  We present a general error-correcting scheme for quantum annealing that
allows for the encoding of a logical qubit into an arbitrarily large number of
physical qubits. Given any Ising model optimization problem, the encoding
replaces each logical qubit by a complete graph of degree $C$, representing the
distance of the error-correcting code. A subsequent minor-embedding step then
implements the encoding on the underlying hardware graph of the quantum
annealer. We demonstrate experimentally that the performance of a D-Wave Two
quantum annealing device improves as $C$ grows. We show that the performance
improvement can be interpreted as arising from an effective increase in the
energy scale of the problem Hamiltonian, or equivalently, an effective
reduction in the temperature at which the device operates. The number $C$ thus
allows us to control the amount of protection against thermal and control
errors, and in particular, to trade qubits for a lower effective temperature
that scales as $C^{-\eta}$, with $\eta \leq 2$. This effective temperature
reduction is an important step towards scalable quantum annealing.
