Local non-CSS quantum error correcting code on a three-dimensional lattice

TL;DR

This paper introduces a new family of 3D local quantum error-correcting codes with topological order, duality relations, and finite-temperature classical memory properties, expanding the understanding of quantum code design.

Contribution

It presents a novel non-CSS quantum error-correcting code on a 3D lattice with local stabilizers, duality with Ising models, and finite-temperature topological order.

Findings

Codes have local interactions and topological order.

Existence of a strong-weak duality with Ising models.

Some topological order persists at finite temperature.

Abstract

We present a family of non-CSS quantum error-correcting code consisting of geometrically local stabilizer generators on a 3D lattice. We study the Hamiltonian constructed from ferromagnetic interaction of overcomplete set of local stabilizer generators. The degenerate ground state of the system is characterized by a quantum error-correcting code whose number of encoded qubits are equal to the second Betti number of the manifold. These models (i) have solely local interactions; (ii) admit a strong-weak duality relation with an Ising model on a dual lattice; (iii) have topological order in the ground state, some of which survive at finite temperature; and (iv) behave as classical memory at finite temperature.