Shor code

In quantum computing, the Shor code or nine qubit Shor 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 1995.¹² 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.² 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.³

The Shor code encodes one logical qubit in 9 physical qubits. To construct the code, we first transform encode a state ${\textstyle \alpha |0\rangle +\beta |1\rangle }$ to an encoding of three qubits, as⁴$${\displaystyle |0\rangle \to |+++\rangle }$$and$${\displaystyle |1\rangle \to |---\rangle ,}$$where ${\textstyle |\pm \rangle =(|0\rangle \pm |1\rangle )/{\sqrt {2}}}$. In order to obtain the desirer Shor logical ${\textstyle |0_{\rm {L}}\rangle }$ and ${\textstyle |1_{\rm {L}}\rangle }$ we use concatenation, that is, each of the three qubits is multiplied into a three qubit block, given by⁴$${\displaystyle |0_{\rm {L}}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )}$$and$${\displaystyle |1_{\rm {L}}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle ).}$$

Qubits for three blocks (0,1,2), (3,4,5) and (6,7,8), where each block is protected from bit-flips and the three blocks are protected together from a phase flip on any of the blocks. Thus the Shor code can correct any bit and/or phase flip errors in any single qubit. It can also correct two bit flips as long as the errors occur in separate blocks.⁴

Due to discretization of errors it can be shown that any unitary transformation on a single qubit can be corrected just by correcting bit flips and phase flips errors.⁴

One can define logical Pauli gates for the Shor code, where the logical Pauli Z gate is given by

$${\displaystyle Z_{\mathrm {L} }=X\otimes X\otimes X\otimes X\otimes X\otimes X\otimes X\otimes X\otimes X,}$$

where ${\textstyle X}$ is the single qubit Pauli X gate. In the same manner a logical Pauli X is given by

$${\displaystyle X_{\mathrm {L} }=Z\otimes Z\otimes Z\otimes Z\otimes Z\otimes Z\otimes Z\otimes Z\otimes Z,}$$where ${\textstyle Z}$ is the single qubit Pauli Z gate.⁴

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, the Shor code fail if two qubits are affected, this happens with probability⁵⁶$${\displaystyle P_{2}(p)=1-(1-p)^{9}-9p(1-p)^{8}\approx 36p^{2},}$$When ${\textstyle P_{2}(p)}$ is larger than p itself (where we neglected terms with power larger than p³), 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 10⁻⁴.⁵

The Shor code is a [[9,1,3]] code (9 qubits, 1 logical qubit, distance 3), the later number indicates that it can correct at most a single qubit error.² In the stabilizer formalism, the Shor code has 8 generators (6 bit flip and 2 phase flip parity checks):⁴

${\displaystyle {\begin{array}{c|ccccccccc}&1&2&3&4&5&6&7&8&9\\\hline g_{1}&Z&Z&I&I&I&I&I&I&I\\g_{2}&I&Z&Z&I&I&I&I&I&I\\g_{3}&I&I&I&Z&Z&I&I&I&I\\g_{4}&I&I&I&I&Z&Z&I&I&I\\g_{5}&I&I&I&I&I&I&Z&Z&I\\g_{6}&I&I&I&I&I&I&I&Z&Z\\g_{7}&X&X&X&X&X&X&I&I&I\\g_{8}&I&I&I&X&X&X&X&X&X\\\end{array}}}$

As the Shor code has only X stabilizers and Z stabilizers (does not mix X and Z in the stabilizer), it is then considered a CSS code.²