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# Quantum Error Correction Codes

## Contents

Thus, measuring the eigenvalues of the generators of S tells us information about the error that has occurred. Cory, M. Quantum circuit of the Shor code Let E {\displaystyle E} be a quantum channel that can arbitrarily corrupt a single qubit. 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 More about the author

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. Langer, R. Quantum circuit of the bit flip code Let | ψ ⟩ = α 0 | 0 ⟩ + α 1 | 1 ⟩ {\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } Furthermore, for the 7-qubit code, transversal Hadamard performs a logical Hadamard, and the transversal π/4 rotation performs a logical  − π/4 rotation.

## Quantum Error Correction For Beginners

As of late 2004, estimates for this threshold indicate that it could be as high as 1-3% [4], provided that there are sufficiently many qubits available. The first demonstration was with NMR qubits.[4] Subsequently, demonstrations have been made with linear optics,[5] trapped ions,[6][7] and superconducting (transmon) qubits.[8] Other error correcting codes have also been implemented, such as Then measure the ancilla in the basis of ∣ + ⟩ and ∣ − ⟩ = ∣0⟩ − ∣1⟩. 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

• B.
• Laflamme, W.
• In fact, most interactions with the environment will act just like a measurement and will collapse the state.
• We perform a multi-qubit measurement that does not disturb the quantum information in the encoded state but retrieves information about the error.
• A number of different techniques have been developed.

Frunzio, S. Cerf, Ulrik L. The advantage of this procedure is that it measures just M and nothing more. Quantum Error Correction Book F.

Unfortunately, the practical requirements for this result are not nearly so good. D. Generated Tue, 25 Oct 2016 01:30:05 GMT by s_wx1196 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.8/ Connection Generated Tue, 25 Oct 2016 01:30:05 GMT by s_wx1196 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection

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 $\left|\overline{00}\right\rangle + \left|\overline{11}\right\rangle$. Quantum Code 7 In general, a gate coupling pairs of qubits allows errors to spread in both directions across the coupling. Rev. It is possible to correct for both types of errors using one code, and the Shor code does just that.

## Stabilizer Codes And Quantum Error Correction.

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 Nebendahl, D. Quantum Error Correction For Beginners Then we complete the operation with a further transversal gate which depends on the outcome of the measurement. Steane Code Lett. 81, 2152–2155 (1998), doi:10.1103/PhysRevLett.81.2152 ^ T.

Frunzio, S. http://caribtechsxm.com/quantum-error/quantum-computation-quantum-error-correcting-codes-and-information-theory.php Calderbank, Peter W. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. E. 5 Qubit Code

If we want to do computation on a state using noisy gates, we need to know how to perform operations on states that are already encoded. D. L. http://caribtechsxm.com/quantum-error/quantum-error-correction-with-degenerate-codes-for-correlated-noise.php The set of such eigenvalues can be represented as an (n − k)-dimensional binary vector known as the error syndrome.

S ⊥  is the set of Pauli operators that commute with all elements of the stabilizer. 5-qubit Quantum Error Correction Nigg, L. If U = i σ y {\displaystyle U=i\sigma _{y}} then both a bit flip error and a sign flip error occur.

## Reed, L.

This theorem seems to present an obstacle to formulating a theory of quantum error correction. Blakestad, J. 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. Bit Flip Memory Error Chuang (2000). "Quantum Computation and Quantum Information".

Comments: Latex, 23 pages, 1 figure. Monz, V. Barreiro, T. navigate to this website In fact, the Shor code corrects arbitrary single-qubit errors.

Comput. Somaroo, "Experimental Quantum Error Correction," Phys. If we cannot stop it from interacting with the environment, it will be no better than a classical 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.

Cerf and U. On a stabilizer code, therefore, logical Pauli operations can be performed via a transversal Pauli operation on the physical qubits.