Circuits: An abstract viewpoint

TL;DR

This paper explores the fundamental mathematical properties of classical and quantum circuits to understand their computational essence and compare their similarities and differences.

Contribution

It provides an abstract, mathematical framework for analyzing circuits, focusing on essential properties while omitting extraneous structures.

Findings

Identifies core mathematical properties of circuits

Clarifies the similarities between classical and quantum circuits

Highlights differences in their abstract structures

Abstract

Our primary purpose is to isolate the abstract, mathematical properties of circuits -- both classical Boolean circuits and quantum circuits -- that are essential for their computational interpretation. A secondary purpose is to clarify the similarities and differences between the classical and quantum situations. The general philosophy in this note is to include the mathematically essential aspects of circuits but to omit any of the additional structures that are usually included for convenience. We shall, however, retain the assumption that circuits are finite; this assumption does no harm to the applicability of our approach and is necessary for some of our work.