Boolean Functions, Projection Operators and Quantum Error Correcting Codes

TL;DR

This paper establishes a fundamental link between Boolean functions and projection operators in Hilbert space, providing a unified framework for designing various quantum error correcting codes, including new classes and extensions.

Contribution

It introduces a novel mathematical framework connecting Boolean functions with projection operators, enabling the construction of diverse quantum error correcting codes.

Findings

Constructed an infinite class of quantum codes extending the ((5,6,2)) code.

Unified framework applies to both additive and non-additive codes.

Extended the framework to operator quantum error correcting codes.

Abstract

This paper describes a fundamental correspondence between Boolean functions and projection operators in Hilbert space. The correspondence is widely applicable, and it is used in this paper to provide a common mathematical framework for the design of both additive and non-additive quantum error correcting codes. The new framework leads to the construction of a variety of codes including an infinite class of codes that extend the original ((5,6,2)) code found by Rains [21]. It also extends to operator quantum error correcting codes.