Where does QEC fit in the larger picture of the execution of a fault tolerant quantum algorithm?
For QEC, there is an initial preparation of the qubit state and the ancillary qubits; then a "noisy channel"; then a check and error correcting. It seems the algorithm would be the noisy channel, but then the final checking and correcting stage would project the state back to its original state negating any the algorithm did. I'm missing something essential here.
Based on what I've understood from your question, I think the interpretation of the noisy channel as the quantum algorithm may be causing unnecessary confusion. QEC should be present in the fault tolerant computing paradigm as a background operation capable of guaranteeing that qubits retain coherence of their quantum states for practical amounts of time. As such, in terms of how you lay out the notion, the quantum algorithm that is being executed can't exactly be associated to the quantum channel. The channel is essentially the conglomerate of error mechanisms (decoherence etc.) by which quantum states are corrupted. Thus, what QEC does is ensure that the qubits the algorithm is working with are in the correct quantum states. You can think of it as a background process or black box that is always running to ensure that the quantum algorithm will not fail due to the incorrectness of its basic information units (the qubits themselves).
Naturally, the follow up to this question is how can QEC correct these quantum states when the implementation of quantum codes is based on quantum gates, which suffer from the aforementioned mechanisms themselves. This leads to a debate over which QECC family (turbo, LDPC, polar...) is better suited for this purpose, but this begins to deviate from the topic of the question.
In an algorithm, there is no noisy channel per se. There are faulty gates causing errors each time they are applied.
However, it's not completely disconnected, as you could model each faulty gate followed by an error on the output qubit(s) of the gate. This error model is the equivalent of a noisy channel. It would act on single qubits for single qubit gates, while actin on 2 qubits for 2 qubit gates (possibly showing correlated errors).
The job of QECC in a faul tolerant design is to correct for errors on qubit gates while keeping in mind that doing so requires some more possibly faulty gates.
