No Strings Attached: Boundaries and Defects in the Cubic Code

TL;DR

This paper extends Haah's cubic code to systems with boundaries and defects, revealing how these features influence topological excitations and enabling new encoding schemes with improved logical qubit scaling.

Contribution

It introduces boundary and defect configurations in the cubic code, demonstrating their effects on excitations and proposing new encodings with linear and superlinear scaling.

Findings

Boundaries and defects can induce local string-like operators.

Defects enhance mobility of fractonic excitations.

New encodings achieve linear scaling of logical qubits.

Abstract

Haah's cubic code is the prototypical type-II fracton topological order. It instantiates the no string-like operator property that underlies the favorable scaling of its code distance and logical energy barrier. Previously, the cubic code was only explored in translation-invariant systems on infinite and periodic lattices. In these settings, the code distance scales superlinearly with the linear system size, while the number of logical qubits within the degenerate ground space exhibits a complicated functional dependence that undergoes large fluctuations within a linear envelope. Here, we extend the cubic code to systems with open boundary conditions and crystal lattice defects. We characterize the condensation of topological excitations in the vicinity of these boundaries and defects, finding that their inclusion can introduce local string-like operators and enhance the mobility of otherwise fractonic excitations. Despite this, we use these boundaries and defects to define new encodings where the number of logical qubits scales linearly without fluctuations, and the code distance scales superlinearly, with the linear system size. These include a subsystem encoding with open boundary conditions and a subspace encoding using lattice defects.