Quantum error correction

Quantum error correction (QEC) is a set of techniques used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorized as essential to achieve fault tolerant quantum computing that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum state preparation, and faulty measurements. Effective quantum error correction would allow quantum computers with low qubit fidelity to execute algorithms of higher complexity or greater circuit depth.

Classical error correction often employs redundancy. The simplest albeit inefficient approach is the repetition code. A repetition code stores the desired (logical) information as multiple copies, and—if these copies are later found to disagree due to errors introduced to the system—determines the most likely value for the original data by majority vote. For instance, suppose a bit is copied in a given state (for example, the state of "on" also known as one) three times. Suppose further that noise in the system introduces an error that corrupts the three-bit state so that one of the three copied bits becomes zero ("off") but the other two remain equal to one. Assuming that errors are independent and occur with some sufficiently low probability p, it is most likely that the error is a single-bit error and the intended message is three bits in the one state. 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. In this example, the logical information is a single bit in the one state and the physical information are the three duplicate bits. Creating a physical state that represents the logical state is called encoding and determining which logical state is encoded in the physical state is called decoding. Similar to classical error correction (QEC), QEC codes do not always correctly decode logical qubits, but instead reduce the effect of noise on the logical state.

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 (logical) information of one logical 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.

In classical error correction, syndrome decoding is used to diagnose which error was the likely source of corruption on an encoded state. An error can then be reversed 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. Depending on the QEC code used, syndrome measurement can determine the occurrence, location and type of errors. In most QEC codes, the type of error is either a bit flip, or a sign (of the phase) flip, or both (corresponding to the Pauli matrices X, Z, and Y). 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). To correct the error, the Pauli operator corresponding to the type of error is used on the corrupted qubit to revert the effect of the error.

The syndrome measurement provides information about the error that has happened, but not about the information 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.

The repetition code works in a classical channel, because classical bits are easy to measure and to repeat. 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. 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 ${\displaystyle \vert \psi \rangle }$ through a noisy channel ${\displaystyle {\mathcal {E}}}$. Let us moreover assume that this channel either flips the state of the qubit, with probability ${\displaystyle p}$, or leaves it unchanged. The action of ${\displaystyle {\mathcal {E}}}$ on a general input ${\displaystyle \rho }$ can therefore be written as ${\displaystyle {\mathcal {E}}(\rho )=(1-p)\rho +p\cdot X\rho X}$.

Let ${\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle }$ be the quantum state to be transmitted. With no error-correcting protocol in place, the transmitted state will be correctly transmitted with probability ${\displaystyle 1-p}$. 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 ${\displaystyle \vert 0\rangle \rightarrow \vert 0_{\rm {L}}\rangle \equiv \vert 000\rangle }$ and ${\displaystyle \vert 1\rangle \rightarrow \vert 1_{\rm {L}}\rangle \equiv \vert 111\rangle }$. The input state ${\displaystyle \vert \psi \rangle }$ is encoded into the state ${\displaystyle \vert \psi '\rangle =\alpha _{0}\vert 000\rangle +\alpha _{1}\vert 111\rangle }$. This mapping can be realized for example using two CNOT gates, entangling the system with two ancillary qubits initialized in the state ${\displaystyle \vert 0\rangle }$. The encoded state ${\displaystyle \vert \psi '\rangle }$ is what is now passed through the noisy channel.