Why and When Pausing is Beneficial in Quantum Annealing

Recent empirical results using quantum-annealing hardware have shown that midanneal pausing has a surprisingly beneficial impact on the probability of finding the ground state for a variety of problems. A theoretical explanation of this phenomenon has thus far been lacking. Here we provide an analysis of pausing using a master-equation framework, and derive conditions for the strategy to result in a success probability enhancement. The conditions, which we identify through numerical simulations and then prove to be sufficient, require that relative to the pause duration the relaxation rate is large and decreasing right after crossing the minimum gap, small and decreasing at the end of the anneal, and is also cumulatively small over this interval, in the sense that the system does not thermally equilibrate. This establishes that the observed success probability enhancement can be attributed to incomplete quantum relaxation, i.e., is a form of beneficial nonequilibrium coupling to the environment.

Figure 1

Example schedules with the following parameter choices: ${\mu }_{\theta }={\mu }_g=0.5$, ${\alpha }_g=0.5$, $E_0=15/\pi$ GHz, $\mathrm{\Delta }=0.001$, and ${\alpha }_{\theta }=1/100$.

Figure 2

Example of annealing parameter $s$ against dimensionless time $\tau =t/t_f$. The pause position $s_p$ and duration $s_d$ are chosen as $s_p=0.5$ and $s_d=0.5$.

Figure 3

Populations of instantaneous eigenstates during an anneal with total time $t_f=100\mathrm{ns}$. $\delta \theta$ denotes the jump of the annealing angle $\theta$ across the minimum gap region. In our model, the magnitude of this jump is directly determined by the boundary condition.

Figure 4

Success probability (without pausing) calculated in both closed- and open-system settings with boundary conditions: (a) $\theta (1)=\pi /2$; (b) $\theta (1)=\pi$. Open-system simulations are done with all three variants of MEs described in Sec. 3, with an Ohmic bath spectral density [Eq. (31)]. Here and below the Ohmic bath parameters are chosen as typical of flux qubits (e.g., Refs. [30, 44, 46, 47, 48, 49]): $2\pi \eta g^2={10}^{-4}$, $T=16$ mK, and ${\omega }_c/2\pi =4$ GHz. The results of the open-system simulations overlap. Note the different vertical axis scales in (a),(b).

Figure 5

Success probability for pausing schedules as shown in Fig. 2. The boundary condition for every plot is $\theta (0)=0$ and $\theta (1)=\pi /2$ [for $\theta (1)=\pi$, the plots are of similar shapes, with the lowest success probability being 0]. For each curve the pausing duration $s_d$ is fixed and the pausing position $s_p$ is varied. (a) Closed system. The inset is an enlargement around $s_p=0.5$. (b) AME. (c) AME success probabilities for different $t_f^{\prime }$ values with $s_d=4$. (d) Final success probabilities from three MEs with $s_d=1$. For (a),(b), and (d), the original total annealing time $t_f$ is set to 100 ns and the Ohmic bath parameters are chosen as $2\pi \eta g^2={10}^{-4}$, $T=16$ mK, and ${\omega }_c/2\pi =4$ GHz. The dashed lines in (b),(c) indicate the thermal ground-state population of the final Hamiltonian.

Figure 6

$\overline{\mathrm{\Gamma }}(s)$ [the dimensionless relaxation rate in Eq. (38b) as a function of $s$, i.e., with $\tau$ replaced by $s$] for an Ohmic bath with different cutoff frequencies: ${\omega }_c/2\pi =0.5$ GHz, monotonically decreasing $\overline{\mathrm{\Gamma }}(s)$ (left); ${\omega }_c/2\pi =1$ GHz, nonmonotonic (middle); ${\omega }_c/2\pi =4$ GHz, monotonically increasing $\overline{\mathrm{\Gamma }}(s)$ (right).

Figure 7

Success probability versus pausing position for the three Ohmic bath cases shown in Fig. 6, in the same order from left to right. The existence of a maximum in the left and middle panels shows that an optimal pausing exists for the corresponding decay rates. The qubit frequency $\mathrm{\Omega }(s)$ [Eq. (2)] changes between $1.88\mathrm{GHz}$ and $4.77\mathrm{MHz}$ during the anneal. Other parameters are $\eta g_z^2=\eta g_y^2={10}^{-4}/2\pi$ and $T=16$ mK. The analytic solution refers to Eq. (42a). For master-equation simulations, the schedules are chosen according to Fig. 1. The pausing duration is fixed at $s_d=4$. All three panels share the same legend.

Figure 8

(a) Annealing angle for the $p$-spin model with $n=20$, $p=19$ [25]. The inset is an enlargement around the peak of angular progression, shown in (b). In (c) we show the annealing angle for the D-Wave $16$-qubit gadget [44]. The inset illustrates the angular progression.

Figure 9

Closed-system success probability versus total annealing time. The gap is chosen according to Eq. (7) with parameters ${\mu }_g=0.5$, ${\alpha }_g=0.5$, $E_0=15/\pi$ GHz and $\mathrm{\Delta }=0.001$.

Figure 10

Graphical proof of Eq. (F2). The partial derivative is obtained in the limit of $\mathrm{\Delta }\to 0$.

Figure 11

Heat map of ${\log }_{10}ϵ$ for different ${\omega }_c$ and $T$ values for the Ohmic bath spectrum [Eq. (31)], covering the entire parameter region shown in Fig. 6. The gap $\overline{\mathrm{\Omega }}(1)\approx 1.88\mathrm{GHz}$ is chosen in accordance with parameters used in Fig. 1. The maximum of $ϵ$ appears near the line ${\omega }_c=2\pi \overline{\mathrm{\Omega }}(1)$.