Comparing Relaxation Mechanisms in Quantum and Classical Transverse-Field Annealing

Annealing schedule control provides opportunities to better understand the manner and mechanisms by which putative quantum annealers operate. By appropriately modifying the annealing schedule to include a pause (keeping the Hamiltonian fixed) for a period of time, we show that it is possible to more directly probe the dissipative dynamics of the system at intermediate points along the anneal and examine thermal relaxation rates, for example, by observing the repopulation of the ground state after the minimum spectral gap. We provide a detailed comparison of experiments from a D-Wave device, simulations of the quantum adiabatic master equation, and a classical analogue of quantum annealing, spin-vector Monte Carlo, and we observe qualitative agreement, showing that the characteristic increase in success probability when pausing is not a uniquely quantum phenomena. We find that the relaxation in our system is dominated by a single timescale, which allows us to give a simple condition for when we can expect pausing to improve the time to solution, the relevant metric for classical optimization. Finally, we also explore in simulation the role of temperature whilst pausing as a means to better distinguish quantum and classical models of quantum annealers.

Figure 1

Example schedule with a pause of length $t_p=0.5\mu \mathrm{s}$ inserted into an anneal with anneal time $t_a=1\mu \mathrm{s}$. The pause occurs at $s_p=0.5$ (i.e., midway through the interpolation). We sometimes refer to the anneal after the pause as the ramp. We use units $h=1$.

Figure 2

Depiction of instance ${\mathcal{I}}_{12}^0$ used in this study. Shown are two Chimera unit cells, with the indexed nodes corresponding to the 12 qubits used in the instance. Red (blue) edges connecting nodes correspond to ferromagnetic (antiferromagnetic) Ising couplings between qubits, with the thickness of the edge in direct proportion to the magnitude of the coupling. Explicit values for the couplings are given in Appendix pp1.

Figure 3

Spectrum (first 20 levels) of ${\mathcal{I}}_{12}^0$. Minimum gap of approximately $\mathrm{\Delta }=0.15$ GHz at $s_{\mathrm{\Delta }}=0.44$. The specified DW operating temperature is 0.25 GHz. We use units $h=1$.

Figure 4

Illustration of the energy level crossing predicted by first-order perturbation theory. At this order in perturbation theory, the ground-state energy $E_0(s)$ remains unchanged while the first excited state energy $E_1(s)$ is lowered as a function of the perturbative parameter $1-s$, resulting in an energy level crossing at $s_{\ast }$.

Figure 5

Semiclassical picture of the dynamics. Left (right) corresponds to early (late) in the anneal. The green circle corresponds to the position of the system in the energy landscape.

Figure 6

Varying the pause time for the DW (top), SVMC TF (middle), and the AME (bottom). All simulations are performed at 12 mK. SVMC TF has an anneal “time” of ${10}^4$ sweeps, and the AME and DW of $1\mu \mathrm{s}$. Note that, for the sake of comparison, the blue curves in each plot have a pause equal to the anneal time $t_a=t_p$, and the other two curves are 1 and 2 orders of magnitude larger. Error bars for the DW are $2\sigma$, computed over at least five independent samples (each sample consisting of ${10}^4$ anneals). Error bars in SVMC TF are $2\sigma$ over ${10}^4$ samples. Note that the AME data is “exact.” The vertical dashed line denotes the position of the minimum gap $s_{\mathrm{\Delta }}$. Note that the $y$ scale is different in all three plots.

Figure 7

Ground-state probability with pause time for the DW (top), SVMC TF (middle), and the AME (bottom). We fix the pause location in each plot to be in the region where the pause has a noticeable effect, taking $s_p=0.57$ for the DW, and $s_p=0.52$ for SVMC and the AME. The curve fits give excellent agreement with $(\alpha ,\beta ,\gamma )=(0.73,0.38,0.047)$ for SMVC and $(\alpha ,\beta ,\gamma )=(0.95,0.26,0.019)$ for the AME. For the DW, we had to use a function with two decay constants (${\gamma }_1,{\gamma }_2)=(0.043,8.5\times {10}^{-4}$). The top plot shows individual DW anneal results (from ${10}^4$ samples), and the AME is “exact.” Error bars for SVMC are $2\sigma$ over ${10}^4$ samples.

