A Schematic Definition of Quantum Polynomial Time Computability

by   Tomoyuki Yamakami, et al.

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, we present such a schematic definition, composed of a small set of initial quantum functions and a few construction rules that dictate how to build a new quantum function from the existing quantum functions. We prove that our schematic definition precisely characterizes all functions that can be computable with high success probabilities on well-formed quantum Turing machines in polynomial time or equivalently, uniform families of polynomial-size quantum circuits. Our new, schematic definition is quite simple and intuitive and, more importantly, it avoids the cumbersome introduction of the well-formedness condition imposed on a quantum Turing machine model as well as of the uniformity condition necessary for a quantum circuit model. Our new approach can further open a door to the descriptional complexity of other functions and to the theory of higher-type quantum functionals.


page 1

page 2

page 3

page 4


Revisiting the simulation of quantum Turing machines by quantum circuits

Yao (1993) proved that quantum Turing machines and uniformly generated q...

Simple circuit simulations of classical and quantum Turing machines

We construct reversible Boolean circuits efficiently simulating reversib...

A Turing machine simulation by P systems without charges

It is well known that the kind of P systems involved in the definition o...

Quantum Random Access Stored-Program Machines

Random access machines (RAMs) and random access stored-program machines ...

Quantum Kolmogorov complexity and quantum correlations in deterministic-control quantum Turing machines

This work presents a study of Kolmogorov complexity for general quantum ...

Oracle Computability and Turing Reducibility in the Calculus of Inductive Constructions

We develop synthetic notions of oracle computability and Turing reducibi...

Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers

A common lens to theoretically study neural net architectures is to anal...

Please sign up or login with your details

Forgot password? Click here to reset