Local stabilizer codes in three dimensions without string logical operators

TL;DR

This paper introduces 3D local stabilizer codes without string logical operators, advancing the development of self-correcting quantum memory by eliminating one-dimensional logical operators.

Contribution

It presents new 3D stabilizer code models with no string logical operators and introduces the concept of logical string segments to analyze their properties.

Findings

Codes lack string logical operators, enabling self-correction.

Logical string segments can be deformed into short stabilizer-bound segments.

Surface-like logical operators have boundary stabilizer violations.

Abstract

We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have string-like logical operators, which make the codes non-self-correcting. We introduce a notion of "logical string segments" to avoid difficulties in defining one dimensional objects in discrete lattices. We prove that every string-like logical operator of our code can be deformed to a disjoint union of short segments, and each segment is in the stabilizer group. The code has surface-like logical operators whose partial implementation has unsatisfied stabilizers along its boundary.