Oracles and Query Lower Bounds in Generalised Probabilistic Theories

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 pri...

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Bibliographic Details
Published in:Foundations of physics Vol. 48; no. 8; pp. 954 - 981
Main Authors: Barnum, Howard, Lee, Ciarán M., Selby, John H.
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2018
Springer Nature B.V
Subjects:
Classical and Quantum Gravitation
Classical Mechanics
Computation
Foundational Aspects of Quantum Information
History and Philosophical Foundations of Physics
Interference
Lower bounds
Philosophy of Science
Physics
Physics and Astronomy
Quantum Physics
Queries
Relativity Theory
Statistical Physics and Dynamical Systems
Query complexity
Generalised probabilistic theories
Higher order interference
Oracles
ISSN:0015-9018, 1572-9516
Online Access:Get full text
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Summary: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 k th level of Sorkin’s hierarchy is ⌈ 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.
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ISSN:0015-9018
1572-9516
DOI:10.1007/s10701-018-0198-4