Spectra of Gauge Code Hamiltonians

TL;DR

This paper investigates the spectral gap of gauge code Hamiltonians, especially the 3D gauge color code, using Perron-Frobenius theory to analyze eigenvalues and explore conditions for gapped spectra in quantum memory models.

Contribution

It introduces a method to analyze spectral gaps of gauge code Hamiltonians and provides numerical results for the 3D gauge color code, linking stabilizer properties to spectral gaps.

Findings

Numerical estimation of the spectral gap for large 3D gauge color code instances.

Constraints on eigenvalues of Hamiltonian blocks using Perron-Frobenius theory.

A proposed relation between bounded stabilizers and the presence of a spectral gap.

Abstract

We study the spectral gap of frustrated spin (qubit) Hamiltonians constructed from quantum subsystem (gauge) codes. Such a Hamiltonian can be block diagonalized, with blocks labelled by eigenvalues of extensively many integrals of motion (stabilizers of the code.) Of particular interest is the 3D gauge color code, whose Hamiltonian has been conjectured to act as a quantum memory at finite temperature. Using Perron-Frobenius theory we constrain the location of first and second eigenvalues among the blocks of the Hamiltonian. This allows us to numerically find the gap of some large instances of the 3D gauge color code, which is compared to other frustrated spin Hamiltonians. Finally, we suggest a relation between bounded stabilizers and gapped spectra.