Quantum error correction

Quantum error correction is used in quantum computer science to protect quantum information from errors due to decoherence and quantum noise. Quantum corrections are fundamental when running fault-tolerant quantum computations, which not only fix faults in stored quantum information, but also faulty quantum gates, as well as erroneous measurements.

General

The classical error correction used redundancy. The easiest way is to store the information several times, and if those copies later differentiate to select the majority. Suppose we copy a bit three times. Let us further assume that a fault condition of the three bits changed so that one bit assumes the value zero, however the other two the value one. We also assume that disturbances are independent and occur with a certain probability p. It is very likely that the error is one bit and the transmitted message includes three ones. There is also a likelihood that a double failure occurs, and the transmitted message includes three zeros, but the result is less likely than the first.

Copy quantum information, according to the no- cloning theorem is not possible, and therefore is an obstacle to the formulation of a theory of quantum error correction dar. But it is possible to transmit the information from a qubit to a system of a plurality of entangled qubits. Peter Shor first discovered this method by developing a code for quantum error correction, which transferred the information from one qubit to an entangled system of nine qubits. A code based on quantum error correction protects quantum information against errors of limited form.

Classical error-correcting codes using a syndrome measurement to determine which error destroys an encrypted state. We then make a mistake reversed by corrective surgery, based on the syndrome we use. Quantum error correction used syndrome measurement. We perform a measurement on several qubits of which does not disturb the quantum information in an encrypted state, but retrieves information about the error. A syndrome measurement can determine whether a qubit has been damaged, and if so, what was damaged. Moreover, tells us the result of the operation not only which physical qubit was affected, but also on which of the possible ways it was affected. The latter is at first glance counterintuitive: As faults occur randomly, as may be the result of disturbance only be a few different options? In most of the code is the result of either an inversion of the bits, or a reversal of the sign ( phase ), or both ( according to the Pauli matrices X, Z, and Y). The reason is that a measurement of the syndrome has a quantum effect projective measurement. Thus, even if the error due to the disturbance is arbitrary, it can be expressed as a superposition of, (Physics) by simple operations origin of the error (which is here represented by the Pauli matrices and the identity ). The syndrome measurement "forces" the qubit to "decide" to a specific " Pauli error", and the measurement tells us what. Now we can use the same Pauli operator re- apply on the affected qubit to revert the effect of the error.

The syndrome measurement tells us as much as possible about the error occurred, but nothing about the value of the is - otherwise stored in the qubit measurement would destroy any superposition of the qubit and other qubits in the quantum computer

The bit flip code

The serial code works in a classical channel, because qubits are easy to measure and restore, but in a quantum channel, it is no longer possible due to the no- cloning theorem, which forbids the production of identical copies of an arbitrary quantum state. A single qubit can not therefore be copied three times as in the above example, and any measurement would change information in the qubit. Nevertheless, there is another method for quantum computers, which is called 3 qubit bit flip code. This method uses entanglement and syndrome measurement, and achieves the same results as the serial code. We take a qubit. The first step of the three qubit bit flip code is the qubit with two other qubits using two CNOT gates with input. to entangle

The result looks like this: This is just a tensor product of three qubits, and different from the cloning of a state. Now these qubits are sent through separate, equally constructed channels. Now the qubit has been, for example, in the first reverse channel, and the result would be as follows. To determine the inversion in any of the three possible qubit requires a disease diagnosis, which involves four projection operators:

This can be obtained by:

So we know that the error syndrome corresponds to. This 3 qubit bit flip code can correct an error if a bit flip error in the channel occurs. He is like a function of a 3 -bit serial codes in classical computers.

The sign flip code

The bit flip is the only type of error in a classical computer, but another possible errors in quantum computers. This is the sign flip code. By transmitting in a channel between the sign and can also be reversed. For example, in the state of a qubit may be converted by reversing the sign in.

The original state of the qubit

Is in the state of

Converted.

The improvement of the error according to the sign flip code is identical to the flip bit code.

The Shor code

The error correction code applied to channels, may either reverse or invert the sign bit. It is also possible to combine the two codes in a code. The Shor code is a method which can correct any qubit errors.

The first, fourth and seventh qubit are for the sign flip code, while the three groups (1,2,3), (4,5,6 ) and ( 7,8,9) are designed to code the bit flip. With the state of a qubit of the Shor code is transformed into a product of qubits 9, wherein

When a bit - flip errors from occurring on a qubit, a syndrome analysis on each set of states (1,2,3), (4,5,6 ) and ( 7,8,9), and the running error corrected.

If the 3 -bit flip- groups (1,2,3), (4,5,6 ) and ( 7,8,9 ) as the three inputs to be considered, the Shor code circuit may a sign flip code can be reduced. That is, fix the Shor code also sign- flip error on a single qubit.

The Shor code can also correct any errors (bit flip and sign flip ) on a single qubit. If any error is an arbitrary unitary transformation U which acts on a qubit

The Pauli matrices are a group of 2x2 Hermitian and unitary matrices. U is equal to I is, the state is unchanged. If, then, a bit flip error in the channel occurred, if, then, the sign must have been reversed, and according to both a bit flip and a sign flip. Then, the error correction will correct the error as above. But the Shor code works only in the case of a one - qubit error.

Models

Over time, researchers have come up with different code models.

• Peter Shor's 9- qubit code, also known as the Shor code, encrypted one logical qubit into physical qubits 9, and may correct any error on a single qubit.
• Andrew Steane found a code the same thing with 7 instead of 9 qubits makes, see Steane code.
• Raymond Laflamme found a class of 5- qubit codes which do the same, and which have to be fault-tolerant and the property.
• A generalization of this concept are the CSS code, named after its inventors: AR Calderbank, Peter Shor and Andrew Steane. According to the quantum Hamming limitation, the encryption of a single logical qubits and a possibility of any error correction in a single qubit requires a minimum of 5 physical qubits.
• A more general class of codes are stabilizer codes, discovered by Daniel Gottesman (A Class of Quantum Error - Correcting Codes Saturating the Quantum Hamming Bound ), and AR Calderbank, Eric Rains, Peter Shor, and NJA Sloane (Quantum Error Correction and Orthogonal Geometry, Quantum Error Correction via codes over GF ( 4) ); these are called additive codes.
• A newer idea is Alexei Kitaevs Topological quantum codes and the general idea of topological quantum computer.

However, these codes allow quantum computing with arbitrary length and content of the limit theorem, founded by Michael Ben- Or and Dorit Aharonov, which claims that we can correct any errors if you quantum codes concatenated how the CSS code, that is, each re-encrypt logical qubit with the same code, and so on, many stages of logarithmic " delivers " the error rate of individual quantum gates below a certain threshold; you would otherwise try the syndromes to measure and correct the errors, more new errors would be incorporated as corrected.

2004 was estimated for this limit that it could be 1-3 % high Quantum Computing with Very Noisy Devices, stating that a sufficient number of qubits are available.