# Quantum Turing Machines Computations and Measurements

## TL;DR

This paper introduces a comprehensive definition of quantum Turing machines (QTMs) that incorporates infinite computations, superpositions, and limits, aiming to unify existing models and define a class of quantum computable functions.

## Contribution

It extends and unifies previous QTM models by allowing arbitrary quantum inputs, superpositions, and infinite computations with limit-based outputs, and proposes a robust observation protocol.

## Key findings

- Defined a new model of QTMs with infinite computations and superpositions.
- Established a class of quantum computable functions mapping quantum states to probability distributions.
- Proposed an observation protocol that preserves outcome probabilities.

## Abstract

Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not being fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example being the intrinsic infinite nature of any quantum computation. In this paper we propose a definition of QTM, which extends and unifies the notions of Deutsch and Bernstein and Vazirani. In particular, we allow both arbitrary quantum input, and meaningful superpositions of computations, where some of them are "terminated" with an "output", while others are not. For some infinite computations an "output" is obtained as a limit of finite portions of the computation. We propose a natural and robust observation protocol for our QTMs, that does not modify the probability of the possible outcomes of the machines. Finally, we use QTMs to define a class of quantum computable functions---any such function is a mapping from a general quantum state to a probability distribution of natural numbers. We expect that our class of functions, when restricted to classical input-output, will be not different from the set of the recursive functions.