Quantum error suppression with commuting Hamiltonians: Two-local is too local

TL;DR

This paper proves that 2-local commuting Hamiltonians cannot effectively protect quantum information from errors due to their limited entanglement and degeneracy splitting, limiting their use in quantum error suppression.

Contribution

The authors demonstrate that 2-local commuting Hamiltonians are ineffective for quantum error suppression, introducing a new no-hiding theorem and analyzing degeneracy splitting effects.

Findings

Degeneracy splitting in 2-local Hamiltonians is proportional to perturbation magnitude.

Ground subspaces of 2-local Hamiltonians have only short-range entanglement.

Quantum information cannot be effectively hidden in correlations of 2-local commuting Hamiltonians.

Abstract

We consider error suppression schemes in which quantum information is encoded into the ground subspace of a Hamiltonian comprising a sum of commuting terms. Since such Hamiltonians are gapped they are considered natural candidates for protection of quantum information and topological or adiabatic quantum computation. However, we prove that they cannot be used to this end in the 2-local case. By making the favorable assumption that the gap is infinite we show that single-site perturbations can generate a degeneracy splitting in the ground subspace of this type of Hamiltonians which is of the same order as the magnitude of the perturbation, and is independent of the number of interacting sites and their Hilbert space dimensions, just as in the absence of the protecting Hamiltonian. This splitting results in decoherence of the ground subspace, and we demonstrate that for natural noise models the coherence time is proportional to the inverse of the degeneracy splitting. Our proof involves a new version of the no-hiding theorem which shows that quantum information cannot be approximately hidden in the correlations between two quantum systems, and should be of independent interest. The main reason that 2-local commuting Hamiltonians cannot be used for quantum error suppression is that their ground subspaces have only short-range (two-body) entanglement.