Quantum error correction

Quantum error correction is in the quantum computer science used to quantum information against errors due to decoherence and quantum noise protected. Quantum error corrections are fundamental when performing error-tolerant quantum calculations, which not only eliminate disturbances in stored quantum information, but also faulty quantum gates and faulty measurements.

General

Classic error correction uses redundancy . The easiest way is to store the information several times and if these copies later differ, to select the majority. Suppose we copy a bit three times. We further assume that a disturbance changes the state of the three bits so that one bit takes on the value zero, but the other two take the value one. We also assume that perturbations are independent and occur with a certain probability  p . It is very likely that the error is in one bit and the message sent contains three ones. There is also a chance that a double error will occur and the message sent contains three zeros, but this result is less likely than the first.

Copying quantum information is not possible according to the no-cloning theorem and therefore represents an obstacle to formulating a theory of quantum error correction. But it is possible to transfer the information from one qubit to an entangled system of several qubits. Peter Shor was the first to discover this method by developing a code for quantum error correction that 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 use a syndrome measurement to determine which error is destroying an encrypted state. We then undo a mistake using corrective surgery based on the syndrome. Quantum error correction also uses syndrome measurement. We carry out a measurement on several qubits, 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, which one has been damaged. In addition, the result of the operation tells us not only which physical qubit was affected, but also which of the possible ways it was affected. The latter is not obvious at first glance: Since disturbances occur arbitrarily, how can the consequence of the disturbance be just one of a few different possibilities? In most codes, the result is either an inversion of the bit or an inversion 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 the projective effect of a quantum measurement . Even if the error was arbitrary as a result of the disturbance, it can be expressed as a superposition (physics) of simple operations - the origin of the error (which is given here by the Pauli matrices and the identity ). The syndrome measurement “forces” the qubit to “decide” on a special “Pauli error”, and the measurement tells us which one. Now we can apply the same Pauli operator again to the affected qubit to reverse the effect of the error.

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

The bit flip code

A serial code (also: repetition code) can be used in a classic channel, since bits are easy to measure and recover. In a quantum channel, however, this is no longer possible due to the no-cloning theorem, which prohibits the creation of identical copies of any quantum state. A single qubit cannot be copied three times as in the example above, and any measurement would change information in the qubit. However, there is another method for quantum computers, which is called the 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 3 qubit bit flip code is the qubit with two other qubits using two CNOT gates with input . to entangle ${\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle }$ ${\displaystyle |0\rangle }$

Quantum circuit of the bit flip code

The result looks like this: This is just a tensor product of three qubits, and different from cloning a state. Now these qubits are sent through separate, identically constructed channels. Now the qubit was, for example, in the first channel reversed, and the result would look like this: . To determine the inversion in any of the three possible qubits, one needs a syndrome diagnosis that includes four projection operators: ${\displaystyle |\psi '\rangle =\alpha _{0}|000\rangle +\alpha _{1}|111\rangle .}$ ${\displaystyle |\psi '_{r}\rangle =\alpha _{0}|100\rangle +\alpha _{1}|011\rangle }$

${\displaystyle P_{0}=|000\rangle \langle 000|+|111\rangle \langle 111|}$

${\displaystyle P_{1}=|100\rangle \langle 100|+|011\rangle \langle 011|}$

${\displaystyle P_{2}=|010\rangle \langle 010|+|101\rangle \langle 101|}$

${\displaystyle P_{3}=|001\rangle \langle 001|+|110\rangle \langle 110|}$

These can be obtained through:

${\displaystyle \langle \psi '_{r}|P_{0}|\psi '_{r}\rangle =0}$

${\displaystyle \langle \psi '_{r}|P_{1}|\psi '_{r}\rangle =1}$

${\displaystyle \langle \psi '_{r}|P_{2}|\psi '_{r}\rangle =0}$

${\displaystyle \langle \psi '_{r}|P_{3}|\psi '_{r}\rangle =0}$

So you know that the error syndrome corresponds with . This 3 qubit bit flip code can correct an error if a bit flip error occurs in the channel. It is like a function of a 3-bit serial code in classic computers. ${\displaystyle P_{1}}$

The sign flip code

Quantum circuit of the Sign Flip Code

The bit flip is the only type of error in classical computers, but a sign flip can also occur in quantum computers. The sign flip code is used to deal with this type of error. By transmitting in one channel, the sign between and can also be reversed. For example, a qubit in the state may be converted into by reversing the sign . ${\displaystyle |0\rangle }$ ${\displaystyle |1\rangle }$ ${\displaystyle |-\rangle =(|0\rangle -|1\rangle )/{\sqrt {2}}}$ ${\displaystyle |+\rangle =(|0\rangle +|1\rangle )/{\sqrt {2}}}$

The original state of the qubit

${\displaystyle |\psi \rangle =\alpha _{0}|+\rangle +\alpha _{1}|-\rangle }$

will be in the state

${\displaystyle |\psi '\rangle =\alpha _{0}|+++\rangle +\alpha _{1}|---\rangle .}$

transformed.

The improvement of the error after the Sign Flip Code is identical to the Bit Flip Code.

The Shor Code

Quantum circuit of the Shor code

The error correction code applied to channels may either reverse the bit or reverse the sign. It is also possible to combine both codes in one code. The Shor Code is just one method that can correct any qubit error.

