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Oracles and Query Lower Bounds in Generalised Probabilistic Theories.
Howard Barnum1,2, Ciarán M Lee3, John H Selby4,5
11Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark.
Foundations of Physics
|November 6, 2018
Summary
We explore how interference impacts computational power in generalized probabilistic theories. Theories with four physical principles have an oracle model, suggesting higher-order interference could offer computational power beyond quantum computing.
Area of Science:
- Foundations of quantum mechanics
- Theoretical computer science
- Information theory
Background:
- Generalized probabilistic theories offer a framework to explore computation beyond classical and quantum limits.
- Understanding the role of interference in computation is crucial for developing new computational paradigms.
Purpose of the Study:
- To investigate the link between interference and computational power within generalized probabilistic theories.
- To establish conditions for a well-defined oracle model in these theories.
- To explore the potential for higher-order interference to surpass quantum computational capabilities.
Main Methods:
- Defined an oracle model for theories satisfying causality, purification, strong symmetry, and informationally consistent composition.
- Proved a subroutine theorem for oracles, a necessary condition for the oracle model's validity.
- Utilized Sorkin's hierarchy of interference behaviors to analyze computational lower bounds.
Main Results:
- Any theory satisfying the four physical principles possesses a well-defined oracle model.
- A "no-information" lower bound for a learning problem scales with the level of interference in Sorkin's hierarchy.
- The established lower bound suggests quantum oracles may not be as powerful as general probabilistic oracles.
Conclusions:
- Searches for higher-order interference are motivated by fundamental physics and the pursuit of computational resources beyond quantum computation.
- The findings open avenues for exploring novel computational paradigms based on advanced interference phenomena.
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