On the other hand, consider measuring, the parity of the first two bits in each computational basis state. In that spirit, we will define, and measurements respectively as “ “, “ ” and “ “.įor example, if we measure, we get a different result for and in the no-error case, so that collapses the encoded state. This process is referred to in the language of Pauli measurements as “measuring Pauli Z” and is equivalent to performing a computational basis measurement.
Conversely, if we measure we are in the −1 eigenspace of Z.
Thus if we measure the qubit and obtain we are in the +1 eigenspace (the set of all vectors that are formed of sums of eigenvectors with only positive or only negative eigenvalues) of the operator. The Pauli-Z matrix clearly has two eigenvectors and with eigenvalues ☑. Let’s pause a moment to focus en Pauli measurement. Now, let’s define the following bit-flip errors E and their actions : Errors (E) Let’s get back to the idea of encoding repeated data bits and let’s define In most codes, the effect is either a bit flip, or a sign (of the phase) flip, or both (corresponding to Pauli X, Z, and Y matrices): 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. 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. We perform a multi-qubit measurement that does not disturb the quantum information in the encoded state but retrieves information about the error.Ī syndrome measurement can determine whether a qubit has been corrupted, and if so, which one. Just like classical error correction, QEC also employs syndrome measurements.
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. It is nevertheless possible to spread the information of one qubit onto a highly entangled state of several qubits. It implies that if we measure each individual qubit and take a majority vote by analogy to classical code above, then we have lost the precise information that we are trying to protect.Īt first sight, the no-cloning theorem seems to present an obstacle to formulating a theory of quantum error correction. The no-cloning theorem has profound implications in quantum computing. It states that, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. Conversely, the no-deleting theorem is the time-reversed dual to this theorem. It proves the impossibility of a simple perfect non-disturbing measurement scheme. In physics, the no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. We then reverse an error by applying a corrective operation based on the syndrome. That is, if any of the three bits are flipped, then we can recover the state of the logical bit by taking a majority vote.Ĭlassical error correcting codes use a syndrome measurement to diagnose which error corrupts an encoded state. We have here a simple repetition code that protects against any one bit flip error. Let be a “logical bit”, encoding the data bit 0 and let encode the data bit 1. In classical computing, if one wants to protect a bit against errors, it can often suffice is to store the information multiple times. It 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 is used in quantum computing to protect quantum information from errors due to decoherence and quantum noise. To be protected, quantum devices have to be kept at extremely cold temperatures (a few milikelvins) and shielded from electromagnetic radiation. Building a quantum device in the real world means having to deal with errors: any qubit stored unprotected or transmitted through a communications channel will inevitably come out changed.