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

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, $T_{\mathrm{eff}}\sim C^{-\eta }$, with $0<\eta \le 2$. 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 also be 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 a reduction of the effective-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.

Figure 1

Empirical results for the antiferromagnetic $K_4$, after encoding, followed by ME and decoding. (a) DW2KQ success probabilities $P_C\left(\alpha \right)$ for 13 nesting levels $C$. Increasing $C$ generally increases $P_C\left(\alpha \right)$ at fixed $\alpha$. (b) Rescaled $P_C\left(\alpha \right)\mapsto P_C(\alpha /{\mu }_C)$ data, exhibiting data collapse. Here, ${\mu }_C$ is chosen for each $C$ such that the quality of the data collapse is maximized. Using the values of ${\mu }_C$ found from data collapse, (c) shows the scaling of the energy boost ${\mu }_C$ vs the amount of physical resources used ($N_{\mathrm{phys}}\sim C^2$). Empirical data are consistent with a power-law scaling ${\mu }_C\sim C^{\eta }$, where $\eta =0.68\pm 0.6$. In all panels, $N_{\mathrm{phys}}\in [8,728]$. The error bars in (a) and (b) correspond to the median, 25th, and 75th percentiles of $P_C\left(\alpha \right)$ computed for each embedding. See Appendix pp2 for more details on the determination of ${\mu }_C$ and the associated error bars.

Figure 2

(a) Success probabilities for a random $K_{16}$ instance with up to $C=3$ nesting levels. Note the monotonic and nonmonotonic improvements of $P_C$ in the scaling (small-$\alpha$) and penalty-limited (large-$\alpha$) regions. (b) Same as (a), but with success probabilities accounting for classical repetition via Eq. (11). Classical repetition outperforms NQAC in the penalty-limited region, while there is an optimal nesting level ($C=2$) for NQAC in the scaling region. (c),(d) The normalized success probabilities (c) without and (d) with classical repetition for the $K_{16}$ ensemble. (e),(f) The same for the $K_{24}$ ensemble. The behavior of the example instance in (a) and (b) is representative for the class of random instances it belongs to.

Figure 3

Results for the (a)–(c) $K_8$ ensemble, (d)–(f) $K_{16}$ ensemble, and (g)–(i) $K_{24}$ ensemble, of 100 instances each. (a),(d),(g) The effective inverse temperature ${\beta }_{C,\mathrm{eff}}$. (b),(e),(h) The collapsed inverse temperature ${\beta }_{C,\mathrm{eff}}$. (c),(f),(i) The gradient overlap $O(p_{\mathrm{DW}},p_T)$. In all plots, we display the median values with the error bars ($25\mathrm{th}$ and $75\mathrm{th}$ percentiles) showing the variation over the ensembles.

Figure 4

Scaling of the inverse effective-temperature reduction coefficient ${\mu }_C$ as a function of $C$ for the four ensembles of instances studied. ${\mu }_C$ was computed as the rescaling coefficient that gives the data collapse in Figs. 3, 3 and 3. Empirical data are consistent with a power-law scaling ${\mu }_C\sim C^{\eta }$, where $\eta \left(K_4\right)=0.66\pm 0.033\eta \left(K_8\right)=0.71\pm 0.15$, $\eta \left(K_{16}\right)=0.84\pm 0.25$, $\eta \left(K_{24}\right)=0.87\pm 0$. See Appendix pp2 for more details on the determination of ${\mu }_C$ and the associated error bars.

Figure 5

(a) Annealing schedule for the DW2KQ (with $\hslash =1$ units). As a reference, we include the operating temperatures of the device, corresponding to 14.1 mK for the DW2KQ. (b) “Chimera” hardware graph of the DW2KQ device in Burnaby used in this work. Available qubits are shown in green and unavailable qubits are shown in red. Programmable couplers are shown as black lines connecting qubits.