  The no-programming theorem prohibits the existence of a Universal
Programmable Quantum Processor. This statement has several implications in
relation to quantum computation, but also to other tasks of quantum information
processing, making this construction a central notion in this context.
Nonetheless, it is well known that even when the strict model is not
implementable, it is possible to conceive of it in an approximate sense.
Unfortunately, the minimal resources necessary for this aim are still not
completely understood. Here, we investigate quantitative statements of the
theorem, improving exponentially previous bounds on the resources required by
such a hypothetical machine. The proofs exploit a new connection between
quantum channels and embeddings between Banach spaces which allows us to use
classical tools from geometric Banach space theory in a clean and simple way.
