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. 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. 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 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 we 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. 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, 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 | ψ ⟩ {displaystyle vert psi angle } through a noisy channel E {displaystyle {mathcal {E}}} . Let us moreover assume that this channel either flips the state of the qubit, with probability p {displaystyle p} , or leaves it unchanged. The action of E {displaystyle {mathcal {E}}} on a general input ρ {displaystyle ho } can therefore be written as E ( ρ ) = ( 1 − p ) ρ + p   X ρ X {displaystyle {mathcal {E}}( ho )=(1-p) ho +p X ho X} . Let | ψ ⟩ = α 0 | 0 ⟩ + α 1 | 1 ⟩ {displaystyle |psi angle =alpha _{0}|0 angle +alpha _{1}|1 angle } be the quantum state to be transmitted. With no error correcting protocol in place, the transmitted state will be correctly transmitted with probability 1 − p {displaystyle 1-p} . 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 | 0 ⟩ → | 0 L ⟩ ≡ | 000 ⟩ {displaystyle vert 0 angle ightarrow vert 0_{L} angle equiv vert 000 angle } and | 1 ⟩ → | 1 L ⟩ ≡ | 111 ⟩ {displaystyle vert 1 angle ightarrow vert 1_{L} angle equiv vert 111 angle } . The input state | ψ ⟩ {displaystyle vert psi angle } is encoded into the state | ψ ′ ⟩ = α 0 | 000 ⟩ + α 1 | 111 ⟩ {displaystyle vert psi ' angle =alpha _{0}vert 000 angle +alpha _{1}vert 111 angle } . This mapping can be realized for example using two CNOT gates, entangling the system with two ancillary qubits initialized in the state | 0 ⟩ {displaystyle vert 0 angle } . The encoded state | ψ ′ ⟩ {displaystyle vert psi ' angle } is what is now passed through the noisy channel. The channel acts on | ψ ′ ⟩ {displaystyle vert psi ' angle } by flipping some subset (possibly empty) of its qubits. No qubit is flipped with probability ( 1 − p ) 3 {displaystyle (1-p)^{3}} , a single qubit is flipped with probability 3 p ( 1 − p ) 2 {displaystyle 3p(1-p)^{2}} , two qubits are flipped with probability 3 p 2 ( 1 − p ) {displaystyle 3p^{2}(1-p)} , and all three qubits are flipped with probability p 3 {displaystyle p^{3}} . Note that a further assumption about the channel is made here: we assume that E {displaystyle {mathcal {E}}} 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.