I'm having a question regarding quantum error correction.
Using a large number of imperfect (but already very good) quantum gates, it is in theory possible to build an equivalent, error-corrected gate. What I don't understand, however, is how it precisely scales when I want to do computation using algorithms with a large input space.
To give a precise question:
Let's say I can create many individual CNOT gates with an individual probability of success of $\eta=99.99\%$. How many of them do I need to implement Shor's algorithm to factor a $1024$ bits integer with an overall probability of success of $50\%.$ ?
Once I reach the point where I can build a single error-corrected gate between two qubits, have I won the fight against decoherence or will it just be harder and harder to correct the errors as the input space scales up ?