{"id":1930,"date":"2018-04-06T09:49:04","date_gmt":"2018-04-06T08:49:04","guid":{"rendered":"https:\/\/www.quantum-bits.org\/?p=1930"},"modified":"2022-08-12T17:17:08","modified_gmt":"2022-08-12T16:17:08","slug":"quantum-error-correction","status":"publish","type":"post","link":"https:\/\/www.quantum-bits.org\/?p=1930","title":{"rendered":"Quantum error correction"},"content":{"rendered":"

Building a quantum device in the real world means having to deal with errors<\/strong>: any qubit stored unprotected or transmitted through a communications channel will inevitably<\/strong> come out changed.<\/p>\n

To be protected, quantum devices have to be kept at extremely cold temperatures (a few milikelvins) and shielded from electromagnetic radiation.<\/p>\n

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Quantum error correction is used in quantum computing to protect quantum information from errors due to decoherence<\/a> and quantum noise<\/a>.<\/p>\n

It is essential if one is to achieve fault-tolerant quantum computation<\/strong> that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.<\/p>\n

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In classical<\/strong> computing, if one wants to protect<\/strong> a bit against errors<\/strong>, it can often suffice is to store the information multiple times<\/strong>.<\/p>\n

Let \\overline{0}=000 be a “logical bit”, encoding the data bit 0 and let \\overline{1}=111 encode the data bit 1.<\/p>\n

We have here a simple repetition code that protects against any one bit flip error. 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<\/strong>. <\/p>\n

Classical error correcting codes use a syndrome measurement <\/strong>to diagnose which error corrupts an encoded state. We then reverse an error by applying a corrective operation<\/strong> based on the syndrome.<\/p>\n

No-cloning theorem<\/strong><\/p>\n

In physics, the no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. It proves the impossibility<\/strong> of a simple perfect non-disturbing measurement scheme. Conversely, the no-deleting<\/a> theorem is the time-reversed dual to this theorem. It states that, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies.<\/p>\n

The no-cloning theorem has profound implications in quantum computing. It implies that if we measure each individual qubit and take a majority vote by analogy to classical code above, then we have lost<\/strong> the precise information that we are trying to protect.<\/p>\n

At first sight, the no-cloning theorem seems to present an obstacle to formulating a theory of quantum error correction.<\/p>\n

It is nevertheless possible to spread the information of one qubit<\/strong> onto a highly entangled<\/strong> state of several qubits. Peter Shor<\/a> 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.<\/p>\n

QEC (Quantum Error Correction)<\/strong><\/p>\n

Just like classical error correction, QEC 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.<\/p>\n

A syndrome measurement<\/strong> 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.<\/p>\n

The syndrome measurement tells us as much as possible about the error that has happened, but nothing <\/strong>at all about the value <\/strong>that is stored in the logical qubit\u2014as otherwise the measurement would destroy any quantum superposition of this logical qubit with other qubits in the quantum computer.<\/p>\n

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):<\/p>\n

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Let’s get back to the idea of encoding repeated data bits and let’s define<\/p>\n