The first, fourth and seventh qubits are for the sign flip code, while the groups of three (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code . The Shor Code transforms the state of a qubit into a product of 9 qubits , where ${\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle }$ ${\displaystyle |\psi '\rangle =\alpha _{0}|0_{S}\rangle +\alpha _{1}|1_{S}\rangle }$

${\displaystyle |0_{S}\rangle =(|000000000\rangle +|000000111\rangle +|000111000\rangle +|000111111\rangle +|111000000\rangle +|111000111\rangle +|111111000\rangle +|111111111\rangle )/{\sqrt {8}}}$

${\displaystyle |1_{S}\rangle =(|000000000\rangle -|000000111\rangle -|000111000\rangle +|000111111\rangle -|111000000\rangle +|111000111\rangle +|111111000\rangle -|111111111\rangle )/{\sqrt {8}}}$

When a bit flip error occurs on a qubit, syndrome analysis is performed on each group of states (1,2,3), (4,5,6), and (7,8,9), and the Mistake corrected.

If the 3-bit flip groups (1,2,3), (4,5,6), and (7,8,9) are viewed as three inputs, then the Shor code circuit can point to a Sign Flip code will be reduced. That said, the Shor Code can also repair sign-flip errors on a single qubit

The Shor Code can also correct any error (bit flip and sign flip) on a single qubit. If any error is any unitary transformation acting on a qubit ${\displaystyle U}$

${\displaystyle U|\psi \rangle =|\psi _{e}\rangle .}$

${\displaystyle |\psi \rangle }$is the original state of the individual qubit that is affected. can be described in the form ${\displaystyle U}$

${\displaystyle U=C_{0}I+C_{1}\sigma _{x}+C_{2}\sigma _{y}+C_{3}\sigma _{z}}$

wherein , , , and complex coefficients, is the identity, and the Pauli matrices are given by ${\displaystyle C_{0}}$ ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle C_{3}}$ ${\displaystyle I}$

${\displaystyle \sigma _{x}={\biggl (}{\begin{matrix}0&1\\1&0\end{matrix}}{\biggr )};}$

${\displaystyle \sigma _{y}={\biggl (}{\begin{matrix}0&-i\\i&0\end{matrix}}{\biggr )};}$

${\displaystyle \sigma _{z}={\biggl (}{\begin{matrix}1&0\\0&-1\end{matrix}}{\biggr )}}$

The Pauli matrices are a group of 2 × 2 Hermitian and unitary matrices. If it is , then that means the state is unchanged. If is, then a bit flip error has occurred in the channel, if is, then the sign must have 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 only works in the event of a 1-qubit error. ${\displaystyle U=I}$ ${\displaystyle U=\sigma _{x}}$ ${\displaystyle U=\sigma _{z}}$ ${\displaystyle U=i\sigma _{y}}$

Models

Over time, various code models have been discovered by researchers.

• Peter Shors 9-qubit-code, also known as the Shor code, encodes 1 logical qubit into 9 physical qubits and can correct any errors in a single qubit.
• Andrew Steane found a code that does the same with 7 instead of 9 qubits (Steane code).
• Raymond Laflamme found a class of 5-qubit codes which do the same thing and which also have the property of being fault tolerant .
• A generalization of this concept are the CSS codes , named after the inventors: AR Calderbank , Peter Shor and Andrew Steane . According to the quantum Hamming limitation, the encryption of a single logical qubit with a possibility for any error correction in a single qubit requires a minimum of 5 physical qubits.
• A more general class of codes are the stabilizer codes discovered by Daniel Gottesman and by A. R. Calderbank , Eric Rains , Peter Shor , and N. J. A. Sloane ; these are called additive codes .
• A more recent idea is Alexei Kitaev's topological quantum codes and the more general idea of topological quantum computers .

However, these codes allow quantum computing of any length and are part of the limit value theorem , founded by Michael Ben-Or and Dorit Aharonov , which claims that you can correct all errors if you concatenate quantum codes, like the CSS codes, that is encrypt each logical qubit again with the same code, and so on, on logarithmically many levels - "supplies" the error rate of individual quantum gates below a certain limit value; If, on the other hand, one were to try to measure the syndromes and correct errors for higher error rates, more new errors would flow in than are corrected.

In 2004, Emanuel Knill estimated that this limit value could be 1–3%, provided that a sufficient number of qubits are available.

literature

• DA Lidar, TA Brun: Quantum Error Correction , Cambridge University Press 2013

Individual evidence

1. ↑ Michael A. Nielsen, Isaac L. Chuang: Quantum Computation and Quantum Information . Cambridge University Press, 2000
2. ↑ Peter W. Shor: Scheme for reducing decoherence in quantum computer memory . In: Physical Review A . tape 52 , no. 4 , October 1995, p. R2493-R2496 , doi : 10.1103 / PhysRevA.52.R2493 .
3. ^ Gottesman: A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound . arxiv : quant-ph / 9604038 1996
4. ↑ Calderbank et al. a .: Quantum Error Correction and Orthogonal Geometry . arxiv : quant-ph / 9605005 1996
5. ↑ Calderbank et al. a .: Quantum Error Correction via Codes over GF (4) . arxiv : quant-ph / 9608006 1996
6. ^ E. Knill: Quantum computing with realistically noisy devices . In: Nature , Volume 434, 2005, pp. 39-44, arxiv : quant-ph / 0410199 2004