-qubit measurement that does not disturb the quantum information in the encoded state but retrieves information about the error.

## Stabilizer Codes And Quantum Error Correction

By using this site, you agree to the Terms of Use and Privacy Policy. The matrix becomes [ 0 0 0 0 1 0 0 1 1 1 1 0 1 1 0 0 | 1 0 0 0 0 1 0 0 1 1 A noiseless quantum channel or the encoded use of noisy quantum channel are two different ways to build up entanglement between a sender and receiver.

It is equal to zero in this case so we need not do anything. R2493–R2496, Oct 1995. So we compactly summarize the stabilizer error-correcting conditions: a stabilizer code can correct any errors E 1 , E 2 {\displaystyle E_{1},E_{2}} in E {\displaystyle {\mathcal {E}}} if E 1 † Quantum Error Correction Book By using this site, you agree to the Terms of Use and Privacy Policy.

The theory of quantum error-correcting codes has been developed to counteract noise introduced in this way. Quantum Code 7 In addition, CSS codes have some very useful properties which make them excellent choices for fault-tolerant quantum computation. The rate pair is ( k / n , c / n ) {\displaystyle \left(k/n,c/n\right)} for a code with the above parameters. https://en.wikipedia.org/wiki/Quantum_convolutional_code Using the stabilizer formalism limits the available states but there is still a lot of interesting freedom.

Let F ( Π Z + ) {\displaystyle F(\Pi ^{\mathbb {Z} ^{+}})} denote the set of Pauli sequences with finite support. Quantum Code Burner 7.0 1 Available at http://arxiv.org/abs/0806.4214 Mark M. To diagnose bit flips in any of the three possible qubits, syndrome diagnosis is needed, which includes four projection operators: P 0 = | 000 ⟩ ⟨ 000 | + | Measuring the syndrome collapses this operation into a correctible error.

## Quantum Code 7

Then we complete the operation with a further transversal gate which depends on the outcome of the measurement. https://en.wikipedia.org/wiki/Wikipedia:Cleanup_Taskforce/Quantum_error_correction A more general class of codes (encompassing the former) are the stabilizer codes discovered by Daniel Gottesman ([1]), and by A. Stabilizer Codes And Quantum Error Correction It uses entanglement and syndrome measurements and is comparable in performance with the repetition code. 5 Qubit Code A procedure due to Knill (for any stabilizer code) teleports the data qubit through an ancilla consisting of two blocks of the QECC containing an encoded Bell state $\left|\overline{00}\right\rangle + \left|\overline{11}\right\rangle$.

Brun, Igor Devetak, and Min-Hsiu Hsieh. Check This Out Each measurement gives us one bit of the error syndrome, which we then decipher classically to determine the actual error. In general, a quantum error-correcting code is a subspace of a Hilbert space designed so that any of a set of possible errors can be corrected by an appropriate quantum operation. The encoding circuit for a quantum convolutional code has a much lower complexity than an encoding circuit needed for a large block code. Bit Flip Memory Error

Generated Tue, 06 Dec 2016 04:29:36 GMT by s_ac16 (squid/3.5.20) Wiley-IEEE Press, 1999. Let us define a map N : ( Z 2 ) 2 n → Π n {\displaystyle \mathbf {N} :\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \Pi ^{n}} as follows: N ( u ) ≡ http://johnlautner.net/quantum-error/quantum-error-correction-ppt.html C This article has been rated as C-Class on the project's quality scale.

A procedure due to Steane uses (forCSS codes) one ancilla in a logical $\left|\overline{0}\right\rangle$ state of the same code and one ancilla in a logical $\left|\overline{0}\right\rangle + \left|\overline{1}\right\rangle$ state. Steane Code Wilde and Todd A. Rev.

## 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.

R. Thus should be distinguished from the encoding operation which maps HlogK into Hn, determining the imbedding of C. W. Fault Tolerant Quantum Computation Quantum Convolutional Coding with Shared Entanglement: General Structure.

A number of different techniques have been developed. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Skip to main content Quantiki Toggle navigation Browse News Forums Video Abstracts Journal Articles RSS Feeds Journal Articles News Quantum convolutional codes are similar because some of the qubits feed back into a repeated encoding unitary and give the code a memory structure like that of a classical convolutional code. have a peek here Brun.

An operation consisting only of single-qubit gates is automatically transversal. The matrix becomes [ 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 | 0 0 0 0 1 0 0 0 0 1 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. 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.

Available at http://arxiv.org/abs/quant-ph/0602129 R. I'm not promising anything, though! A syndrome measurement can determine whether a qubit has been corrupted, and if so, which one. The set Π = { I , X , Y , Z } {\displaystyle \Pi =\left\{I,X,Y,Z\right\}} consists of the Pauli operators: I ≡ [ 1 0 0 1 ] ,

The matrix becomes [ 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 | 0 0 0 0 1 0 0 0 0 1 A 78, 012341 (2008). For example, in the case where the first qubit is flipped, the result would be | ψ r ′ ⟩ = α 0 | 100 ⟩ + α 1 | 011 Row operations do not affect the error-correcting properties of the code but are crucial for arriving at the optimal decomposition from the fundamental theorem of symplectic geometry.

Theorem 2 Let S be a stabilizer with n − k generators, and let S ⊥  = {E ∈ Pn s.t. [E, M] = 0 ∀M ∈ S}.