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## Quantum Error Correction For Beginners

## Stabilizer Codes And Quantum Error Correction.

## Frank Gaitan (2008). "Quantum Error Correction and Fault Tolerant Quantum Computing".

## Contents |

It is possible to **correct for both types of errors** using one code, and the Shor code does just that. The smallest distance-3 CSS code is the 7-qubit code, a [[7, 1, 3]] QECC created from the classical Hamming code (consisting of all sums of classical strings 1111000, 1100110, 1010101, and 1111111). Jacobs and J. Huck, J. news

A quantum error correcting code protects quantum information against errors of a limited form. Calderbank, Peter Shor and Andrew Steane. Itano, J. 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.

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. Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors. Definition 2 The distance d of an ((n, K)) is the smallest weight of a nontrivial Pauli operator E ∈ Pn s.t.

- If we want to preserve a quantum state for a long time without doing any calculations, or if we want to send it through a noisy communications channel, we can just
- Suppose that the state of qubit 8 at time 5 has implications for the states of both qubit 8 and qubit 11 at time 6.
- Cerf, Ulrik L.
- If U = i σ y {\displaystyle U=i\sigma _{y}} then both a bit flip error and a sign flip error occur.
- Freedman, Michael H.; Meyer, David A.: Projective plane and planar quantum codes.
- Chwalla, M.
- It is possible to correct for both types of errors using one code, and the Shor code does just that.
- DiCarlo, S.

Text is available **under the Creative Commons Attribution-ShareAlike License;** additional terms may apply. It uses entanglement and syndrome measurements and is comparable in performance with the repetition code. Cerf, Ulrik L. Quantum Error Correction Book Thus, measuring the eigenvalues of the generators of S tells us information about the error that has occurred.

In general, a gate coupling pairs of qubits allows errors to spread in both directions across the coupling. Stabilizer Codes And Quantum Error Correction. If U = σ z {\displaystyle U=\sigma _{z}} , a sign flip error occurs. 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. Niset, G.

Quantum error correction also employs syndrome measurements. Quantum Code 7 Comments: 46 pages, with large margins. D. All of them share some basic features: they involve creation and verification of specialized ancilla states, and use transversal gates which interact the data block with the ancilla state.

Share Comment Leave a comment Quantum computers are largely theoretical devices that could perform some computations exponentially faster than conventional computers can. https://arxiv.org/abs/0904.2557 Hennrich and R. Quantum Error Correction For Beginners However, the initial construction of the cat state is not fault-tolerant, so a single gate error then could eventually produce two errors in the data block. Steane 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

To diagnose bit flips in any of the three possible qubits, syndrome diagnosis is needed, which includes four projection operators: P 0 = | 000 ⟩ ⟨ 000 | + | navigate to this website Calderbank, Eric Rains, Peter Shor, and N. The sign flip code[edit] Quantum circuit of the phase flip code Flipped bits are the only kind of error in classical computer, but there is another possibility of an error with 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. 5 Qubit Code

Using the stabilizer formalism limits the available states but there is still a lot of interesting freedom. Note that this gives us a finite generating set of gates. R. More about the author A non-degenerate code is one for which different elements of the set of correctable errors produce linearly independent results when applied to elements of the code.

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. 5-qubit Quantum Error Correction Despite being efficiently simulable, most stabilizer states on a large number of qubits exhibit maximal bipartite entanglement[Dahlsten and Plenio, QIC 2006]. If distinct of the set of correctable errors produce orthogonal results, the code is considered pure.[3] Models[edit] Over time, researchers have come up with several codes: Peter Shor's 9-qubit-code, a.k.a.

M. Let E phase {\displaystyle E_{\text{phase}}} be a quantum channel that can cause at most one phase flip. Because the ancillas in Steane and Knill error correction are more complicated than the cat state, it is especially important to verify the ancillas before using them. Bit Flip Memory Error Found.

This theorem seems to present an obstacle to formulating a theory of quantum error correction. Stabilizer codes have a special relationship to a finite subgroup Cn of the unitary group U(2n) frequently called the Clifford group. The system returned: (22) Invalid argument The remote host or network may be down. click site This means that the Shor code can also repair sign flip error for a single qubit.[2] The Shor code also can correct for any arbitrary errors (both bit flip and sign

Rev. Those types of measurements, in a real system, can be very hard to do. The disadvantage is that it is not transversal, and thus not fault-tolerant. Shor’s insight was that it’s possible to measure relationships between qubits without measuring the values stored by the qubits themselves.

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 } The sign flip code[edit] Quantum circuit of the phase flip code Flipped bits are the only kind of error in classical computer, but there is another possibility of an error with Franson, "Demonstration of quantum error correction using linear optics," Phys. And for reasonably sized quantum computers, that fraction can be arbitrarily large — although the larger it is, the more qubits the computer requires. “There were many, many different proposals, all

Price, W.