#ALEXEI KITAEV QUANTUM ERROR CORRECTION CODE#

Calderbank, Peter Shor and Andrew Steane.

A generalisation of this concept are the CSS codes, named for their inventors: A.

#ALEXEI KITAEV QUANTUM ERROR CORRECTION CODE#

Andrew Steane found a code which does the same with 7 instead of 9 qubits, see Steane code.the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for one bit flip and one phase flip error. Over time, researchers have come up with several codes:

The crucial point is that 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. So, 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. 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 reason is that the measurement of the syndrome has the projective effect of a quantum measurement. 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). 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. This is because a quantum error correcting code is designed such that a certain operation, called syndrome measurement can determine whether a qubit has been corrupted, and if so, which one. Now, if noise or decoherence corrupts one qubit, the information is not lost. Therefore it was a relief when Peter Shor realized that, even if the information cannot be copied, the information of one qubit (now called a logical qubit) can be spread onto several ( physical) qubits by using a quantum error correcting code. However, this is not possible with quantum information, as it cannot be copied: see no-cloning theorem. if, say, one copy says, the bit is a 0, and two others claim it to be a 1, then probably all three were a 1 and the first bit got corrupted. IntroductionĬlassical 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 i.e. Quantum error correction techniques are methods to protect quantum information from errors due to decoherence and other noise. Quantum states cannot be cloned (Wootters and Zurek, 1982), so the repetition encoding cannot be employed.[taken from wikipedia - needs major work) Similarly, quantum error correction aims to detect and correct errors of stored quantum information. Error correction employs the error detection outcome and performs the appropriate steps to correct it, thus reconstructing the original information.Ĭlassical error correction uses redundancy to spread information in many copies so that errors can be detected, for example by majority voting, and then corrected. The error detection process is based on an active monitoring of the system and the possibility of identifying errors without destroying the encoded information. To overcome errors we need to detect and correct them. To obtain a reliable outcome from a computation we need to be certain that the processed information remains resilient to errors at all times. Storing or manipulating information with a real physical system is naturally subject to errors. This opens the door for employing a large variety of many-body effects to combat errors. Beyond the inherited error correcting characteristics, topological systems protect the encoded information with the presence of the Hamiltonian that energetically penalises transformations between states. In particular, he defined a Hamiltonian whose eigenstates are also states of a quantum error correcting code. The birth of topological quantum computation took place when Alexei Kitaev (2003) made the ingenious step of turning a quantum error correcting code into a many-body interacting system.

Alexei kitaev quantum error correction