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# Quantum Error Correcting Codes

## Contents

Raymond Laflamme and collaborators found a class of 5-qubit codes which do the same, which also have the property of being fault-tolerant. 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}}} Lassen, M. Quantum error correction From Wikipedia, the free encyclopedia Jump to: navigation, search Quantum error correction is used in quantum computing to protect quantum information from errors due to decoherence and other weblink

B. Itano, J. 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. A, Vol. 54, No. 2, pp. 1098-1106, 1996 DOI: 10.1103/PhysRevA.54.1098 Citeas: arXiv:quant-ph/9512032 (or arXiv:quant-ph/9512032v2 for this version) Submission history From: Peter W.

## Quantum Error Correction For Beginners

We perform a multi-qubit measurement that does not disturb the quantum information in the encoded state but retrieves information about the error. Suppose we copy a bit three times. The encoded $\left|\overline{0}\right\rangle$ for this code consists of the superposition of all even-weight classical codewords and the encoded $\left|\overline{1}\right\rangle$ is the superposition of all odd-weight classical codewords. Knill, R.

• It uses entanglement and syndrome measurements and is comparable in performance with the repetition code.
• Nielsen and Isaac L.
• Knill, C.
• Any qubit stored unprotected or one transmitted through a communications channel will inevitably come out at least slightly changed.
• If an error is detected, the protocol can trace it back to its origin and correct it.

Wineland, "Realization of quantum error correction," Nature 432, 602-605 (2004), doi:10.1038/nature03074 ^ P. Pittman, B. A classical [n, k, d] linear code (n physical bits, k logical bits, classical distance d) can be defined in terms of an (n − k) × n binary parity check matrix H --- every classical codeword Quantum Error Correction Book M.

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. Stabilizer Codes And Quantum Error Correction. Blakestad, J. R. https://arxiv.org/abs/quant-ph/9512032 If this condition is satisfied, t separate single-qubit or single-gate failures are required for a distance 2t + 1 code to fail.

A transversal operation is one in which the ith qubit in each block of a QECC interacts only with the ith qubit of other blocks of the code or of special Quantum Code 7 J. S ⊥  is the set of Pauli operators that commute with all elements of the stabilizer. 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

## Stabilizer Codes And Quantum Error Correction.

Cerf, Ulrik L. this page D. Quantum Error Correction For Beginners It is frequently useful to consider other representations of stabilizer codes. 5 Qubit Code Britton, W.

We get a similar result in the case where the noise is a general quantum operation on each qubit which differs from the identity by something of size O(ε). have a peek at these guys If distinct of the set of correctable errors produce orthogonal results, the code is considered pure.[3] Models Over time, researchers have come up with several codes: Peter Shor's 9-qubit-code, a.k.a. L. By using this site, you agree to the Terms of Use and Privacy Policy. Steane Code

A transversal operation has the virtue that an error occurring on the 3rd qubit in a block, say, can only ever propagate to the 3rd qubit of other blocks of the A 71, 052332 (2005), doi:10.1103/PhysRevA.71.052332 ^ J. Note that the square brackets specify that the code is a stabilizer code, and that the middle term k refers to the number of encoded qubits, and not the dimension 2k check over here Let C1 be an [n, k1, d1] code and let C2 be an [n, k2, d2] code.

H. 5-qubit Quantum Error Correction The procedure is transversal, so an error on a single qubit in the initial cat state or in a single gate during the interaction will only produce one error in the For H2, we perform the same procedure, but each 1 is instead replaced by X.

## Copying quantum information is not possible due to the no-cloning theorem.

Share Comment Leave a comment Quantum computers are largely theoretical devices that could perform some computations exponentially faster than conventional computers can. Once you perform a measurement on the qubits, however, the superposition collapses, and the qubits take on definite values. The weight wt(P) of a Pauli operator P ∈ Pn is the number of qubits on which it acts as X, Y, or Z (i.e., not as the identity). Bit Flip Memory Error Rev.

For instance, a standard bound on classical error correction is the Hamming bound (or sphere-packing bound), but the analogous quantum Hamming bound k/n ≤ 1 − (t/n)log3 − h(t/n) for [[n, k, 2t + 1]] codes (when n is large) is Your cache administrator is webmaster. Thus, it is sufficient in general to check that the error-correction conditions hold for a basis of errors. http://vealcine.com/quantum-error/quantum-error-correcting-codes-from-the-compression-formalism.php Despite being efficiently simulable, most stabilizer states on a large number of qubits exhibit maximal bipartite entanglement[Dahlsten and Plenio, QIC 2006].

For instance, if Cab = δab, then the various erroneous subspaces are orthogonal to each other. Since these two operations are completely separate, it can also correct Y errors as both a bit flip and a phase error. 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}}} Cambridge University Press.

Itano, J. When this is true, C1 and C2 define an [[n, k1 + k2 − n, d]] stabilizer code, where d ≥ min(d1, d2). Particular caution is necessary, as computational gates can cause errors to propagate from their original location onto qubits that were previously correct. Another useful representation is to map the single-qubit Pauli operators I, X, Y, Z to the finite field GF(4), which sets up a connection between stabilizer codes and a subset of

The 1st, 4th and 7th qubits are for the sign flip code, while the three group of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code. 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). Submitted to Phys. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

The solution is to use transversal gates whenever possible. Nevertheless, in quantum computing there is another method, namely the three qubit bit flip code. 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 The system returned: (22) Invalid argument The remote host or network may be down.

Your cache administrator is webmaster. Then we complete the operation with a further transversal gate which depends on the outcome of the measurement. Chuang (2000). "Quantum Computation and Quantum Information". The codewords of the QECC are by definition in the  + 1-eigenspace of all elements of the stabilizer, but an error E acting on a codeword will move the state into the