Logic Functions and Quantum Error Correcting Codes

TL;DR

This paper explores the deep connections between logic functions and quantum error correcting codes (QECCs), providing new unified constructions, proofs of existence, and conditions for Boolean basis states in QECCs over finite fields.

Contribution

It introduces a unified framework for constructing QECCs using graphs, projectors, and logic functions, and establishes new conditions and proofs related to Boolean basis states.

Findings

Unified construction of QECCs over GF(p)

Proof of existence of graphical QECCs using Boolean functions

Conditions for Boolean basis states in QECCs

Abstract

In this paper, based on the relationship between logic functions and quantum error correcting codes(QECCs), we unify the construction of QECCs via graphs, projectors and logic functions. A construction of QECCs over a prime field GF(p) is given, and one of the results given by Ref[8] can be viewed as a corollary of one theorem in this paper. With the help of Boolean functions, we give a clear proof of the existence of a graphical QECC in mathematical view, and find that the existence of an [[n,k,d]] QECC over GF(p) requires similar conditions with that depicted in Ref[9]. The result that under the correspondence defined in Ref[17], every [[n,0,d]] QECC over GF(2) corresponding to a simple undirected graph has a Boolean basis state, which is closely related to the adjacency matrix of the graph, is given. After a modification of the definition of operators, we find that some QECCs constructed via projectors depicted in Ref[11] can have Boolean basis states. A necessary condition for a Boolean function being used in the construction via projectors is given. We also give some examples to illustrate our results.