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# Quantum Error Correction Codes

## Contents

An example of a stabilizer code is the 5-qubit code, a [[5, 1, 3]] code whose stabilizer can be generated by \$\begin{matrix} X \otimes Z \otimes Z \otimes X \otimes I, \\ I Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors. Blatt, "Experimental Repetitive Quantum Error Correction," Science 332, 1059-1061 (2011), doi:10.1126/science.1203329 ^ M. Suppose further that a noisy error corrupts the three-bit state so that one bit is equal to zero but the other two are equal to one. check over here

Lett. 81, 2152–2155 (1998), doi:10.1103/PhysRevLett.81.2152 ^ T. This is the reason the world at a human scale looks classical - big objects are very likely to interact at least a little bit with their environment, so they are The best rigorous proofs of the threshold to date show that the threshold is at least 2 × 10 − 5 (meaning one error per 50, 000 operations). The system returned: (22) Invalid argument The remote host or network may be down.

## Quantum Error Correction For Beginners

Frank Gaitan (2008). "Quantum Error Correction and Fault Tolerant Quantum Computing". Schaetz, M. It is assumed that measurements and classical computations can be performed quickly and reliably, and that quantum gates can be performed between arbitrary pairs of qubits in the computer, irrespective of

The syndrome measurement tells us as much as possible about the error that has happened, but nothing at all about the value that is stored in the logical qubit—as otherwise the The original state of the encoded qubit can then be restored by a simple unitary transformation. Instead of the unencoded ∣ + ⟩ state, we must use a more complex ancilla state ∣00…0⟩ + ∣11…1⟩ known as a 'cat' state. Quantum Error Correction Book 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.

We therefore have to resort to more complicated techniques. Stabilizer Codes And Quantum Error Correction. Stabilizer Codes In order to better manipulate and discover quantum error-correcting codes, it is helpful to have a more detailed mathematical structure to work with. DiCarlo, S. 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.

S ⊥  is the set of Pauli operators that commute with all elements of the stabilizer. Quantum Code 7 Sun, L. Particular caution is necessary, as computational gates can cause errors to propagate from their original location onto qubits that were previously correct. w ⋅ v = 0 for all v ∈ C.

## Stabilizer Codes And Quantum Error Correction.

Classical error correcting codes use a syndrome measurement to diagnose which error corrupts an encoded state. 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. Quantum Error Correction For Beginners The most widely-used structure gives a class of codes known as stabilizer codes. Steane Code Somaroo, "Experimental Quantum Error Correction," Phys.

Langer, R. check my blog However for larger N an exponentially growing number of states are possible. Cerf, Ulrik L. 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 5 Qubit Code

Thus, for the 7-qubit code, the full logical Clifford group is accessible via transversal operations. This is just the simplest quantum code. Knill, C. http://vealcine.com/quantum-error/quantum-error-correction-with-degenerate-codes-for-correlated-noise.php 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.

Nebendahl, D. 5-qubit Quantum Error Correction R. According to the quantum Hamming bound, encoding a single logical qubit and providing for arbitrary error correction in a single qubit requires a minimum of 5 physical qubits.

## pt is the threshold for fault-tolerant quantum computation.

• These include EPR and GHZ states.
• Of these, only the assumption of independent errors is at all necessary, and that can be considerably relaxed to allow short-range correlations and certain kinds of non-Markovian environments.
• Many more are known, and there is a well-developed theory of quantum error-correcting codes.
• D.
• The Pauli group Pk, however, can be performed transversally on any stabilizer code.
• In general, a gate coupling pairs of qubits allows errors to spread in both directions across the coupling.

the equation ⟨ψi∣E∣ψj⟩ = C(E)δij fails. 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}}} We give a simple circuit which takes the initial state with four extra qubits in the state |0> to the encoded state. Bit Flip Memory Error Indeed, the set S ⊥  \ S of undetectable errors is a boon in this case, as it allows us to perform these gates.

Optimized simulations of fault-tolerant protocols suggest the true threshold may be as high as 5%, but to tolerate this much error, existing protocols require enormous overhead, perhaps increasing the number of Niset, G. L. have a peek at these guys 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

The simplest method, due to Shor, is very general but also requires the most overhead and is frequently the most susceptible to noise. S. In addition, CSS codes have some very useful properties which make them excellent choices for fault-tolerant quantum computation. B.

We use the notation [[n, k, d]] to a refer to such a stabilizer code. Then the Pauli operators of weight t or less form a basis for the set of all errors acting on t or fewer qubits, so a QECC which corrects these Pauli Revised April 1996 to give more intuition and an example.