Thermalization time bounds for Pauli stabilizer Hamiltonians

TL;DR

This paper establishes a lower bound on the spectral gap of the Davies generator for commuting Pauli Hamiltonians, which are key in quantum memory models, providing insights into their thermalization times and stability.

Contribution

It introduces a general lower bound on the spectral gap for these Hamiltonians, linking it to energy barriers and system size, with implications for quantum memory stability.

Findings

Spectral gap bound scales as N^{-1} exponential in inverse temperature and energy barrier

At low temperatures, the bound accurately predicts the gap's asymptotic behavior

Conditions are discussed under which the bound can be improved to a constant

Abstract

We prove a general lower bound to the spectral gap of the Davies generator for Hamiltonians that can be written as the sum of commuting Pauli operators. These Hamiltonians, defined on the Hilbert space of $N$-qubits, serve as one of the most frequently considered candidates for a self-correcting quantum memory. A spectral gap bound on the Davies generator establishes an upper limit on the life time of such a quantum memory and can be used to estimate the time until the system relaxes to thermal equilibrium when brought into contact with a thermal heat bath. The bound can be shown to behave as $\lambda \geq {\cal O}(N^{-1}\exp(-2\beta \, \overline{\epsilon}))$, where $\overline{\epsilon}$ is a generalization of the well known energy barrier for logical operators. Particularly in the low temperature regime we expect this bound to provide the correct asymptotic scaling of the gap with the system size up to a factor of $N^{-1}$. Furthermore, we discuss conditions and provide scenarios where this factor can be removed and a constant lower bound can be proven