Dissipative Landau-Zener problem and thermally assisted Quantum Annealing

We revisit here the issue of thermally assisted Quantum Annealing by a detailed study of the dissipative Landau-Zener problem in the presence of a Caldeira-Leggett bath of harmonic oscillators, using both a weak-coupling quantum master equation and a quasiadiabatic path-integral approach. Building on the known zero-temperature exact results [Wubs et al., Phys. Rev. Lett. 97, 200404 (2006)], we show that a finite temperature bath can have a beneficial effect on the ground-state probability only if it couples also to a spin direction that is transverse with respect to the driving field, while no improvement is obtained for the more commonly studied purely longitudinal coupling. In particular, we also highlight that, for a transverse coupling, raising the bath temperature further improves the ground-state probability in the fast-driving regime. We discuss the relevance of these findings for the current quantum-annealing flux qubit chips.

Figure 1

Probability $P_{\mathrm{gs}}(v,T)$ of remaining in the ground state of the Hamiltonian ${\widehat{H}}_{\mathrm{Q}}\left(t\right)$ when evolving with Eq. (3) in the presence of a thermal bath at temperature $T$ versus the driving velocity $\hslash v/{\mathrm{\Delta }}^2$, for a purely longitudinal ${\widehat{\sigma }}^z$ coupling (a) and for a purely transversal ${\widehat{\sigma }}^x$ coupling (b) to the environment. The bath properties are fully determined by the choice of its spectral function; here we have considered an ohmic bath with a cutoff frequency ${\omega }_c=10\mathrm{\Delta }/\hslash$ and a coupling $\alpha =2\times {10}^{-3}$ (see Sec. 2 for precise definitions). The black solid line is the coherent-evolution result $P_{\mathrm{gs}}^{\mathrm{LZ}}\left(v\right)$ and the black dashed line in panel (b) is the probability $P_{\mathrm{gs}}(v,T=0)$ obtained by means of the exact analytical result at zero temperature [24], while the various points are QME data for different bath temperatures. Black arrows indicate the direction of increasing $T$.

Figure 2

Check of the validity of the QME approach at zero temperature for a pure transversal noise, $\theta =\pi /2$. The probability to follow the ground state at zero temperature $P_{\mathrm{gs}}(T=0)$ versus (a) the driving velocity $\hslash v/{\mathrm{\Delta }}^2$ and (b) the coupling strength $\alpha$. The lines are the exact predictions obtained using Eq. (11), while the points correspond to our QME in Eqs. (8) (without RWA). Here $\hslash {\omega }_c=10\mathrm{\Delta }$.

Figure 3

Check of the validity of the QME approach at zero temperature. Probability to follow the ground state $P_{\mathrm{gs}}(v,T=0)$ for fixed driving velocities $\hslash v/{\mathrm{\Delta }}^2$ versus the noise coupling direction $\theta$ for $\alpha =0.02$ (a) and $\alpha =0.2$ (b). Lines are the exact results of Ref. [24]; points correspond to our QME in Eqs. (8).

Figure 4

(a) $P_{\mathrm{gs}}(v,T)$ as a function of the driving $\hslash v/{\mathrm{\Delta }}^2$ for pure longitudinal noise, $\theta =0$, at finite temperature $k_{B}T=25\mathrm{\Delta }$ and different values of the ohmic coupling strength $\alpha$. Lines correspond to QME results, points to QUAPI. The black solid line shows, as a guide to the eye, $P_{\mathrm{gs}}^{\mathrm{LZ}}\left(v\right)\equiv P_{\mathrm{gs}}(v,T=0)$. (b) $P_{\mathrm{gs}}$ versus the ohmic coupling strength at a fixed driving velocity $\hslash v=0.2{\mathrm{\Delta }}^2$ and different bath temperatures. (c), (d) Detail of the time-evolving ${\mathrm{P}}_{\mathrm{gs}}\left(t\right)$ for “slow” [(c) $\hslash v/{\mathrm{\Delta }}^2=0.2]$ and “fast” [(d) $\hslash v/{\mathrm{\Delta }}^2=2]$ drivings, for various couplings $\alpha$ and $k_{B}T=3\mathrm{\Delta }$. Data in (b)–(d) obtained using QUAPI, and $\hslash {\omega }_c=10\mathrm{\Delta }$ throughout.

Figure 5

Single oscillator approach. Ground state probability ${\mathrm{P}}_{\mathrm{gs}}\left(t\right)$ versus time for (a) $\hslash v/{\mathrm{\Delta }}^2=6$ and (b) $\hslash v/{\mathrm{\Delta }}^2=0.6$. Here $\lambda =0.5\mathrm{\Delta }$ and $\theta =\pi /2$. The black line is the coherent evolution ${\mathrm{P}}_{\mathrm{gs}}\left(t\right)$ in absence of dissipation; the blue and red lines are ${\mathrm{P}}_{\mathrm{gs}}\left(t\right)$ at $T=0$ and $k_{B}T=2\hslash \mathrm{\Omega }$, respectively. In both cases $\hslash \mathrm{\Omega }=50\mathrm{\Delta }$. Vertical lines at $\pm t^{*}$, the resonance times in Eq. (15), highlight the excitation/relaxation mechanisms.

Figure 6

Comparison of QME results obtained with and without the RWA for a longitudinal bath coupling with $\alpha =0.02$ and $k_{B}T=10\mathrm{\Delta }$. The points denote numerically exact QUAPI data. The RWA line shows a quite clear overshooting above the coherent evolution result which is an artifact of the approximation.