Feeding the multitude: A polynomial-time algorithm to improve sampling

A wide variety of optimization techniques, both exact and heuristic, tend to be biased samplers. This means that when attempting to find multiple uncorrelated solutions of a degenerate Boolean optimization problem a subset of the solution space tends to be favored while, in the worst case, some solutions can never be accessed by the algorithm used. Here we present a simple postprocessing technique that improves sampling for any optimization approach, either quantum or classical. More precisely, starting from a pool of a few optimal configurations, the algorithm generates potentially new solutions via rejection-free cluster updates at zero temperature. Although the method is not ergodic and there is no guarantee that all the solutions can be found, fair sampling is typically improved. We illustrate the effectiveness of our method by improving the exponentially biased data produced by the D-Wave 2X quantum annealer [S. Mandrà et al., Phys. Rev. Lett. 118, 070502 (2017)], as well as data from three-dimensional Ising spin glasses. As part of the study, we also show that sampling is improved when suboptimal states are included and discuss sampling at a finite fixed temperature.

Figure 1

Number of solutions generated using cluster updates $\left(N_{\mathrm{out}}^{\mathrm{GS}}\right)$ as a function of ground states found using the D-Wave 2X quantum annealer $\left(N_{\mathrm{in}}^{\mathrm{GS}}\right)$ for different system sizes $N$ and known degeneracies $G$: (a) $N=512$, (b) $N=648$, (c) $N=800$, and (d) $N=968$. Each point represents an individual instance. Any points to the left of the $N_{\mathrm{out}}^{\mathrm{GS}}=N_{\mathrm{in}}^{\mathrm{GS}}$ (solid diagonal line) mean that additional minimizing configurations are found. For points that lie on the line, either all solutions were already known or the cluster updates produced no improvement.

Figure 2

Relative improvement $[N_{\mathrm{out}}^{\mathrm{GS}}/N_{\mathrm{in}}^{\mathrm{GS}}]$ averaged over instances as a function of known degeneracy $G$ for different system sizes $N$ after postprocessing the minimizing configurations obtained with the D-Wave 2X quantum annealer. Square brackets represent a disorder average over all instances.

Figure 3

Number of solutions generated using cluster updates $\left(N_{\mathrm{out}}^{\mathrm{GS}}\right)$ as a function of ground states found using the D-Wave 2X quantum annealer $\left(N_{\mathrm{in}}^{\mathrm{GS}}\right)$ for different system sizes $N$ and known degeneracies $G$: (a) $N=512$, (b) $N=648$, (c) $N=800$, and (d) $N=968$. Each point represents an individual instance. Note that both ground states and first-excited states were fed into the cluster update in this experiment. Including a more diverse set of configurations and keeping track of minimizing configurations results in a better sampling of the ground-state manifold, as can be seen for cases where only one ground-state configuration was available and many more ground states were obtained by adding first-excited states into the state pool (data points on the vertical axis).

Figure 4

Number of solutions generated using cluster updates $\left(N_{\mathrm{out}}^{\mathrm{GS}}\right)$ as a function of only excited states found using the D-Wave 2X quantum annealer $\left(N_{\mathrm{in}}^{\mathrm{ES}}\right)$ for different system sizes $N$ and known degeneracies $G$: (a) $N=512$, (b) $N=648$, (c) $N=800$, and (d) $N=968$. Each point represents an individual instance. For all points that lie above the horizontal axis, the resampled excited states produced new minimizing configurations.

Figure 5

Relative improvement $[[N_{\mathrm{out}}^{\mathrm{GS}}/p{N}_{\mathrm{in}}^{\mathrm{GS}}]]$ averaged over instances and ten independent runs as a function of known degeneracy $G$ for different system sizes $N$ and only using a fraction $p$ of available states after postprocessing the minimizing configurations obtained with the D-Wave 2X quantum annealer: (a) $p=0.25$, (b) $p=0.50$, and (c) $p=0.75$. The data are normalized such that for $[[N_{\mathrm{out}}^{\mathrm{GS}}/p{N}_{\mathrm{in}}^{\mathrm{GS}}]]=1$ all known solutions in the initial ground-state pool are found. Double-square brackets represent averages over instances and ten independent random trials.

Figure 6

Maximum-minimum hamming distance versus the system size $N$ with degeneracy $G=24$ for the chimera graph. For each instance the minimum Hamming distance between each new ground state found and the original pool of configurations is calculated. The data shown are the maximum of these minima. The box represents the $25\%–50\%$ confidence interval, the centerline is the median, the bars are the $5\%–95\%$ confidence interval, and the points are outliers. Data are shown from initial pools consisting of (a) only ground states, (b) only excited states, and (c) both ground states and excited states. Outliers represent nontrivial solutions with large Hamming distance from the initial pool found by the method.

