Kolmogorov complexity of unitary transformations in quantum computing

TL;DR

This paper introduces a new measure of Kolmogorov complexity for unitary transformations in quantum computing, quantifying the minimal information needed to describe quantum operations.

Contribution

It defines a novel Kolmogorov complexity for unitaries based on qubit complexity, extending classical notions to quantum transformations.

Findings

Provides a complexity bound for unitary transformations.

Discusses the optimality of the proposed complexity measure.

Connects the complexity to quantum circuit description length.

Abstract

We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the context of quantum computing, it corresponds to the least possible amount of data to define and describe a quantum circuit or quantum computer program. Our Kolmogorov complexity of unitary transformation is built upon Kolmogorov "qubit complexity" of Berthiaume, W. Van Dam and S. Laplante via mapping from unitary transformations to positive operators, which are subsequently "purified". We discuss the optimality of our notion of Kolmogorov complexity in a broad sense and obtain a simple complexity bound.