  Nested quantum annealing correction (NQAC) is an error correcting scheme for
quantum annealing that allows for the encoding of a logical qubit into an
arbitrarily large number of physical qubits. The encoding replaces each logical
qubit by a complete graph of degree $C$. The nesting level $C$ represents the
distance of the error-correcting code and controls the amount of protection
against thermal and control errors. Theoretical mean-field analyses and
empirical data obtained with a D-Wave Two quantum annealer (supporting up to
$512$ qubits) showed that NQAC has the potential to achieve a scalable
effective temperature reduction, $T_{\rm eff} \sim C^{-\eta}$, with $\eta \leq
2$. We confirm that this scaling is preserved when NQAC is tested on a D-Wave
2000Q device (supporting up to $2048$ qubits). In addition, we show that NQAC
can be also used in sampling problems to lower the effective temperature of a
quantum annealer. Such effective temperature reduction is relevant for
machine-learning applications. Since we demonstrate that NQAC achieves error
correction via an effective reduction of the temperature of the quantum
annealing device, our results address the problem of the "temperature scaling
law for quantum annealers", which requires the temperature of quantum annealers
to be reduced as problems of larger sizes are attempted to be solved.
