I was looking at the assumptions behind the Microsoft Azure resource estimator, which is mostly built on Litinsky's paper.
Calling $Q_{\text{Alg}}$ the number of logical qubits required by the end-user algorithm, they claim that the total number of logical qubits (more specifically "tiles") required by the algorithm is (Eq (D1) on page 29):
$$Q = 2 Q_{\text{Alg}}+\sqrt{8 Q_{\text{Alg}}}+1$$
It comes from the "fast block" scheme of Litinsky, shown below. This number excludes all the distillation part of the computer used to distill the magic states (they are acknowledged with another equation which I won't discuss).
My question: Am I correct by saying that this number neglects the count of the purified magic-state used in the computation, as it will occupy one of the tiles below. I insist on the fact I talk about the purified magic-state that we get at the end of the distillation process, I am not talking about the distillation factory.
In practice, if we inject one $T$-gate per clock cycle, it would be negligible in the total count, but "conceptually" the total number of tiles we need should then be written as:
$$Q = 2 (Q_{\text{Alg}}+\color{red}1)+\sqrt{8 (Q_{\text{Alg}}+\color{red}1)}+1$$
However, if I wanted to do more than one state injection per timestep (if some $T$-gates layer commute for instance), I should write
$$Q = 2 (Q_{\text{Alg}}+\color{red}N_{\parallel})+\sqrt{8 (Q_{\text{Alg}}+\color{red}N_{\parallel})}+1$$
where $N_{\parallel}$ is the maximum number of parallel $T$-gate implemented in the algorithm (and in this case it might not necessarily be negligible).
If the answer is "it doesn't matter, it is negligible", I would like to understand why in the case of $N_{\parallel}>1$, but overall I would like to check that my conceptual understanding is correct, i.e. that I am correct about the "exact" formulas (this is the most important point of my question).
Would you agree with me or I am missing something??
