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Nested quantum annealing correction (NQAC) is an error correcting scheme forquantum annealing that allows for the encoding of a logical qubit into anarbitrarily large number of physical qubits. The encoding replaces each logicalqubit by a complete graph of degree $C$. The nesting level $C$ represents thedistance of the error-correcting code and controls the amount of protectionagainst thermal and control errors. Theoretical mean-field analyses andempirical data obtained with a D-Wave Two quantum annealer (supporting up to$512$ qubits) showed that NQAC has the potential to achieve a scalableeffective temperature reduction, $T_{\rm eff} \sim C^{-\eta}$, with $\eta \leq2$. We confirm that this scaling is preserved when NQAC is tested on a D-Wave2000Q device (supporting up to $2048$ qubits). In addition, we show that NQACcan be also used in sampling problems to lower the effective temperature of aquantum annealer. Such effective temperature reduction is relevant formachine-learning applications. Since we demonstrate that NQAC achieves errorcorrection via an effective reduction of the temperature of the quantumannealing device, our results address the problem of the "temperature scalinglaw for quantum annealers", which requires the temperature of quantum annealersto be reduced as problems of larger sizes are attempted to be solved.

Scalable effective temperature reduction for quantum annealers via nested quantum annealing correction