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# Quantum Error Code Correction

## Contents

Cambridge University Press. ^ W.Shor, Peter (1995). "Scheme for reducing decoherence in quantum computer memory". Math. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a single qubit. The 7-qubit code is much studied because its properties make it particularly well-suited to fault-tolerant quantum computation. weblink

Your cache administrator is webmaster. Suppose we copy a bit three times. Definition 2 The distance d of an ((n, K)) is the smallest weight of a nontrivial Pauli operator E ∈ Pn s.t. A quantum error correcting code protects quantum information against errors of a limited form.

## Stabilizer Codes And Quantum Error Correction.

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. Itano, J. Copying quantum information is not possible due to the no-cloning theorem. Lassen, M.

A newer idea is Alexei Kitaev's topological quantum codes and the more general idea of a topological quantum computer. We need to add some additional gate outside the Clifford group to allow universal quantum computation; a single gate will suffice, such as the single-qubit π/8 phase rotation diag(1, exp(iπ/4)). Chiaverini, D. 5-qubit Quantum Error Correction Furthermore, these calculations make a number of assumptions about the physical properties of the computer.

H. 5 Qubit Code Peter Shor first discovered this method of formulating a quantum error correcting code by storing the information of one qubit onto a highly entangled state of nine qubits. Experimental realization There have been several experimental realizations of CSS-based codes. Copying quantum information is not possible due to the no-cloning theorem.

Please try the request again. Quantum Code 7 Sloane ([2], [3]); these are also called additive codes. Now these qubits will be sent through a channel E bit {\displaystyle E_{\text{bit}}} where we assume that at most one bit flip may occur. R is a quantum operation and (R ∘ Ea)(∣ψ⟩) = ∣ψ⟩ for all Ea ∈ E, ∣ψ⟩ ∈ C.

1. Laflamme, W.
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3. A 71, 052332 (2005), doi:10.1103/PhysRevA.71.052332 ^ J.
4. For instance, a qubit in the state | − ⟩ = ( | 0 ⟩ − | 1 ⟩ ) / 2 {\displaystyle |-\rangle =(|0\rangle -|1\rangle )/{\sqrt {2}}} may have its
5. And that’s what’s really got people excited.
6. We can imagine the various possible errors taking the subspace C into other subspaces of Hn, and we want those subspaces to be isomorphic to C, and to be distinguishable from
7. Somaroo, "Experimental Quantum Error Correction," Phys.
8. Differentiation of sine in Fourier domain Can I use my client's GPL software?

## 5 Qubit Code

The system returned: (22) Invalid argument The remote host or network may be down. F. Stabilizer Codes And Quantum Error Correction. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Steane Code Ser., Chapman & Hall/CRC, Boca Raton, FL, 2002.

Lett. 81, 2152–2155 (1998), doi:10.1103/PhysRevLett.81.2152 ^ T. have a peek at these guys Note that the error syndrome does not tell us anything about the encoded state, only about the error that has occurred. Nigg, L. D. Quantum Error Correction Book

In fact, his error-correction code was a response to skepticism about the feasibility of implementing his factoring algorithm. However, in order to have a stabilizer code at all, the generators produced by the above procedure must commute. 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 check over here To do this, we could use nine qubits instead of three: |0> -> (|000> + |111>) (|000> + |111>) (|000> + |111>) |1> -> (|000> - |111>) (|000> - |111>) (|000>

Any qubit stored unprotected or one transmitted through a communications channel will inevitably come out at least slightly changed. Fault-tolerant Quantum Computation But until now, codes that could make do with limited measurements could correct only a limited number of errors — one roughly equal to the square root of the total number Chuang (2000). "Quantum Computation and Quantum Information".

## Specifically: Definition 1 Let Hn be a 2n-dimensional Hilbert space (n qubits), and let C be a K-dimensional subspace of Hn.

We encode a 0 as 000 and a 1 as 111. Zurek, T. 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 $\left|\overline{00}\right\rangle + \left|\overline{11}\right\rangle$. Bit Flip Memory Error Frequently, we are interested in codes that correct any error affecting t or fewer physical qubits.

Stabilizer codes have a special relationship to a finite subgroup Cn of the unitary group U(2n) frequently called the Clifford group. Jacobs and 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. http://vealcine.com/quantum-error/quantum-error-correction-ppt.php Calderbank, Peter Shor and Andrew Steane.

M. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. If we only wish to detect errors, a distance d code can detect errors on up to d − 1 qubits.