# Quantum Error Correction Wikipedia

## Contents |

Blakestad, J. This position has a focus on the mathematics of quantum information. Crucial to most designs for quantum computers is quantum error correction, which helps preserve the fragile quantum states on which quantum computation depends. Lecture Notes, MSRI Workshop on Quantum Computation. ^ Igor Devetak (2005). "The private classical capacity and quantum capacity of a quantum channel". weblink

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 Let us employ the shorthand u = ( z | x ) {\displaystyle u=\left(z|x\right)} and v = ( z ′ | x ′ ) {\displaystyle v=\left(z^{\prime }|x^{\prime }\right)} where z {\displaystyle Proof. The first step of the three qubit bit flip code is to entangle the qubit with two other qubits using two CNOT gates with input | 0 ⟩ {\displaystyle |0\rangle } https://en.wikipedia.org/wiki/Quantum_error_correction

## Stabilizer Codes And Quantum Error Correction.

Universal fault-tolerance is known to be possible for any stabilizer code, but in most cases the more complicated type of construction is needed for all but a few gates. Steane, “Error correcting codes in quantum theory,” Phys. A 5-qubit code is the smallest possible code which protects a single logical qubit against single-qubit errors. In a paper they’re presenting at the Association for Computing Machinery’s Symposium on Theory of Computing in June, researchers from MIT, Google, the University of Sydney, and Cornell University present a

Quantum convolutional codes are similar because some of the qubits feed back into a repeated encoding unitary and give the code a memory structure like that of a classical convolutional code. The first quantum error-correcting codes are strikingly similar to classical block codes in their operation and performance. The product is in the Clifford group, and is only performed if the measurement outcome is 1. Quantum Error Correction Book It is frequently useful to consider other representations of stabilizer codes.

We employ the notation u = ( z | x ) , v = ( z ′ | x ′ ) {\displaystyle \mathbf {u} =\left(\mathbf {z} |\mathbf {x} \right),\mathbf {v} =\left(\mathbf The simplest method, due to Shor, is very general but also requires the most overhead and is frequently the most susceptible to noise. Conversely, we are also interested in the problem of setting upper bounds on achievable values of (logK)/n and d/n. view publisher site A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect.

arXiv:quant-ph/9604015. Steane Code The theory of fault-tolerant quantum computation **tells us how to perform operations** on states encoded in a quantum error-correcting code without compromising the code's ability to protect against errors. re-encode each logical qubit by the same code again, and so on, on logarithmically many levels—provided the error rate of individual quantum gates is below a certain threshold; as otherwise, the Lett. 81, 2152–2155 (1998), doi:10.1103/PhysRevLett.81.2152 ^ T.

## 5 Qubit Code

Perform a controlled-M operation from the ancilla to the state being measured. Chiaverini, D. Stabilizer Codes And Quantum Error Correction. Continuing, we have = ∑ a n ∈ T δ p n Pr { E a n } Pr S { ∃ E b n : b n ∈ T δ Quantum Code 7 A finite-depth decoding circuit corresponding to the stabilizer G {\displaystyle {\mathcal {G}}} exists by the algorithm given in (Grassl and Roetteler 2006).

Brun. "Extra shared entanglement reduces memory demand in quantum convolutional coding." Phys. have a peek at these guys **T. **In either case, the final state still tells us nothing about the data beyond the eigenvalue of M. D. 5 Qubit Quantum Error Correction

Unfortunately, the Clifford group by itself does not have much computational power: it can be efficiently simulated on a classical computer. Andrew Steane found a code which does the same with 7 instead of 9 qubits, see Steane code. re-encode each logical qubit by the same code again, and so on, on logarithmically many levels—provided the error rate of individual quantum gates is below a certain threshold; as otherwise, the http://vealcine.com/quantum-error/quantum-error-correction-usc.php The set of stabilizer codes is exactly the set of codes which can be created by a Clifford group encoder circuit using ∣0⟩ ancilla states.

More sophisticated techniques for fault-tolerant error correction involve less interaction with the data but at the cost of more complicated ancilla states. Bit Flip Memory Error Leuchs, N. A more general class of codes (encompassing the former) are the stabilizer codes of Daniel Gottesman.

## Wilde.

- Note that the following procedure can be used to measure (non-fault-tolerantly) the eigenvalue of any (possibly multi-qubit) Pauli operator M: Produce an ancilla qubit in the state ∣ + ⟩ = ∣0⟩ + ∣1⟩.
- Note that the Gilbert-Varshamov bound simply states that codes at least this good exist; it does not suggest that better codes cannot exist.
- For H2, we perform the same procedure, but each 1 is instead replaced by X.

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 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. The simplest way is to store the information multiple times, and—if these copies are later found to disagree—just take a majority vote; e.g. Quantum Code Reel The receiver measures all the generators in G {\displaystyle {\mathcal {G}}} and corrects for errors as he receives the online encoded qubits.

This typical set consists of the likely errors in the sense that ∑ a n ∈ T δ p n Pr { E a n } ≥ 1 − ϵ , Math. 1 (2001), no. 3, 325–332. Researchers have found many examples of classical codes satisfying this constraint, but most classical codes do not. this content Pittman, B.

Itano, J. Cerf and U. 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). S ⊥ is the set of Pauli operators that commute with all elements of the stabilizer.

The encoding circuit is online if it acts on a few blocks of qubits at a time. D. Inf. 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.

But, Harrow says, some redundancy in the hardware will probably be necessary to make the scheme efficient. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a single qubit. Huck, J. We occasionally omit tensor product symbols in what follows so that A 1 ⋯ A n ≡ A 1 ⊗ ⋯ ⊗ A n . {\displaystyle A_{1}\cdots A_{n}\equiv A_{1}\otimes \cdots \otimes

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. Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors. Further reading[edit] Mark M.