Quantum error correction
Suppose further that noise in the system introduces an error that corrupts the three-bit state so that one of the copied bits becomes zero (off) but the other two remain equal to one.
This theorem seems to present an obstacle to formulating a theory of quantum error correction.
Depending on the QEC code used, syndrome measurement can determine the occurrence, location and type of errors.
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 combination of basis operations called the error basis (which is given by the Pauli matrices and the identity).
This approach does not work for a quantum channel in which, due to the no-cloning theorem, it is not possible to repeat a single qubit three times.
To overcome this, a different method has to be used, such as the three-qubit bit-flip code first proposed by Asher Peres in 1985.
[3] This technique uses entanglement and syndrome measurements and is comparable in performance with the repetition code.
In the case of the simple three-qubit repetition code, the encoding consists in the mappings
This mapping can be realized for example using two CNOT gates, entangling the system with two ancillary qubits initialized in the state
Formally, this correcting procedure corresponds to the application of the following map to the output of the channel:
It is possible to correct for both types of errors on a logical qubit using a well-designed QEC code.
This means that the Shor code can also repair a sign flip error for a single qubit.
In other words, the Shor code can correct any combination of bit or phase errors on a single qubit.
Several proposals have been made for storing error-correctable quantum information in bosonic modes.
[10][11] One insight offered by these codes is to take advantage of the redundancy within a single system, rather than to duplicate many two-level qubits.
Then if the dominant error mechanism of the system is the stochastic application of the bosonic lowering operator
[10][12] Measuring the odd parity will allow correction by application of an appropriate unitary operation without knowledge of the specific logical state of the qubit.
Cat code, realized by Ofek et al.[13] in 2016, defined two sets of logical states:
Similar to the binomial code, if the dominant error mechanism of the system is the stochastic application of the bosonic lowering operator
, the error takes the logical states from the even parity subspace to the odd one, and vice versa.
Single-photon-loss errors can therefore be detected by measuring the photon number parity operator
If distinct of the set of correctable errors produce orthogonal results, the code is considered pure.
[21] Subsequently, demonstrations have been made with linear optics,[22] trapped ions,[23][24] and superconducting (transmon) qubits.
[25] In 2016 for the first time the lifetime of a quantum bit was prolonged by employing a QEC code.
[13] The error-correction demonstration was performed on Schrödinger-cat states encoded in a superconducting resonator, and employed a quantum controller capable of performing real-time feedback operations including read-out of the quantum information, its analysis, and the correction of its detected errors.
The system uses an active syndrome extraction technique to diagnose errors and correct them while calculations are underway without destroying the logical qubits.
By encoding quantum information in the nuclear spin of a phosphorus atom embedded in silicon and employing advanced pulse control techniques, they demonstrated enhanced error resilience.
[39] In 2022, research at University of Engineering and Technology Lahore demonstrated error cancellation by inserting single-qubit Z-axis rotation gates into strategically chosen locations of the superconductor quantum circuits.
[40] The scheme has been shown to effectively correct errors that would otherwise rapidly add up under constructive interference of coherent noise.
This is a circuit-level calibration scheme that traces deviations (e.g. sharp dips or notches) in the decoherence curve to detect and localize the coherent error, but does not require encoding or parity measurements.
