Quantum error correction QEC is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, bit flip error correction models quantum preparation, and faulty measurements. Classical error correction employs redundancy. 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.
Suppose we copy a bit three times. 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.
If we assume that noisy errors are independent and occur with some probability p, it is most likely that the error is a single-bit error and the transmitted message is three ones.
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. Copying quantum information is not possible due to the no-cloning theorem.
This theorem seems to present an obstacle to formulating a theory of quantum error correction. But it is possible to spread the information of one qubit onto a highly entangled state of several physical qubits. Peter Shor first discovered this method of formulating a quantum error correcting code by storing the information of one qubit onto a highly entangled state of nine qubits.
A quantum error correcting code protects quantum information against errors of a limited form. Classical bit flip error correction models correcting codes use a syndrome measurement to diagnose which error corrupts an encoded state. We then reverse an error by applying a corrective operation based on the syndrome.
Quantum error correction also employs syndrome measurements. We perform a multi-qubit measurement that does not disturb the quantum information in the encoded state but retrieves information about the error. A syndrome measurement can determine whether a qubit has been corrupted, and if so, which one. What is more, the outcome of this operation the syndrome tells us not only which physical qubit was affected, but also, in which of several possible ways it was affected.
The latter is counter-intuitive at first sight: Since noise is arbitrary, how can the effect of noise be one of only few distinct possibilities? In most codes, the effect is either a bit flip, or a sign of the phase flip, or bit flip error correction models corresponding to the Pauli matrices XZand Y. The reason is that the measurement of the syndrome has the projective effect of a quantum measurement.
So even if the error due to the noise was arbitrary, it can be expressed as a superposition of basis operations—the error basis which is here given by the Pauli bit flip error correction models and the identity. The syndrome measurement "forces" the qubit to "decide" for a certain specific "Pauli error" to "have happened", and the syndrome tells us which, so that we can let the same Bit flip error correction models operator act again on the corrupted qubit to revert the effect bit flip error correction models the error.
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 bit flip error correction models the logical qubit—as otherwise the measurement would destroy any quantum superposition of this logical qubit with other qubits in the quantum computer.
The repetition code works in a classical channel, because classical bits are easy to measure and to repeat. However, in a quantum channel, it is no longer possible, due to the no-cloning theoremwhich forbids the creation of identical copies of an arbitrary unknown quantum state.
So a single qubit cannot be repeated three times as bit flip error correction models the bit flip error correction models example, as any measurement of the qubit will change its wave function. Nevertheless, in quantum computing there is another method, namely the three qubit bit flip code. It uses entanglement and syndrome measurements and is comparable in performance with the repetition code.
We can however improve on this number by encoding the state into a greater number of qubits, in such a way that errors in the corresponding logical qubits can be detected and corrected. Note that a further assumption about the channel is made here: The problem is now how to detect and correct such errors, without at the same time corrupting the transmitted state.
One can then detect whether a qubit was flipped, without also querying for the values being transmittedby asking whether one of the qubits differs from the others. This amounts to performing a measurement with four different outcomes, corresponding to the following four projective measurements:.
This reveals which qubits are different from which others, without at the same time giving information about the state of the qubits themselves. Formally, this correcting procedure corresponds to the application of the following map to the output of the channel:. Flipped bits are the only kind of error in classical computer, but there is another possibility of an error with quantum computers, the sign flip.
In the Hadamard basis, bit flips become sign flips and sign flips become bit flips. The error channel may induce either a bit flip, a sign flip, or both. It is possible to correct for both types of errors using one code, and the Shor code does just that. In fact, the Shor code corrects arbitrary single-qubit errors. The 1st, 4th and 7th qubits are for the sign flip code, while the three group of qubits 1,2,34,5,6and 7,8,9 are designed for the bit flip code.
If a bit flip error happens to a qubit, the syndrome analysis will be performed on each set of states 1,2,34,5,6and 7,8,9then correct the error. If the three bit flip group 1,2,34,5,6and 7,8,9 are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code.
This means that the Shor code can also repair sign flip error for a single qubit. The Bit flip error correction models code also can correct for any arbitrary errors both bit flip and sign flip to a single qubit.
If U is equal to I, then no error occurs. Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors. Several proposals have been made for storing error-correctable quantum information in bosonic modes.
Unlike a two-level system, an oscillator has infinitely many energy levels in a single physical system. For example, the cat code [3] was followed shortly after by Gottesman-Kitaev-Preskill GKP states, [4] and more recently by the binomial code. Written in the Fock basis, the simplest binomial encoding is. Since the codewords involve only even photon number, and the error states involve only odd photon number, errors can be detected by measuring the photon number parity of the system.
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.
If distinct of the set of correctable errors produce orthogonal results, the code is considered pure. That these codes allow indeed for bit flip error correction models computations of arbitrary length is the content of the threshold theoremfound by Michael Ben-Or and Dorit Aharonovwhich asserts that you can correct for all bit flip error correction models if you concatenate quantum codes such as the CSS codes—i.
There have been several experimental realizations of CSS-based codes. The first demonstration was with NMR qubits. Other error correcting codes have also bit flip error correction models implemented, such as one bit flip error correction models at correcting for photon loss, the dominant error source in photonic qubit schemes.
From Wikipedia, the free encyclopedia. Nielsen and Isaac L. Explicit use of et al. Transactions on Information Theory,Vol. Quantum computer Qubit DiVincenzo's criteria Quantum information Quantum machine Quantum programming Timeline of quantum computing Cloud-based quantum computing. Quantum capacity Classical capacity Entanglement-assisted classical capacity Quantum channel Quantum network Quantum cryptography Quantum key distribution Quantum energy teleportation Quantum teleportation Superdense coding LOCC Entanglement distillation.
Universal quantum simulator Deutsch—Jozsa algorithm Grover's algorithm Quantum Fourier transform Shor's algorithm Simon's problem Quantum phase estimation algorithm Quantum counting algorithm Quantum annealing Algorithmic cooling Quantum algorithm for linear systems of equations Amplitude amplification. Quantum circuit Quantum gate One-way quantum computer cluster state Adiabatic quantum computation Topological quantum computer.
Trapped ion quantum computer Optical lattice. Charge qubit Flux qubit Phase qubit Transmon. Retrieved from " https: Quantum information science Quantum computing Fault-tolerant computer systems.
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