Figure 7

Maximum-minimum hamming distance versus the system size $N$ with degeneracy $G=96$ for the chimera graph. For each instance the minimum Hamming distance between each new ground state found and the original pool of configurations is calculated. The data shown are the maximum of these minima. The box represents the $25\%–50\%$ confidence interval, the centerline is the median, the bars are the $5\%–95\%$ confidence interval, and the points are outliers. Data are shown from initial pools consisting of (a) only ground states, (b) only excited states, and (c) both ground states and excited states. Outliers represent nontrivial solutions with large Hamming distance from the initial pool found by the method.

Figure 8

Number of solutions generated using cluster updates $\left(N_{\mathrm{out}}^{\mathrm{GS}}\right)$ as a function of only excited states found using simulated annealing $\left(N_{\mathrm{in}}^{\mathrm{ES}}\right)$ for different system sizes $N$ and known degeneracies $G$ for three-dimensional systems: (a) $N=64\left(L=4\right)$, (b) $N=125\left(L=5\right)$, and (c) $N=216\left(L=6\right)$. Each point represents an individual instance. For all points that lie above the horizontal axis, the resampled excited states produced new minimizing configurations. Note that for $N=64$ most instanced were solved, i.e., the pool of excited states is small.

Figure 9

Relative improvement $[[N_{\mathrm{out}}^{\mathrm{GS}}/p{N}_{\mathrm{in}}^{\mathrm{GS}}]]$ averaged over instances and ten independent runs as a function of known degeneracy $G$ for different system sizes $N$ and only using a fraction $p$ of available states after postprocessing the minimizing configurations obtained with simulated annealing in three space dimensions: (a) $p=0.25$, (b) $p=0.50$, and (c) $p=0.75$. The data are normalized such that for $[[N_{\mathrm{out}}^{\mathrm{GS}}/p{N}_{\mathrm{in}}^{\mathrm{GS}}]]=1$ all known solutions in the initial ground-state pool are found. Double-square brackets represent averages over instances and ten independent random trials.

Figure 10

Histograms of the change in energy $\delta E$ between states found by the D-Wave 2X quantum annealer and the same states postprocessing them with the cluster updates for different system sizes $N=8\times c^2$ for problems with Gaussian-distributed couplers: (a) $N=512$, (b) $N=648$, and (c) $N=800$. In most cases lower-energy configurations were obtained using the cluster algorithm with the improvement becoming more pronounced for the larger system sizes.

Figure 11

Histogram of the distribution of $\mathrm{\Delta }E=|\mathrm{\Delta }{E}^{\left(1\right)}|=|\mathrm{\Delta }{E}^{\left(2\right)}|$ of cluster moves around the average thermodynamic energy $\langle E\rangle$ at different temperatures $T$ in three space dimensions $\left(N=216\right)$ during a finite-temperature Monte Carlo simulation that includes cluster updates. In general, the changes in energy are small compared to the average energy with rare large rearrangements. Both panels show the same data set with (b) zooming into the tails.

Figure 12

Comparison of ${\mathrm{\Theta }}_{\text{max}}$ (maximum absolute difference of the empirical cumulative distribution with respect to the expected distribution; see the text) for the distribution of states found by the D-Wave 2X (DW2X) device at a given energy $E=E_{\text{gs}}+\mathrm{\Delta }E$ for $N=512$ and $N=648$. The baseline for ${\mathrm{\Theta }}_{\text{max}}$ is computed by using a random uniform distribution. As one can see, the bias in sampling for the D-Wave 2X device persists up to the $20\text{th}$ level when ${\mathrm{\Theta }}_{\text{max}}$ for the quantum annealer is comparable to the baseline.

Figure 13

(a) Binned number of times a ground-state configuration is found using the D-Wave 2X quantum annealer sorted by rank compared to (b) the cluster-update postprocessed data. Data for instances on a chimera lattice with $N$ sites and $G=3\times 2^k$ ground states are shown. The horizontal axis is normalized by $G$ for easier display of the data. While the magnitudes of the vertical axes are different in both panels, in both cases the range spans four orders of magnitudes. This allows for a direct visual comparison of the slopes. The raw data show a clear bias in (a) that is reduced considerably after applying the cluster updates shown in (b).

Figure 14

Number of hits for $N={10}^5$ uniformly distributed random numbers among $3\times 2^k$ different values, with $k=5$. As expected, the histogram of number of hits follows the inverse of the error function.

Figure 15

Normalized number of hits for $N={10}^5$ and $N={10}^6$ uniformly distributed random numbers among $3\times 2^k$ different values, with $k=5$. As shown in the figure, the normalized histogram becomes flatter by increasing the number of random numbers.