Thermalization time bounds for Pauli stabilizer Hamiltonians

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