# Quantum Error Correcting

## Contents |

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? The solution is to use transversal gates whenever possible. Study: With Medicaid, ER visits remain high for two years International Policy Lab issues second annual call for proposals to faculty and researchers Sloan School of Management Economist Bengt Holmström’s Nobel H. weblink

F. A particularly useful fact is that a transversal CNOT gate (i.e., CNOT acting between the ith qubit of one block of the QECC and the ith qubit of a second block Barrett, **R. **B. https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

Leibfried, T. 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. Also, note that a single phase error in the cat state will cause the final measurement outcome to be wrong (even and odd switch places), so we should repeat the measurement Itano, J.

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Furthermore, these calculations make a number of assumptions about the physical properties of the computer. Prog. In that case, let us consider tensor products of the Pauli matrices $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}, X=\begin{pmatrix}0&1\\1&0\end{pmatrix}, Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ Define the Pauli group Pn as the group consisting of tensor products of I, X, Quantum Error Correction Book Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply.

The most widely-used structure gives a class of codes known as stabilizer codes. 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 The signs of states in a quantum superposition are important, so we need to be able to correct sign errors as well as bit flip errors. http://news.mit.edu/2015/quantum-error-correction-0526 These include EPR and GHZ states.

Sun, L. Quantum Code 7 E. 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 Freedman, Michael H.; **Meyer, David A.: Projective plane** and planar quantum codes.

## Stabilizer Codes And Quantum Error Correction.

Cornell University Library We gratefully acknowledge support fromthe Simons Foundation and The Alliance of Science Organisations in Germany, coordinated by TIB, MPG and HGF arXiv.org > quant-ph > arXiv:quant-ph/9512032 Search or Therefore, if p is below the threshold pt, we can achieve an arbitrarily good error rate ε per logical gate or timestep using only poly(logε) resources, which is excellent theoretical scaling. Quantum Error Correction For Beginners In quantum error correction, Harrow explains, “These measurement always have the form ‘Does A disagree with B?’ Except it might be, instead of A and B, A B C D E 5 Qubit Code Each row of the parity check matrix can be converted into a Pauli operator by replacing each 0 with an I operator and each 1 with a Z operator.

Then we interact the ancilla with the encoded data qubits using gates from our stock of transversal gates and perform a fault-tolerant measurement. http://vealcine.com/quantum-error/quantum-error-correcting-codes-from-the-compression-formalism.php If an error is modeled by a unitary transform U, which will act on a qubit | ψ ⟩ {\displaystyle |\psi \rangle } , then U {\displaystyle U} can be described However, quantum error correction and fault-tolerant computation is now a much larger field and many new codes, techniques, and methodologies have been developed to implement error correction for large scale quantum J. Steane Code

A quantum error correcting code protects quantum information against errors of a limited form. DiCarlo, S. The computational complexity of the encoder is frequently a great deal lower than that of the decoder. check over here Then the bit flip code from above can recover | ψ ⟩ {\displaystyle |\psi \rangle } by transforming into the Hadamard basis before and after transmission through E phase {\displaystyle E_{\text{phase}}}

If we cannot stop it from interacting with the environment, it will be no better than a classical computer. 5-qubit Quantum Error Correction Each measurement gives us one bit of the error syndrome, which we then decipher classically to determine the actual error. By using this site, you agree to the Terms of Use and Privacy Policy.

## This theorem seems to present an obstacle to formulating a theory of quantum error correction.

It is possible to correct for both types of errors using one code, and the Shor code does just that. H. Sloane ([2], [3]); these are also called additive codes. Bit Flip Memory Error 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

We could repeat this procedure to get an n3-qubit code, and so forth. Let E phase {\displaystyle E_{\text{phase}}} be a quantum channel that can cause at most one phase flip. 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. this content 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