Understanding Quantum Tunneling through Quantum Monte Carlo Simulations

The tunneling between the two ground states of an Ising ferromagnet is a typical example of many-body tunneling processes between two local minima, as they occur during quantum annealing. Performing quantum Monte Carlo (QMC) simulations we find that the QMC tunneling rate displays the same scaling with system size, as the rate of incoherent tunneling. The scaling in both cases is $O({\mathrm{\Delta }}^2)$, where $\mathrm{\Delta }$ is the tunneling splitting (or equivalently the minimum spectral gap). An important consequence is that QMC simulations can be used to predict the performance of a quantum annealer for tunneling through a barrier. Furthermore, by using open instead of periodic boundary conditions in imaginary time, equivalent to a projector QMC algorithm, we obtain a quadratic speedup for QMC simulations, and achieve linear scaling in $\mathrm{\Delta }$. We provide a physical understanding of these results and their range of applicability based on an instanton picture.

Figure 1

Upper panel: Typical instanton–anti-instanton trajectory with periodic boundary condition in imaginary time. Along such trajectories the path samples both the $|\uparrow ⟩$ and $|\downarrow ⟩$ states. The magnetization $m$ exhibits two jumps corresponding to instantons (tunneling events) in imaginary time (red arrows). Lower panel: with open boundary condition only one instanton is required and the tunneling probability thus increased. On the right we sketch the double-well potential.

Figure 2

Path integral QMC transition time $\xi$ as a function of the inverse temperature for a chain at $\mathrm{\Gamma }=0.7$. At high temperatures $\xi$ is independent of the system size $L$ and suppressed as $\exp (-E_{\text{barrier}}/T)$. At low temperatures, quantum tunneling becomes more efficient and depends only on $L$.

Figure 3

Average QMC tunneling time $\xi$ as a function of system size $L$ (open symbols) for a fully connected graph at $\beta =8$ for various values of the transverse field $\mathrm{\Gamma }$. Fits of the times for $8\le L\le 16$ are shown as solid lines. To compare to physical QA we also show $\alpha (\mathrm{\Gamma })/\mathrm{\Delta }(\mathrm{\Gamma },L{)}^2$ with closed symbols, obtained by exact diagonalization. Rescaling by $L$-independent constants $\alpha (\mathrm{\Gamma })$ we find identical scaling with system size $L$.

Figure 4

Average PIGS tunneling time $\xi$ as a function of $L$ (open symbols) for a fully connected graph at $\beta =8$ with various values of the transverse field $\mathrm{\Gamma }$. Fits of the times for $8\le L\le 20$ are shown as solid lines. Points proportional to $1/\mathrm{\Delta }(L)$, obtained with exact diagonalization, are shown with closed symbols.