A SCHEMATIC DEFINITION OF QUANTUM POLYNOMIAL TIME COMPUTABILITY

A Schematic Definition of Quantum Polynomial Time Computability

In the past four decades, the notion of quantum polynomial-time computability has been realized by the theoretical models of quantum Turing machines and quantum circuits. Here, we seek a third model, which is a quantum analogue of the schematic (inductive or constructive) definition of (primitive) recursive functions. For quantum functions mapping finite-dimensional Hilbert spaces to themselves...

On function spaces and polynomial-time computability

In Computable Analysis, elements of uncountable spaces, such as the real line R, are represented by functions on strings and fed to Turing machines as oracles; or equivalently, they are represented by infinite strings and written on the tapes of Turing machines [Wei00, BHW08]. To obtain reasonable notions of computability and complexity, it is hence important to choose the “right” representatio...

Characterizing Polynomial Time Computability of Rational and Real Functions

Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real function...

On semantic and type-theoretic aspects of polynomial-time computability

Computability over an Arbitrary Structure. Sequential and Parallel Polynomial Time

We provide several machine-independent characterizations of deterministic complexity classes in the model of computation proposed by L. Blum, M. Shub and S. Smale. We provide a characterization of partial recursive functions over any arbitrary structure. We show that polynomial time computable functions over any arbitrary structure can be characterized in term of safe recursive functions. We sh...