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I was exploring this paper: https://arxiv.org/pdf/1505.07390.pdf The paper describes the syndrome measurement for the [7,1,3] code. I was looking a fault-tolerant syndrome measurement and found this Figure. I was wondering what is $\psi_{L}$ and what is its function to the circuit, and I am thinking if I can remove $\psi_{L}$ and apply CNOT gate from $|0\rangle _{L}$ to $|+\rangle_{L}$ directly? Will it change the property of the circuit especially the fault-tolerant property.

It will be great to share the similar kind of circuit for [7,1,3] steane code for QEC.

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The circuit in the diagram is for fault-tolerantly measuring the error syndrome of the logical qubit $|\psi_L\rangle$. I.e., $|\psi_L\rangle$ is the logical qubit (a qubit encoded into 7 qubits using the Steane code) whose errors you want to correct. So removing it wouldn't make sense.

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