We now face two problems with continuous computations:
1. Applying operations on logical qubits. Applying operations on encoded qubits would only be possible by decoding - applying - encoding again. Not optimal. Also errors can happen in decode encode.
2. Operations on encoded qubits could introduce new more complex errors.


## Operations on logical/physical qubits

We begin by defining how to apply operations to encoded qubits:

![[Pasted image 20260806135601.png]]
![[Pasted image 20260806135610.png]]


## Fault tolerant Gates

Now we need to reduce our error probability.

![[Pasted image 20260806140451.png]]

To make this circuit fault-tolerant, we replace it with:

![[Pasted image 20260806140514.png]]

Let's look at a circuit for CNOT:

![[Pasted image 20260806140629.png]]
![[Pasted image 20260806140639.png]]
![[Pasted image 20260806140650.png]]