Figure 8

Pause time required to reach target success probability $P^{\ast }$ for the DW (top), SVMC TF (middle), and the AME (bottom). The fits (dashed lines) are from a least-square fitting, with exponents given in the legend. Note how the exponents change in the top plot compared to the bottom plots. In the AME and SVMC, the data is generated by taking curves as in Fig. 6, and finding the value on the $x$ axis corresponding to some fixed target $P^{\ast }$. If it lies between data points, we connect the points on either side by a straight line and extrapolate to find the value. Since the raw DW data is noisier when pausing at a fixed location (e.g., as in Fig. 7), we do a similar analysis but take a range around $P^{\ast }\pm 0.01$. For all data points in this range, the value reported here is the median, with error bars being the interquartile range.

Figure 9

SVMC dependence on temperature with random angle updates. In the standard version of SVMC with new angles chosen uniformly randomly in $[0,\pi ]$ dynamics continue late in the anneal; thus, even at high temperatures, a large success probability can be achieved, assuming the pause is late enough. The anneal “time” is ${10}^4$ sweeps, and the pause “time” is fixed at ${10}^6$ sweeps for all curves. The location of the minimum gap $s_{\mathrm{\Delta }}$ is marked with a vertical dashed line.

Figure 10

Temperature dependence for SVMC TF (top) and the AME (bottom). Plots showing dependence of temperature under the pause at $s_p$. In SVMC, the anneal “time” is ${10}^4$ sweeps, and the pause “time” is fixed at ${10}^6$ sweeps for all curves. In the AME we use an anneal of $1\mu \mathrm{s}$ and pause of $100\mu \mathrm{s}$. The location of the minimum gap $s_{\mathrm{\Delta }}$ is marked with a vertical dashed line.

Figure 11

Time to solution for various decay rates, as a function of the pause time. For the plot, we use parameters of the AME as found in Fig. 7 ($P_G=0.95,P_a=0.69,t_a=1\mu \mathrm{s}$). Here ${\gamma }_{\max }^{-1}$ is $0.72\mu \mathrm{s}$ as shown in Table 1. When ${\gamma }^{-1}$ is small enough, an initial decrease in TTS can be observed. For $t_p$ larger than shown, the curves eventually overlap and scale linearly in $t_p$.

Figure 12

Comparison of different annealing rates. Ground-state probability reached by the DW (top) and the SVMC-TF algorithm (bottom) for a fixed pause duration $100\mu \mathrm{s}$ for the DW, and $100k$ sweeps of SVMC TF, but a varied annealing rate.

Figure 13

Comparison between standard SVMC (top) and SVMC with transverse-field updates (bottom). These simulations are both performed at 12 mK with $10k$ sweeps in the ramp. Note that the standard SVMC continues to have updates much later in the anneal (N.B. at $s=0.75$, the ratio $A/B<{10}^{-2}$). Here we use an anneal “time” of ${10}^4$ sweeps for all curves.

Figure 14

Standard update SVMC pause time to target. Pause time required for the standard SVMC algorithm to reach target success probability $P^{\ast }$. As in SVMC TF, the time required increases exponentially in $s_p$ (location of the pause).

Figure 15

Single timescale fit for the DW under the pause. Top: fixing $\alpha =P_0(t_p\to \mathrm{\infty })$ and $\beta =\alpha -P_0(t_p=0)$, we obtain ${\gamma }^{-1}=47\mu \mathrm{s}$ from a nonlinear least-squares fitting procedure. Bottom: using the same functional form, but replacing the single timescale $1/\gamma$ by the dominant timescale from the top plot of Fig. 7.