Computation in generalised probabilisitic theories

Abstract

From the general difficulty of simulating quantum systems using classical systems, and in particular 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 AWPP ⊆ , where AWPP is a classical complexity class (known to be included in PP, hence PSPACE). This work investigates limits on computational power that are imposed by simple physical, or information theoretic, 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 inclusions still hold? We show that given only an assumption of tomographic locality (roughly, that multipartite states and transformations can be characterized 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 postselection on measurement outcomes. Aaronson showed that the corresponding quantum complexity class, PostBQP, is equal to PP. Given only the assumption of tomographic locality, the inclusion in PP still holds for post-selected computation in general theories. Hence in a world with post-selection, quantum theory is optimal for computation in the space of all operational theories. We then consider whether one can obtain relativized complexity results for general theories. It is not obvious how to define a sensible notion of a computational 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 does not include NP.By comparison, relatively little has been learned about the connections between physical principles and computation. It was shown in [7] that a maximally non-local theory has no non-trivial reversible dynamics and, thus, any reversible computation in such a theory can be efficiently simulated on a classical computer. Aside from this result, most previous investigations into computation beyond the usual quantum formalism have centred around non-standard theories involving modifications of quantum theory. These theories often appear to have immense computational power and entail unreasonable physical consequences. For example, non-linear quantum theory appears to be able to solve NP-complete problems in polynomial time [8], as does quantum theory in the presence of closed timelike curves [9,40]. Aaronson has considered other modifications of quantum theory, such as a hidden variable model in which the hist...

Causal and causally separable processes

“…Proof S2 (proposition 2.3). First, observe that the property (18) holds for the case where the specified K consecutive sets exhaust all local experiments ¼ { } n 1, , . This is because, in this case, each of the local experiments in the K th consecutive set is causally preceded by or causally independent from every other local experiment.…”

“…Lemma S1. Let the property (18) hold for = ¢ + K K I, where ¢  K 1. Then it also holds for = ¢ K K .…”

“…1 o o s s 1 , , , 1 , , , , 1 , , , , , , , 1 , , , 1 , , , , 1 , , , 1 , , , , , , , , S1 g n I I I I 1 1 1 , , I I I I I I I 1 1 K K K K K I I I I I where the sum on the right-hand side is over all sets of local experiments that can be the ¢ + ( ) K I th set when the first ¢ K consecutive sets are the specified ones. If equation (18) holds for = ¢ + K K I, all terms in the sum can What remains to be shown is that this probability cannot depend on the settings ¼…”

Generalised phase kick-back: the structure of computational algorithms from physical principles

“…Applying operation S to the particle state corresponds to swapping a pair of indistinguishable particles and so must leave the statistics of any measurement invariant. Therefore S s s | ) | ) = , where s | ) is the initial particle state Theorem(2) tells us that the above diagram corresponds to a kicked-back phase on the control system, as in equation (9). Thus, to every particle type, there exists a corresponding phase transformation, which was the connection discussed in [23].…”

“…In contrast, the framework of operationally defined theories [5][6][7][8][9] provides a clear-cut operational language in which to investigate this problem. Theories within this framework can differ [7] from classical and quantum theories.…”