# How to do error correction after encoded Bell measurement?

Need some help with the concepts of encoded/logical bell measurement.

Please visualize the picture in your mind. Suppose I have a node with 7+7 qubits side by side, left 7 is $$|0_{L}\rangle$$ and right 7 is $$|+_{L}\rangle$$. Suppose $$|0_{L}\rangle$$ = $$|1001011\rangle$$ and $$|+_{L}\rangle$$ = $$|0011011\rangle$$. Now I apply $$CNOT$$ between each pair of left and right qubits and after that $$H$$ gate on the all left side 7 qubits.

Now the problem how do I error correction. This part I am having confusion. I have an idea but I am having doubt.

Do I need to take additional 7 qubits for the left side and apply $$CZ$$ on each one and similarly additional 7 qubits for the right side and apply $$CX$$? This idea came after seeing the bell measurement from teleportation. This part is bothering, since those are logical qubits.

• How come $|+_L\rangle$ is orthogonal to $|0_L\rangle$? They should have an inner product of $1/\sqrt2$. Also, how do these states relate to whatever error correcting code you're wanting to use? Jul 7 at 6:43

## 1 Answer

Well, You should know Logical qubits are having topological feature. those are not spanning in a single state like you said.

Logical qubit has to be written in superposition states. like this. this is arbitrary state for shor code with distance 3.

and if you are wondering about CNOT operation between two Logical qubits read this paper. https://arxiv.org/abs/1111.4022