Resource Quantification for the No-Programing Theorem

Abstract

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.

Asymptotic Performance of Port-Based Teleportation

“…Previously, the best known upper bound on the fidelity [19] had the same dependence on N , but was increasing in d, thus failing to reflect the fact that the task becomes harder with increasing d. Interestingly, a lower bound from [38] on the program register size of a universal programmable quantum processor also yields a converse bound for PBT that is incomparable to the one from [19] and weaker than our bound. Finally we provide a proof of the following 'folklore' fact that had been used in previous works on port-based teleportation.…”

“…Note that the dimension d is part of the denominator instead of the numerator as one might expect in the asymptotic setting. Thus, the bound lacks the right qualitative behavior for large values of d. A different, incomparable, bound can be obtained from a recent lower bound on the program register dimension of a universal programmable quantum processor obtained by Kubicki et al [38],…”

Programmability of covariant quantum channels

“…The former is the case if we consider unitarily covariant channels (see Example 20). The latter is the case if C = T U V and the symmetries we consider are trivial, as [27,47].…”

“…In the literature, where the set of channels C often consists of isometries or even unitaries, it is customary to impose that the program state π T is pure. While this does not affect the upper bounds on the program dimension (allowing mixed states cannot make it more difficult to program a processor compared to pure states), it is in particular an essential assumption for some lower bounds when unitaries are implemented [27,47].…”

“…The question of optimal program dimension (the dimension of the program register) for an approximate universal quantum processor was only recently answered. The works [27] and [47] provided new upper and lower bounds on the program dimension applying methods from Banach space theory and quantum entropies, respectively. In this work, we consider a special class of channels, the covariant channels, i.e.…”