Quantum error correction

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, faulty 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.g. 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 error correcting codes use a "syndrome measurement" to diagnose which error corrupts an encoded state. It then can reverse an error by applying a corrective operation based on the syndrome. Quantum error correction also employs syndrome measurements. It performs 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's 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 both (corresponding to the Pauli matrices "X", "Z", and "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 matrices 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 error correction can let the same Pauli operator act again on the corrupted qubit to revert the effect of 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 in the logical qubit—as otherwise the measurement would destroy any quantum superposition of this logical qubit with other qubits in the quantum computer, which would prevent it from being used to convey quantum information.
Bit flip code.
The repetition code works in a classical channel, because classical bits are easy to measure and to repeat. This stops being the case for a quantum channel in which, due to the no-cloning theorem, it is no longer possible to repeat a single qubit three times. To overcome this, a different method first proposed by Asher Peres in 1985
, such as the so-called "three-qubit bit flip code", has to be used. This technique uses entanglement and syndrome measurements and is comparable in performance with the repetition code.
Consider the situation in which we want to transmit the state of a single qubit formula_1 through a noisy channel formula_2. Let us moreover assume that this channel either flips the state of the qubit, with probability formula_3, or leaves it unchanged. The action of formula_2 on a general input formula_5 can therefore be written as formula_6.
Let formula_7 be the quantum state to be transmitted. With no error correcting protocol in place, the transmitted state will be correctly transmitted with probability formula_8. 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. In the case of the simple three-qubit repetition code, the encoding consists in the mappings formula_9 and formula_10. The input state formula_1 is encoded into the state formula_12. This mapping can be realized for example using two CNOT gates, entangling the system with two ancillary qubits initialized in the state formula_13. The encoded state formula_14 is what is now passed through the noisy channel.
The channel acts on formula_14 by flipping some subset (possibly empty) of its qubits. No qubit is flipped with probability formula_16, a single qubit is flipped with probability formula_17, two qubits are flipped with probability formula_18, and all three qubits are flipped with probability formula_19. Note that a further assumption about the channel is made here: we assume that formula_2 acts equally and independently on each of the three qubits in which the state is now encoded. The problem is now how to detect and correct such errors, "without at the same time corrupting the transmitted state".
Let us assume for simplicity that formula_3 is small enough that the probability of more than a single qubit being flipped is negligible. One can then detect whether a qubit was flipped, "without also querying for the values being transmitted", by 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:formula_22This can be achieved for example by measuring formula_23 and then formula_24. This reveals which qubits are different from which others, without at the same time giving information about the state of the qubits themselves. If the outcome corresponding to formula_25 is obtained, no correction is applied, while if the outcome corresponding to formula_26 is observed, then the Pauli "X" gate is applied to the formula_27-th qubit. Formally, this correcting procedure corresponds to the application of the following map to the output of the channel: 
formula_28Note that, while this procedure perfectly corrects the output when zero or one flips are introduced by the channel, if more than one qubit is flipped then the output is not properly corrected. For example, if the first and second qubits are flipped, then the syndrome measurement gives the outcome formula_29, and the third qubit is flipped, instead of the first two. To assess the performance of this error correcting scheme for a general input we can study the fidelity formula_30 between the input formula_14 and the output formula_32. Being the output state formula_33 correct when no more than one qubit is flipped, which happens with probability formula_34, we can write it as formula_35, where the dots denote components of formula_33 resulting from errors not properly corrected by the protocol. It follows that formula_37This fidelity is to be compared with the corresponding fidelity obtained when no error correcting protocol is used, which was shown before to equal formula_38. A little algebra then shows that the fidelity "after" error correction is greater than the one without for formula_39. Note that this is consistent with the working assumption that was made while deriving the protocol (of formula_3 being small enough).
Sign flip code.
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. Through the transmission in a channel the relative sign between formula_41 and formula_42 can become inverted. For instance, a qubit in the state formula_43 may have its sign flip to formula_44
The original state of the qubit
formula_45
will be changed into the state
formula_46
In the Hadamard basis, bit flips become sign flips and sign flips become bit flips. Let formula_47 be a quantum channel that can cause at most one phase flip. Then the bit flip code from above can recover formula_48 by transforming into the Hadamard basis before and after transmission through formula_47.
Shor code.
The error channel may induce either a bit flip, a sign flip (i.e., a phase 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.
Let formula_50 be a quantum channel that can arbitrarily corrupt a single qubit. The 1st, 4th and 7th qubits are for the sign flip code, while the three group of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code. With the Shor code, a qubit state formula_51 will be transformed into the product of 9 qubits formula_52, where
If a bit flip error happens to a qubit, the syndrome analysis will be performed on each set of states (1,2,3), (4,5,6), and (7,8,9), then correct the error.
If the three bit flip group (1,2,3), (4,5,6), and (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 Shor code also can correct for any arbitrary errors (both bit flip and sign flip) to a single qubit. If an error is modeled by a unitary transform U, which will act on a qubit formula_48, then formula_56 can be described in the form
where formula_58,formula_59,formula_60, and formula_61 are complex constants, I is the identity, and the Pauli matrices are given by
If "U" is equal to "I", then no error occurs. If formula_65, a bit flip error occurs. If formula_66, a sign flip error occurs. If formula_67 then both a bit flip error and a sign flip error occur. Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors.
Bosonic codes.
Several proposals have been made for storing error-correctable quantum information in bosonic modes. Unlike a two-level system, a quantum harmonic oscillator has infinitely many energy levels in a single physical system. Codes for these systems include cat, Gottesman-Kitaev-Preskill (GKP), and binomial codes. 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.
Written in the Fock basis, the simplest binomial encoding is
formula_68
where the subscript L indicates a "logically encoded" state. Then if the dominant error mechanism of the system is the stochastic application of the bosonic lowering operator formula_69 the corresponding error states are formula_70 and formula_71 respectively. 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. Measuring the odd parity will allow correction by application of an appropriate unitary operation without knowledge of the specific logical state of the qubit. However, the particular binomial code above is not robust to two-photon loss.
General codes.
In general, a "quantum code" for a quantum channel formula_72 is a subspace formula_73, where formula_74 is the state Hilbert space, such that there exists another quantum channel formula_75 with
formula_76
where formula_77 is the orthogonal projection onto formula_78. Here formula_75 is known as the "correction operation".
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".
Models.
Over time, researchers have come up with several codes:
That these codes allow indeed for quantum computations of arbitrary length is the content of the quantum threshold theorem, found by Michael Ben-Or and Dorit Aharonov, which asserts that you can correct for all errors if you concatenate quantum codes such as the CSS codes—i.e. re-encode each logical qubit by the same code again, and so on, on logarithmically many levels—"provided" that the error rate of individual quantum gates is below a certain threshold; as otherwise, the attempts to measure the syndrome and correct the errors would introduce more new errors than they correct for.
As of late 2004, estimates for this threshold indicate that it could be as high as 1–3%, provided that there are sufficiently many qubits available.
Experimental realization.
There have been several experimental realizations of CSS-based codes. The first demonstration was with NMR qubits. Subsequently, demonstrations have been made with linear optics, trapped ions, and superconducting (transmon) qubits.
In 2016 for the first time the lifetime of a quantum bit was prolonged by employing a QEC code. The error-correction demonstration was performed on Schrodinger-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 work demonstrated how the quantum-error-corrected system reaches the break-even point at which the lifetime of a logical qubit exceeds the lifetime of the underlying constituents of the system (the physical qubits).
Other error correcting codes have also been implemented, such as one aimed at correcting for photon loss, the dominant error source in photonic qubit schemes.