Computation in generalised probabilistic theories

TL;DR

This paper explores the computational limits of generalized probabilistic theories, showing that under certain physical assumptions, their computational power aligns with known complexity classes like AWPP and PP, even without causality.

Contribution

It establishes that theories with tomographic locality and without assuming causality still contain efficient computation within AWPP and PP, extending quantum computational bounds to more general theories.

Findings

Efficient computation in general theories is contained in AWPP under tomographic locality.

Post-selected computation in general theories is contained in PP with only tomographic locality.

There exists a classical oracle where theories satisfying certain assumptions do not include NP.

Abstract

From the existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is intrinsically more powerful than classical computation. At present, the best upper bound known for the power of quantum computation is that BQP is in AWPP. This work investigates limits on computational power that are imposed by physical principles. To this end, we define a circuit-based model of computation in a class of operationally-defined theories more general than quantum theory, and ask: what is the minimal set of physical assumptions under which the above inclusion still holds? We show that given only an assumption of tomographic locality (roughly, that multipartite states can be characterised by local measurements), efficient computations are contained in AWPP. This inclusion still holds even without assuming a basic notion of causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future measurement choices). Following Aaronson, we extend the computational model by allowing post-selection on measurement outcomes. Aaronson showed that the corresponding quantum complexity class is equal to PP. Given only the assumption of tomographic locality, the inclusion in PP still holds for post-selected computation in general theories. Thus in a world with post-selection, quantum theory is optimal for computation in the space of all general theories. We then consider if relativised complexity results can be obtained for general theories. It is not clear how to define a sensible notion of an oracle in the general framework that reduces to the standard notion in the quantum case. Nevertheless, it is possible to define computation relative to a `classical oracle'. Then, we show there exists a classical oracle relative to which efficient computation in any theory satisfying the causality assumption and tomographic locality does not include NP.