I am not necessarily focused on topological error correction (as the answer already provided reasons in this context). Even in the case of topological error correction I am not sure to understand the answer provided (look at comments below the answer).
Note: I am familiar with concatenated code, I just have very rough basics in surface code theory (which must be why I struggle to understand the already given answer).
[edit2]: Most of my questions are answered in the comments of the answer from Craig Gidney.
If I want to implement an error protected algorithm (I assume my gateset is Clifford+T), I typically need to know.
- The total number of Clifford gates it requires including the logical identities
- The total number of $T$ gates
However, when you look at papers doing resource estimations for algorithms, it frequently occurs to not have access to all those information.
For instance, many examples in the references given here do not provide the total number of Clifford gates.
I totally understand why the number of $T$ gates is important. Those gates are typically the ones for which you need to do magic state distillation and are the most costly in terms of physical resources.
But when you do error correction:
- Every gate will be noisy: you need to know the total number of logical gates in your algorithm (to know what is the logical error probability you should target, and hence the number of physical qubit per logical ones)
- Even if a given $T$ gates requires $\times 100$ more resources than a Clifford one, if you have $\times 1000$ more Clifford gate in your algo, then the dominant cost will be due to the Clifford operations.
- Probably even worse: you might have a lot of identity operation occuring on your qubits. If you look at figure 7 of this paper, the identity operation completely dominate the number of logical gate.
It leads me to the following questions:
- Why frequently the total number of Clifford gates is being ignored in papers doing resource estimation of algorithms.
- Are the logical identities included in the Clifford gate count?
For 1: If there is an argument saying that typically the number of $T$ gates is $\gg$ than the number of Clifford, I would be interested to hear about it.
For 2: I know that a logical identity is mathematically a Clifford operation. But I am not sure if they are taken into account in the counting in practice. As at some point qubits necessarily have to wait, you can be somehow "forced" to apply an identity. It could occur conceptually that the number of identities completely dominates the rest.