Bounds on Effective Hamiltonians for Stabilizer Codes

TL;DR

This paper investigates the limitations on constructing low-locality effective Hamiltonians for stabilizer codes, providing conditions where such Hamiltonians cannot exist, especially for certain well-known code classes.

Contribution

It establishes theoretical conditions that prevent the existence of low-locality effective Hamiltonians for stabilizer codes, simplifying the analysis for specific code families.

Findings

Effective Hamiltonians do not exist under certain conditions.

Results apply to Calderbank-Shor-Steane and surface code stabilizer codes.

Simplifies understanding of Hamiltonian locality constraints.

Abstract

This manuscript introduces various notions of k-locality of stabilizer codes inherited from the associated stabilizer groups. A choice of generators for the group leads to a Hamiltonian with the code in its groundspace, while a Hamiltonian holding the code in its groundspace might be called effective if its locality is less than that of a natural choice of generators (or any choice). This paper establishes some conditions under which effective Hamiltonians for stabilizer codes do not exist. Our results simplify in the cases of Calderbank-Shor-Steane stabilizer codes and topologically-ordered stabilizer codes arising from surface cellulations.