  Physical constraints make it challenging to implement and control many-body
interactions. For this reason, designing quantum information processes with
Hamiltonians consisting of only one- and two-local terms is a worthwhile
challenge. Enabling error suppression with two-local Hamiltonians is
particularly challenging. A no-go theorem of Marvian and Lidar [Physical Review
Letters 113(26), 260504 (2014)] demonstrates that, even allowing particles with
high Hilbert-space dimension, it is impossible to protect quantum information
from single-site errors by encoding in the ground subspace of any Hamiltonian
containing only commuting two-local terms. Here, we get around this no-go
result by encoding in the ground subspace of a Hamiltonian consisting of
non-commuting two-local terms arising from the gauge operators of a subsystem
code. Specifically, we show how to protect stored quantum information against
single-qubit errors using a Hamiltonian consisting of sums of the gauge
generators from Bacon-Shor codes [Physical Review A 73(1), 012340 (2006)] and
generalized-Bacon-Shor code [Physical Review A 83(1), 012320 (2011)]. Our
results imply that non-commuting two-local Hamiltonians have more
error-suppressing power than commuting two-local Hamiltonians. While far from
providing full fault tolerance, this approach improves the robustness
achievable in near-term implementable quantum storage and adiabatic quantum
computations, reducing the number of higher-order terms required to encode
commonly used adiabatic Hamiltonians such as the Ising Hamiltonians common in
adiabatic quantum optimization and quantum annealing.

Quantum Physics