Optimal working point in dissipative quantum annealing

We study the effect of a thermal environment on the quantum annealing dynamics of a transverse-field Ising chain. The environment is modeled as a single Ohmic bath of quantum harmonic oscillators weakly interacting with the total transverse magnetization of the chain in a translationally invariant manner. We show that the density of defects generated at the end of the annealing process displays a minimum as a function of the annealing time, the so-called optimal working point, only in rather special regions of the bath temperature and coupling strength plane. We discuss the relevance of our results for current and future experimental implementations with quantum annealing hardware.

Figure 1

Density of defects vs annealing time for a quantum Ising chain weakly coupled to an Ohmic bath, at different bath temperatures $T$, compared to the ideal coherent evolution (KZ) behavior, $n_{\mathrm{def}}\left(\tau \right)\sim {\tau }^{-1/2}$. The plot highlights the three distinct behaviours we have found: $n_{\mathrm{def}}\left(\tau \right)$ can (i) display a global minimum (green triangles), (ii) a local minimum (blue circles), and (iii) converge monotonically toward a large-$\tau$ thermal plateau (red squares). Here the system-bath coupling constant is kept fixed at $\alpha ={10}^{-2}$.

Figure 2

Density of defects vs annealing time $\tau$ for (a) $\alpha ={10}^{-3}$, (b) $\alpha ={10}^{-2},$ for different effective temperatures $T$, as indicated in the legend. The arrows indicate the direction of increasing temperatures. The trend for high $T$ is of AKZ type, with an emergent OWP. At lower $T$ and/or higher $\alpha$ values, a monotonic trend smoothly appears, with the absence of OWP.

Figure 3

Density of defects vs annealing time $\tau$ for $k_{B}T=J$, at different coupling strengths $\alpha$. Note that, for large enough annealing times, $n_{\mathrm{def}}\left(\tau \right)$ converges toward the thermal value.

Figure 4

(a) Dependence of $n_{\mathrm{opt}}$ on $T$, for various values of $\alpha$. Each curve defines an upper value $T_{\mathrm{up}}\left(\alpha \right)$ at which $n_{\mathrm{opt}}\left(T_{\mathrm{up}}\right)=n_{\infty }\left(T_{\mathrm{up}}\right)$ (marked by stars), and a lower $T_{\mathrm{low}}\left(\alpha \right)$ at which the local minimum defining $n_{\mathrm{opt}}$ disappears. (b) Phase diagram in the $T-\alpha$ plane with $T_{\mathrm{up}}\left(\alpha \right)$ and $T_{\mathrm{low}}\left(\alpha \right)$. A proper OWP only exists for $T>T_{\mathrm{up}}\left(\alpha \right)$. The black solid line is a fit of $T_{\mathrm{up}}\left(\alpha \right)$ using Eq. (17), with $C=2.08$ and $D=12.3$. The red dashed line is a fit with $T_{\mathrm{low}}\left(\alpha \right)=c{\alpha }^{-b}\sqrt{\alpha -{\alpha }_c}$, where we find ${\alpha }_c=5.5·{10}^{-4},c=3.58$, and $b=0.22$. The shaded area alludes to the typical range of temperatures of interest for the D-Wave hardware [7, 8], with $k_{B}T\simeq 12$ mK and $J\gtrsim 80$ mK.

Figure 5

Density of defects vs time for $\tau ={10}^5,k_{B}T=J$ and different system-bath coupling strengths. The arrow at $t_c$ marks the value of $t$ at which the transverse field crosses the critical value, $h\left(t_c\right)=h_c$. For $\alpha ={10}^{-2}$, where the defects density has fully converged (see Fig. 3), $n_{\mathrm{def}}\left(t\right)$ is almost superimposed to the exact instantaneous thermal one computed from Eq. (B4).

Figure 6

Test for the additivity assumption Eq. (20) for the formation of defects. We compare $n_{\mathrm{def}}(t=\tau )$, calculated with the full Bloch-Redfield evolution in Eq. (9) (continuous curves, filled symbols), to the sum of $n_{\mathrm{def}}^{\mathrm{coh}}\left(\tau \right)$ plus the purely dissipative evolution contribution $n_{\mathrm{def}}^{\mathrm{diss}}\left(\tau \right)$ from Eq. (21) (dashed curve, empty symbols).

Figure 7

Comparison between our dissipative QA results (points) and sudden quenches from $h_0=10$ to $h_0=0$ followed by thermal relaxation (lines), for $\alpha =0.01$ and different temperatures. Data obtained in both cases by the QME dynamics described in the paper. Horizontal black lines identify the expected thermal values for each temperature.

Figure 8

Thermodynamics of the defect density in the transverse-field Ising chain for $k_{B}T=J$. The PBC thermodynamics (orange solid lines) is calculated with Eq. (B7), while the OBC thermodynamics (red solid lines and diamonds) corresponds to Eq. (B11). The blue solid circles are PBC-QME relaxation dynamics data for $N/2$ two-level systems for $T_b=2T$.

Figure 9

Difference between density of defects for (a) PBC and (b) OBC and the expected thermal value at the thermodynamic limit, plotted vs the number of sites $N$, for $h=0.5,1$. (a) The scaling is exponential in $N$. (b) The scaling is polynomial in $N$; our best fit gives a convergence with $1/N$.