At the very beginning he says:

> The rule of simulation that I would like to have is that the number of computer elements required to simulate a large physical system is only to be proportional to the space-time volume of the physical system. I don't want to have an explosion. That is, if you say I want to explain this much physics, I can do it exactly and I need a certain-sized computer. If doubling the volume of space and time means I'll need an exponentially larger computer, I consider that against the rules (I make up the rules, I'm allowed to do that).

He emphasizes this again in the section about computing the probabilities:

> We emphasize, if a description of an isolated part of nature with N variables requires a general function of N variables and if a computer stimulates this by actually computing or storing this function then doubling the size of nature (N->2N) would require an exponentially explosive growth in the size of the simulating computer. It is therefore impossible, according to the rules stated, to simulate by calculating the probability. [emphasis mine]

So when he uses the term 'computer' he doesn't mean 'abstract Turing machine', he explicitly means 'realizable/efficient Turing machine'.