Suppose I try to run Shor's algorithm on a big noisy quantum computer.
For a given input I repeat the circuit many times and collect statistics for the measurements of each qubit.
Now based on the measurement statistics, I try to guess a factor (or more the order in the order finding subroutine). Since I allow for polynomial overhead and it is easy to check I can make fairly many guesses.
Since the circuit is fairly deep and the qubits are noisy the measurements of individually qubits will probably be very close to 50%/50%, but if I just have a slight bias 49,9% then the probability gets boosted by Chernoff bounds and since the samples are independent for the measurements of a single qubit.
Furthermore, with substantial classical overhead, I get to try many combinations based on my simulated data.
I sense that the strategy sketched above will never work and we need error correction? But what is the exact reason? Will the bias be so small that I need exponentially many samples or checks? Do I need finer control over the noise to make sure that the noise is unbiased?