Protected Qubits and Chern Simons theories in Josephson Junction Arrays

TL;DR

This paper explores how certain Hamiltonians with specific symmetries can host protected doubly degenerate states, which are potentially useful for fault-tolerant quantum computing, by connecting them to Josephson junction arrays and lattice Chern-Simons theories.

Contribution

The authors identify a class of Hamiltonians with symmetry-protected degeneracy, demonstrate their realization in Josephson junction arrays, and connect them to lattice Chern-Simons theories.

Findings

Protected degeneracy can be realized in Josephson junction arrays.

Low energy modes scale with system size, affecting protection.

Mapping to Z_2 Chern-Simons models explains degeneracy.

Abstract

We present general symmetry arguments that show the appearance of doubly denerate states protected from external perturbations in a wide class of Hamiltonians. We construct the simplest spin Hamiltonian belonging to this class and study its properties both analytically and numerically. We find that this model generally has a number of low energy modes which might destroy the protection in the thermodynamic limit. These modes are qualitatively different from the usual gapless excitations as their number scales as the linear size (instead of volume) of the system. We show that the Hamiltonians with this symmetry can be physically implemented in Josephson junction arrays and that in these arrays one can eliminate the low energy modes with a proper boundary condition. We argue that these arrays provide fault tolerant quantum bits. Further we show that the simplest spin model with this symmetry can be mapped to a very special Z_2 Chern-Simons model on the square lattice. We argue that appearance of the low energy modes and the protected degeneracy is a natural property of lattice Chern-Simons theories. Finally, we discuss a general formalism for the construction of discrete Chern-Simons theories on a lattice.