While seemingly intuitive, the idea of a QFT field as a probability field over space is not correct. It usually comes as a shock to anybody entering QFT when they finally realize the actual "probability space" is infinitely larger..
So as a programmer's example you might have an idea that to simulate a QFT you could have an array of floats over space to describe your "electron field", and you draw in values to reflect a probability distribution and you'll write some update rules to describe how this evolves in time. But this won't work, because it's not how nature works.
What you need is that electron field array (and a photon field array to make any kind of non-trivial observations), but in 4D (space + time), and you need to iterate over all possible values in all array bins for all fields, calculate a magic number for each configuration, and weigh all these magic numbers together to figure out the actual probabilities for any configuration.
Even for a 16x16x16x16 array, that is 65536 bins, even with only 2 levels in each bin this is 2^65536 combinations.
All of practical QFT (theoretical, perturbative or non-perturbative lattice methods) is about doing this calculation with (obviously radical) simplifications.
That we can do these enormous simplifications and still get usable results, can be a very good sign that the underlying reality is in fact simpler than what QFT implies but nobody has figured this out yet.