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Oracles and Query Lower Bounds in Generalised Probabilistic Theories
Foundations of Physics ( IF 1.5 ) Pub Date : 2018-07-12 , DOI: 10.1007/s10701-018-0198-4
Howard Barnum 1, 2 , Ciarán M Lee 3 , John H Selby 4, 5
Affiliation  

We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying four natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle model to be well-defined. The four principles are: causality (roughly, no signalling from the future), purification (each mixed state arises as the marginal of a pure state of a larger system), strong symmetry (existence of a rich set of nontrivial reversible transformations), and informationally consistent composition (roughly: the information capacity of a composite system is the sum of the capacities of its constituent subsystems). Sorkin has defined a hierarchy of conceivable interference behaviours, where the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. Given our oracle model, we show that if a classical computer requires at least n queries to solve a learning problem, because fewer queries provide no information about the solution, then the corresponding “no-information” lower bound in theories lying at the kth level of Sorkin’s hierarchy is $$\lceil {n/k}\rceil $$⌈n/k⌉. This lower bound leaves open the possibility that quantum oracles are less powerful than general probabilistic oracles, although it is not known whether the lower bound is achievable in general. Hence searches for higher-order interference are not only foundationally motivated, but constitute a search for a computational resource that might have power beyond that offered by quantum computation.

中文翻译:

广义概率理论中的预言和查询下界

我们在广义概率理论的操作定义框架内研究干扰和计算能力之间的联系。为了比较该框架内不同理论的计算能力,我们表明任何满足四个自然物理原理的理论都拥有明确定义的预言模型。事实上,我们在此类理论中证明了预言机的子程序定理,这是明确定义预言机模型的必要条件。这四个原则是:因果关系(粗略地说,没有来自未来的信号)、纯化(每个混合状态都作为较大系统的纯状态的边缘出现)、强对称性(存在一组丰富的非平凡可逆变换)以及信息一致的组合(粗略地说:复合系统的信息容量是其组成子系统的容量之和)。索尔金定义了可想象的干扰行为的层次结构,其中层次结构中的顺序对应于多缝实验中具有不可约相互作用的路径数量。给定我们的预言机模型,我们表明,如果经典计算机需要至少 n 个查询来解决学习问题,因为较少的查询不提供有关解决方案的信息,则理论中相应的“无信息”下界位于第 k 层索金的层次结构是 $$\lceil {n/k}\rceil $$⌈n/k⌉。这个下限留下了一种可能性,即量子预言机不如一般概率预言机强大,尽管尚不清楚该下限一般是否可以实现。因此,对高阶干涉的搜索不仅具有根本动机,而且构成了对可能具有超出量子计算提供的能力的计算资源的搜索。
更新日期:2018-07-12
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