Continuous theory of operator expansions of finite dimensional Hilbert spaces, continuous structures of quantum circuits and decidability

TL;DR

This paper develops a continuous logical framework for finite-dimensional Hilbert spaces with unitary operators, linking quantum automata and circuits to decidability and approximation problems in metric groups.

Contribution

It extends previous work by providing a comprehensive continuous theory for quantum circuit structures, including new results on decidability and approximation.

Findings

Established a continuous logical framework for quantum circuit structures

Connected continuous theories with approximation by metric groups

Extended prior results with new sections and corrections

Abstract

We consider continuous structures which are obtained from finite dimensional Hilbert spaces over $\mathbb{C}$ by adding some unitary operators. Quantum automata and circuits are naturally interpretable in such structures. We consider appropriate algorithmic problems concerning continuous theories of natural classes of these structures. We connect them with the topic of approximations by metric groups. This paper extends and corrects the paper A. Ivanov, "Continuous structures of quantum circuits", arXiv: 1406.4635. Replacing version contains a new section (Section 4.4).