Dimension minimization of a quantum automaton

TL;DR

This paper introduces a method to minimize the dimension of quantum automata by leveraging properties of invariant subspaces and unitary transformations, resulting in an efficient algorithm for reducing qubits needed.

Contribution

It presents a novel approach to dimension minimization of quantum automata using invariant subspaces and provides a polynomial-time algorithm for finding minimal automata.

Findings

Dimension reduction is possible under specific conditions on unitary transformations.

An algorithm for minimal quantum automaton construction is developed.

The algorithm's complexity is polynomial in automaton size and input alphabet.

Abstract

A new model of a Quantum Automaton (QA), working with qubits is proposed. The quantum states of the automaton can be pure or mixed and are represented by density operators. This is the appropriated approach to deal with measurements and dechorence. The linearity of a QA and of the partial trace super-operator, combined with the properties of invariant subspaces under unitary transformations, are used to minimize the dimension of the automaton and, consequently, the number of its working qubits. The results here developed are valid wether the state set of the QA is finite or not. There are two main results in this paper: 1) We show that the dimension reduction is possible whenever the unitary transformations, associated to each letter of the input alphabet, obey a set of conditions. 2) We develop an algorithm to find out the equivalent minimal QA and prove that its complexity is polynomial in its dimension and in the size of the input alphabet.