Oracle Separations for RPH

TL;DR

This paper introduces the real polynomial hierarchy (RPH) as an extension of classical models using real tapes, and establishes oracle separation results to compare real and discrete computational complexities, suggesting potential strict inclusions.

Contribution

It develops a technique to transfer oracle separation results from binary to real computation models, providing new insights into the hierarchy and limitations of real and quantum computation.

Findings

RPH is properly contained in PSPACE under certain oracles

Separation between levels of the real polynomial hierarchy

Quantum polynomial time not simulated by real Turing machines

Abstract

While theoretical computer science primarily works with discrete models of computation, like the Turing machine and the wordRAM, there are many scenarios in which introducing real computation models is more adequate. We want to compare real models of computation with discrete models of computation. We do this by means of oracle separation results. We define the notion of a real Turing machine as an extension of the (binary) Turing machine by adding a real tape. Using those machines, we define and study the real polynomial hierarchy RPH. We are interested in RPH as the first level of the hierarchy corresponds to the well-known complexity class ER. It is known that $NP \subseteq ER \subseteq PSPACE$ and furthermore $PH \subseteq RPH \subseteq PSPACE$. We are interested to know if any of those inclusions are tight. In the absence of unconditional separations of complexity classes, we turn to oracle separation. We develop a technique that allows us to transform oracle separation results from the binary world to the real world. As applications, we show there are oracles such that: - $RPH^O$ proper subset of $PSPACE^O$, - $\Sigma_{k+1}^O$ not contained in $\Sigma_kR^O$, for all $k\geq 0$, - $\Sigma_kR^O$ proper subset of $\Sigma_{k+1}R^O$, for all $k\geq 0$, - $BQP^O$ not contained in $RPH^O$. Our results hint that ER is strictly contained in PSPACE and that there is a separation between the different levels of the real polynomial hierarchy. We also bound the power of real computations by showing that NP-hard problems are unlikely to be solvable using polynomial time on a realRAM. Furthermore, our oracle separations hint that polynomial-time quantum computing cannot be simulated on an efficient real Turing machine.