Shor code

In quantum computing, the Shor code or Shor nine qubit code is a foundational code in quantum error correction that protects quantum information against decoherence and operational errors. It was the first quantum error correcting code, introduced by Peter Shor in 1995. It encodes a single logical qubit into a system of nine physical qubits, allowing simultaneous correction of both bit-flip, phase-flip or a joint phase and bit flip errors on any single physical qubit. As the first quantum error-correcting code to demonstrate fault tolerant quantum computing in principle, the Shor code marked a critical step toward the development of reliable quantum computing systems. The Shor code is a simple example of a Bacon–Shor code. These codes have the property that are constructed from local operations and repeating patterns, and introduce the ability to switching encoding dynamically (while the circuit is running) in a fault-tolerant manner.

In quantum computing, the Shor code or Shor nine qubit code is a foundational code in quantum error correction that protects quantum information against decoherence and operational errors. It was the first quantum error correcting code, introduced by Peter Shor in 1995. It encodes a single logical qubit into a system of nine physical qubits, allowing simultaneous correction of both bit-flip, phase-flip or a joint phase and bit flip errors on any single physical qubit. As the first quantum error-correcting code to demonstrate fault tolerant quantum computing in principle, the Shor code marked a critical step toward the development of reliable quantum computing systems. The Shor code is a simple example of a Bacon–Shor code. These codes have the property that are constructed from local operations and repeating patterns, and introduce the ability to switching encoding dynamically (while the circuit is running) in a fault-tolerant manner. The Shor code encodes one logical qubit in 9 physical qubits. To construct the code, we first encode a state \alpha|0\rangle+\beta|1\rangle to an encoding of three qubits, as |0\rangle\to|+++\rangle and |1\rangle\to|---\rangle, where |\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2 . In order to obtain the desired Shor logical |0_{\rm L}\rangle and |1_{\rm L}\rangle we use concatenation, that is, each of the three qubits is multiplied into a three qubit block, given by |0_{\rm L}\rangle=\frac{1}{2\sqrt{2}}(|000\rangle + |111\rangle) \otimes (|000\rangle + |111\rangle ) \otimes (|000\rangle + |111\rangle) and |1_{\rm L}\rangle=\frac{1}{2\sqrt{2}}(|000\rangle - |111\rangle) \otimes (|000\rangle - |111\rangle ) \otimes (|000\rangle - |111\rangle). Z_{\mathrm L}=X\otimes X\otimes X\otimes X\otimes X\otimes X\otimes X\otimes X\otimes X, where X is the single qubit Pauli X gate. In the same manner a logical Pauli X is given by X_{\mathrm L}=Z\otimes Z\otimes Z\otimes Z\otimes Z\otimes Z\otimes Z\otimes Z\otimes Z, where Z is the single qubit Pauli Z gate. According to the threshold theorem a quantum error correction code can correct physical error if the error rate is below a certain threshold. If p is the probability of a random error happening on a single qubit, a conservative estimate can treat any error affecting two or more qubits as uncorrectable; this happens with probability P_2(p)=1-(1-p)^9-9p(1-p)^8\approx 36p^2, When P_2(p) is larger than p itself (where we neglected terms with power larger than p 3 ), it is better to not use Shor code at all. In this case the threshold is approximately p=1/36=2.78%. However including errors in the error correction itself this value can drop to 10 −4 . ==References==

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