Dynamics of open quantum systems by interpolation of von Neumann and classical master equations, and its application to quantum annealing

We propose a method to interpolate dynamics of von Neumann and classical master equations with an arbitrary mixing parameter to investigate the thermal effects in quantum dynamics. The two dynamics are mixed by intervening to continuously modify their solutions, thus coupling them indirectly instead of directly introducing a coupling term. This maintains the quantum system in a pure state even after the introduction of thermal effects and obtains not only a density matrix but also a state vector representation. Further, we demonstrate that the dynamics of a two-level system can be rewritten as a set of standard differential equations, resulting in quantum dynamics that includes thermal relaxation. These equations are equivalent to the optical Bloch equations at the weak coupling and asymptotic limits, implying that the dynamics cause thermal effects naturally. Numerical simulations of ferromagnetic and frustrated systems support this idea. Finally, we use this method to study thermal effects in quantum annealing, revealing nontrivial performance improvements for a spin glass model over a certain range of annealing time. This result may enable us to optimize the annealing time of real annealing machines.

Figure 1

Time evolution of the probability ${\rho }_{11}$ with $T=0$ and $h=\mathrm{\Gamma }=1$; for $\alpha =0.1$ (solid black) and 0.2 (dashed red).

Figure 2

Stationary-state probabilities ${\rho }_{11}$ as a function of $\alpha$ for $T=0$ and $h=\mathrm{\Gamma }=1$. The solid and dashed lines represent the stable and unstable solutions, respectively.

Figure 3

Stable stationary-state probability ${\rho }_{11}$ as a function of $T$, with $h=\mathrm{\Gamma }=1$ and $\alpha =0.1$ (solid line). The dashed and dotted lines represent the probabilities derived from the standard partition function of the total Hamiltonian with $\mathrm{\Gamma }=0$ and 1.

Figure 4

Time evolution of the total probabilities for the states associated with the three energy levels of the HT model for $\alpha =0.1, T=0$, and $s=0.8$. Going from top to bottom, the solid lines represent the ground state (black) and the first (blue) and second (green) excited states of the QUBO Hamiltonian, while the dashed lines represent the probabilities of the stationary state.

Figure 5

Time evolution of the total probabilities for the states associated with the three energy levels of the HT model for $\alpha =0.1, T=0$, and $s\left(t\right)=0.8\sqrt{t/100}$. From top to bottom at the end of the simulation, the solid lines represent the ground state (black) and the first (blue) and second (green) excited states of the QUBO Hamiltonian, while the dashed lines represent the stationary-state probabilities.

Figure 6

Time evolution of the 8-spin QS system's $P_I/P_C$ ratio for (from bottom to top) $\alpha =0$ (black), 0.25 (blue), 0.5 (green), 0.75 (red), and 1 (purple).

Figure 7

QUBO ground-state probability as a function of annealing time for the mixing parameters $\alpha =0,0.1,0.2,\cdots ,1$ under the annealing schedule $s\left(t\right)={(t/\tau )}^{0.4}$.

Figure 8

Stationary-state probability ${\rho }_{11}$ as a function of $\alpha$ at (going from left to right and top to bottom) $T=0.2,0.5,0.9,1.0,1.1,1.2,2,5$, and 100 with $h=\mathrm{\Gamma }=1$. The solid and dashed lines represent the stable and unstable solutions, respectively.

Figure 9

QUBO  ground-state  probability  as a  function of the  scheduling index $\gamma$ for total annealing times of (going from left to right and top to bottom) $\tau =2,5,10,20$,  and  50. Each  figure depicts the average probabilities for 50 randomly generated SK model Hamiltonians with mixing parameters $\alpha =0.1,0.2,\cdots ,1$. The inset figures depict magnified views around the tops of the curves.