Bit flip code

Goal

We want to protect our Qubits against Decoherences and changes due to Quantum Gate errors and environmental noise

Solution

We implement ${|0⟩}_L$ and ${|1⟩}_L$ using three physical qubits, by setting

$$\{\begin{array}{rl}{|0⟩}_L& =|000⟩\\ {|1⟩}_L& =|111⟩\end{array}$$

We apply the Measurements $⟨Z_1{Z}_2⟩$ and $⟨Z_2{Z}_3⟩$ to get information about the system’s state $|\psi ⟩$:

$$(Z_1{Z}_2,Z_2{Z}_3)=\{\begin{array}{rl}(1,1)& \Rightarrow |\psi ⟩=\alpha |000⟩+\beta |111⟩\\ (1,-1)& \Rightarrow |\psi ⟩=\alpha |001⟩+\beta |110⟩\\ (-1,1)& \Rightarrow |\psi ⟩=\alpha |100⟩+\beta |011⟩\\ (-1,-1)& \Rightarrow |\psi ⟩=\alpha |010⟩+\beta |101⟩\end{array}$$

Having gotten these Measurements, we can correct any changes to go back to the valid states of $\{{|0⟩}_L,{|1⟩}_L\}=\{|000⟩,|111⟩\}$, by flipping the appropriate Qubits. This fails if we had $2$ or more errors, which happens with probability $3p^2(1-p)+p^3$, where $p$ is the probability of a Qubit being flipped by error.

The probability of error can be taken to be arbitrary close to zero, using enough ancillary qubits. This is known as the quantum threshold theorem